aa r X i v : . [ m a t h . N T ] M a r SQUARE SERIES GENERATING FUNCTION TRANSFORMATIONS
MAXIE D. SCHMIDTSCHOOL OF MATHEMATICSGEORGIA INSTITUTE OF TECHNOLOGY117 SKILES BUILDING686 CHERRY STREET NWATLANTA, GA 30332
Abstract.
We construct new integral representations for transformations of the ordinarygenerating function for a sequence, h f n i , into the form of a generating function that enu-merates the corresponding “square series” generating function for the sequence, h q n f n i ,at an initially fixed non-zero q ∈ C . The new results proved in the article are given byintegral–based transformations of ordinary generating function series expanded in terms ofthe Stirling numbers of the second kind. We then employ known integral representations forthe gamma and double factorial functions in the construction of these square series trans-formation integrals. The results proved in the article lead to new applications and integralrepresentations for special function series, sequence generating functions, and other relatedapplications. A summary Mathematica notebook providing derivations of key results andapplications to specific series is provided online as a supplemental reference to readers. Notation and Conventions
Most of the notational conventions within the article are consistent with those employedin the references [11, 15]. Additional notation for special parameterized classes of the squareseries expansions studied in the article is defined in Table 1 on page 4. We utilize this notationfor these generalized classes of square series functions throughout the article. The followinglist provides a description of the other primary notations and related conventions employedthroughout the article specific to the handling of sequences and the coefficients of formalpower series: ◮ Sequences and Generating Functions : The notation h f n i ≡ { f , f , f , . . . } isused to specify an infinite sequence over the natural numbers, n ∈ N , where we define N = { , , , . . . } and Z + = { , , , . . . } . The ordinary (OGF) and exponential (EGF)generating functions for a prescribed sequence, h f n i , are defined by the respectiveformal power series in the notation of F f ( z ) := OGF z [[ f , f , f , . . . ]] ≡ ∞ X n =0 f n z n b F f ( z ) := EGF z [[ f , f , f , . . . ]] ≡ ∞ X n =0 f n z n n ! . Date : 2017.03.24-v1.2010
Mathematics Subject Classification.
Key words and phrases.
Square series, generating function, series transformation, integral representation,gamma function, double factorial, Stirling number of the second kind, Jacobi theta function, Ramanujan thetafunction. ◮ Power Series and Coefficient Extraction : Given the (formal) power series, F f ( z ) := P n f n z n , in z that enumerates a sequence taken over n ∈ Z , the notation used to ex-tract the coefficients of the power series expansion of F f ( z ) is defined as f n ≡ [ z n ] F f ( z ). ◮ Iverson’s Convention : The notation [ cond ] δ is used as in the references [11] tospecify the value 1 if the Boolean–valued input cond evaluates to true and 0 otherwise.The notation [ n = k = 0] δ is equivalent to δ n, δ k, , and where δ : Z → { , } is Kronecker’s delta function and δ i,j = 1 if and only if i = j .2. Introduction
Motivation.
Many generating functions and special function series of interest in com-binatorics and number theory satisfy so-termed unilateral or bilateral “ square series ” powerseries expansions of the form e F sq ( f ; q, z ) := X n ∈ S f n q n z n , (1)for some q ∈ C , a prescribed sequence, h f n i , whose ordinary generating function , is analytic onan open disk, z ∈ R ( f ), and for some indexing set S ⊆ Z . Bilateral square series expansionsin the form of (1) arise in the series expansions of classical identities for infinite products suchas in the famous triple product identity ∞ X n = −∞ x n z n = ∞ Y n =1 (1 − x n )(1 + x n − z )(1 + x n − z − ) , (2)and in the quintuple product identity given by ∞ X n = −∞ ( − n q n (3 n − / z n (1 − zq n ) = ( q, − z, − q/z ; q ) ∞ (cid:0) qz , q/z ; q (cid:1) ∞ , (3)for | q | <
1, any z = 0, and where the q –Pochhammer symbols , ( a ; q ) n ≡ (1 − a )(1 − aq ) · · · (1 − aq n − ) and ( a , a , . . . , a r ; q ) n := Q ri =1 ( a i ; q ) n , denote the multiple infinite products on theright-hand-side of (3) [15, § § § Jacobi theta functions , denoted by ϑ i ( u, q ) for i = 1 , , ,
4, or by ϑ i ( u | τ ) when the nome q ≡ exp ( ıπτ ) satisfies Im( τ ) >
0, form another class of square series expansions of spe-cial interest in the applications considered within the article. The classical forms of these thetafunctions satisfy the respective bilateral and corresponding asymmetric, unilateral Fourier–type square series expansions given by [15, § ϑ ( u, q ) = ∞ X n = −∞ q ( n + ) ( − n − / e (2 n +1) ıu = 2 q / ∞ X n =0 q n ( n +1) ( − n sin ((2 n + 1) u ) ϑ ( u, q ) = ∞ X n = −∞ q ( n + ) e (2 n +1) ıu = 2 q / ∞ X n =0 q n ( n +1) cos ((2 n + 1) u ) ϑ ( u, q ) = ∞ X n = −∞ q n e nıu = 1 + 2 ∞ X n =1 q n cos (2 nu ) ϑ ( u, q ) = ∞ X n = −∞ q n ( − n e nıu = 1 + 2 ∞ X n =1 q n ( − n cos (2 nu ) . Additional square series expansions related to these forms are derived from the special cases ofthe Jacobi theta functions defined by ϑ i ( q ) ≡ ϑ i (0 , q ), and by ϑ ( j ) i ( u, q ) ≡ ∂ ( j ) ϑ i ( u , q ) /∂u j ) | u = u or by ϑ ( j ) i ( q ) ≡ ϑ ( j ) i (0 , q ) where the higher-order j th derivatives are taken over any fixed j ∈ Z + . QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 3
Definition 2.1 (Square Series Generating Functions) . For a given infinite sequence, h f n i = { f , f , f , . . . } , its ordinary generating function (OGF) is defined as the formal power series F f ( z ) := ∞ X n =0 f n z n , while its square series expansion , or corresponding square series generating function , is definedby F sq ( f ; q, z ) := X n ≥ f n q n z n , (5)for some q ∈ C such that | q | ≤ R ( f ). We note that the definition of the unilateralsquare series generating function defined by (5) corresponds to taking the indexing set of S ≡ N in the first definition of (1). Most of the examples of bilateral series expansions where S ≡ Z we will encounter as applications in the article are reduced to the sum of two unilateralsquare series generating functions of the form in (5).2.2. Approach.
We employ a new alternate generating–function–based approach to the uni-lateral square series expansions in (5) within this article which lead to new integral transformsthat generate these power series expansions. The approach to expanding the forms of the uni-lateral square series in the article mostly follows from results that are rigorously justified asoperations on formal power series. As a result, these transformations provide new approachesto these series and other insights that are not, by necessity, directly related to the underlyinganalysis or combinatorial interpretations of these special function series.The particular applications of the new results and integral representations proved withinthe article typically fall into one of the following three primary generalized classes of thestarting sequences, h f n i , in the definition of (5) given above: ◮ Geometric-Series-Based Sequence Functions (Section 4) : Variations of theso-termed geometric square series functions over sequences of the form f n ≡ p ( n ) · c n with c ∈ C and a fixed p ( n ) ∈ C [ n ], including the special cases where p ( n ) := 1 , ( an + b ) , ( an + b ) m for some fixed constants, a, b ∈ C , and m ∈ Z + ; ◮ Exponential-Series-Based Generating Functions (Section 5) : The exponentialsquare series functions involving polynomial multiples of the exponential-series-basedsequences, f n ≡ r n /n !, for a fixed r ∈ C ; and ◮ Fourier-Type Sequences and Generating Functions (Section 6) : Fourier-typesquare series functions which involve the sequences f n ≡ sc( αn + β ) · c n with α, β ∈ R ,for some c ∈ C , and where the trigonometric function sc ∈ { sin , cos } .The special expansions of the square series functions defined in Table 1 summarize the gen-eralized forms of the applications cited in Section 2.4 below and in the application sections 4– 6 of the article.2.3. Terminology.
We say that a function with a series expansion of the form in (1) orin (5) has a so-termed square series expansion of the form studied by the article. Withinthe context of this article, the terms “ square series ” and “ square series integral ,” or “ squareseries integral representation ”, refer to the specific constructions derived from the particulargenerating function transformation results established by this article. The usage of theseterms is then applied interchangeably in the context of any number of applications to themany existing well-known classical identities, special theta functions, and other power seriesexpansions of special functions, which are then studied through the new forms of the squareseries transformations and integral representations proved within the article.
MAXIE D. SCHMIDT
Geometric Square Series Functions
Eq. Function Series Parameters (T.1.1) G sq ( q, c, z ) P ∞ n =0 q n c n z n | cz | < (T.1.2) ϑ d ( q, c, z ) P ∞ n = d q n c n z n | cz | < d ∈ Z + (T.1.3) Q a,b ( q, c, z ) P ∞ n =0 q n ( an + b ) c n z n | cz | < a, b ∈ C (T.1.4) ϑ d,m ( q, c, z ) P ∞ n = d n m × q n c n z n | cz | < d, m ∈ N (T.1.5) ϑ d,m ( α, β ; q, c, z ) P ∞ n = d ( αn + β ) m × q n c n z n | cz | < α, β ∈ C ; d, m ∈ N (T.1.6) G sq ( p, m ; q, c ) P ∞ n =0 ( q p ) n ( c · q m ) n | q m cz | < p, m ∈ R + Fourier–Type Square Series Functions
Eq. Function Series Parameters (T.1.7) F sc ( α, β ; q, c, z ) P ∞ n =0 q n sc ( αn + β ) c n z n | cz | < (T.1.8) ϑ ( u, q, z ) q / × P ∞ n =0 q n ( − n sin((2 n +1) u ) q n z n | qz | < u ∈ R (T.1.9) ϑ ( u, q, z ) q / × P ∞ n =0 q n cos((2 n +1) u ) q n z n | qz | < u ∈ R (T.1.10) ϑ ( u, q, z ) q × P ∞ n =0 q n cos((2 n +2) u ) q n z n | q z | < u ∈ R (T.1.11) ϑ ( u, q, z ) − q × P ∞ n =0 q n ( − n cos((2 n +2) u ) q n z n | q z | < u ∈ R Exponential Square Series Functions
Eq. Function Series Parameters (T.1.12) E sq ( q, r, z ) P ∞ n =0 q n r n z n /n ! r, z ∈ C (T.1.13) e E sq ( q, r, z ) P ∞ n =0 q ( n ) r n z n /n ! r, z ∈ C Table 1.
Special Classes of Generalized Square Series Functions
Definitions of the generalized series expansions for the forms of several parameterizedclasses of special square series functions considered by the article.
Examples and Applications.
We first briefly motivate the utility to the sequence–based generating function approach implicit to the square series expansions suggested by thedefinition of (5) through several examples of the new integral representations following asconsequences of the new results established in Section 3. Sections 4 – 6 of the article alsoconsider integral representations for generalized forms of the three primary classes of squareseries expansions listed in Section 2.2, as well as other concrete examples of concrete specialcase applications of our new results. This subsection is placed in the introduction both todemonstrate the stylistic natures of the new square series integral representations we provein later sections of the article and to motivate Theorem 3.3 by demonstrating the breadth ofapplications to special functions offered by our new results.
Series for the Infinite q -Pochhammer Symbol. The next results illustrate several differentseries forms for
Euler’s function , f ( q ) ≡ ( q ) ∞ , given by the respective bilateral and unilateral QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 5 series for the infinite product expansions given by [4, § § q ) ∞ = ∞ Y n =1 (1 − q n ) = 1 − q − q + q + q − q − q + · · · (6)= ∞ X n = −∞ ( − n q n (3 n +1) / = 1 + ∞ X n =1 ( − n (cid:16) q ω ( n ) + q ω ( − n ) (cid:17) , for | q | < ω ( ± k ) = (3 k ∓ k ) /
2, in the last equation denotethe pentagonal numbers [18, A000326]. The cube powers of this product are expanded by
Jacobi’s identity as the following series whenever | q | < § cf . Entry 22] [15, cf . § q ) ∞ = ∞ Y n =1 (1 − q n ) = 1 + ∞ X m =1 ( − m (2 m + 1) q m ( m +1) / . (7)For a sufficiently small real–valued q or strictly complex-valued q such that Im( q ) >
0, eachwith | q | ∈ (0 , § § § q ) ∞ = 1 − q × ∞ X n =0 ( − n q n (3 n +5) / + ∞ X n =1 ( − n q n (3 n +1) / (8a)= 1 − q × F sq (cid:16) f ; q / , − (cid:17) − q × F sq (cid:16) f ; q / , − (cid:17) = 1 − Z ∞ e − t / √ π q (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 dt − Z ∞ e − t / √ π q (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 dt, | q | ∈ (0 , / q ) ∞ = 12 q / ϑ ′ ( √ q ) = 1 + ∞ X n =1 q n ( n +1) / (2 n + 1)( − n (8b)= 1 − q × F sq (cid:16) f ; q / , − (cid:17) = 1 + Z ∞ e − t / √ π q / (cid:16) ( q + 1) cosh (cid:16)p Log( q ) t (cid:17) + 2 q / (cid:17)(cid:16) q + 2 q / cosh (cid:16)p Log( q ) t (cid:17) + 1 (cid:17) dt − Z ∞ e − t / √ π q (cid:16) q / cosh (cid:16)p Log( q ) t (cid:17)(cid:17) q + 2 q / cosh (cid:16)p Log( q ) t (cid:17) + 1 dt, | q | ∈ (cid:16) , − / (cid:17) ( q ) ∞ = 13 / q / × ϑ (cid:16) π/ , q / (cid:17) (8c)= 2 √ × ∞ X n =0 q n ( n +1) / ( − n cos (cid:16) (2 n + 1) · π (cid:17) = 2 √ × F sq (cid:16) f ; q / , − (cid:17) = Z ∞ e − t / √ π √ q cosh (cid:18) √ Log( q ) t √ (cid:19) z +2 q / cosh (cid:18) √ Log( q ) t √ (cid:19) z +2 q / cosh (cid:18) √ Log( q ) t √ (cid:19) z +1 q / z +2 √ q cosh (cid:18) √ Log( q ) t √ (cid:19) z + q / (cid:18) (cid:18) √ Log( q ) t √ (cid:19)(cid:19) z +2 q / cosh (cid:18) √ Log( q ) t √ (cid:19) z +1 dt, MAXIE D. SCHMIDT when the respective sequence forms from the square series functions in the previous equationsare defined to be f ( n ) := q n/ , f ( n ) := q n/ , f ( n ) := (2 n + 3) · q n/ , and f ( n ) := q n/ · cos ((2 n + 1) · π/ The General Two-Variable Ramanujan Theta Function.
For any non-zero a, b ∈ C with | ab | <
1, the general (two-variable) Ramanujan theta function , f ( a, b ), is defined by the bilateralseries expansion [6, §
16, (18.1)] [15, § f ( a, b ) := ∞ X n = −∞ a n ( n +1) / b n ( n − / (9)= 1 + ( a + b ) + ab ( a + b ) + a b ( a + b ) + a b ( a + b ) + · · · , where the second infinite expansion of the series for f ( a, b ) corresponds to the leading powersof ab taken over the triangular numbers , T n ≡ n ( n − /
2, [18, A000217] [12, § XII]. The generalRamanujan theta function is also expanded by the pair of unilateral geometric square seriesfunctions from Table 1 as f ( a, b ) = 1 + ∞ X n =1 h a n ( n +1) / b n ( n − / + a n ( n − / b n ( n +1) / i (10)= 1 + ϑ (cid:16) √ ab, √ ab − , (cid:17) + ϑ (cid:16) √ ab, √ ba − , (cid:17) f ( a, b ) = 1 + ∞ X n =0 h a × a n ( n +3) / b n ( n +1) / + b × a n ( n +1) / b n ( n +3) / i (11)= 1 + a × G sq (cid:16) √ ab, a √ ab, (cid:17) + b × G sq (cid:16) √ ab, b √ ab, (cid:17) . These symmetric forms of the unilateral series satisfy the well known property for the functionthat f ( a, b ) ≡ f ( b, a ) [7, §
1] [14, cf . §
5] and lead to the new integral representations in thenext formula given by f ( a, b ) = 1 + Z ∞ ae − t / √ π − a √ ab cosh (cid:16)p Log( ab ) t (cid:17) a b − a √ ab cosh (cid:16)p Log( ab ) t (cid:17) + 1 dt (12)+ Z ∞ be − t / √ π − b √ ab cosh (cid:16)p Log( ab ) t (cid:17) ab − b √ ab cosh (cid:16)p Log( ab ) t (cid:17) + 1 dt. The full two-parameter form of the Ramanujan theta function, f ( a, b ), defines the expansionsfor many other special function series. The particular notable forms of these modified thetafunction series include the one–variable , or single–argument , form of the function, f ( q ) := f ( − q, − q ) ≡ ( q ) ∞ , which is related to the Dedekind eta function , η ( τ ) ≡ q / (¯ q ) ∞ when¯ q ≡ exp (2 πıτ ) is the square of the nome q , and the special cases of Ramanujan’s functions , ϕ ( q ) ≡ f ( q, q ) and ψ ( q ) ≡ f ( q, q ), considered in Section 4.2 [15, § § § XII].The definition of the general two-variable form of the theta function, f ( a, b ), is also given interms of the classical Jacobi theta function , ϑ ( z, q ) ≡ ϑ ( z | τ ), where a := qe ız , b := qe − ız ,and q ≡ e ıπτ [15, § ϑ ( v, τ ) over the inputs of v := log (cid:0) ab (cid:1) × (4 πı ) − and τ := log ( ab ) × (2 πı ) − in [12, § XII].
New Integral Formulas for Explicit Values of Special Functions.
Similar series related to theJacobi theta functions, ϑ i (0 , q ), lead to further new, exact integral formulas for certain explicitspecial constant values involving the gamma function , Γ( z ), at the rational inputs of Γ(1 / / / QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 7 proved within this article include the following special cases of the results for Ramanujan’sfunctions given in Section 4: ϕ (cid:0) e − π (cid:1) ≡ π / Γ (cid:0) (cid:1) · p √ / = 1 + Z ∞ e − t / √ π " e π (cid:0) e π − cos (cid:0) √ πt (cid:1)(cid:1) e π − e π cos (cid:0) √ πt (cid:1) + 1 dtψ (cid:16) e − π/ (cid:17) ≡ π / Γ (cid:0) (cid:1) · (cid:0) √ (cid:1) / e π/ / = Z ∞ e − t / √ π " cos (cid:0)p π t (cid:1) − e π/ cos (cid:0)p π t (cid:1) − cosh (cid:0) π (cid:1) dt. When q := ± exp ( − kπ ) for some real-valued k >
0, the forms of the Ramanujan thetafunctions, ϕ ( q ) and ψ ( q ), define the entire classes of special constants studied in [22, cf . § cf . § Dedekind eta function , η ( τ ), through it’s relations to the Jacobi theta functions and to theq-Pochhammer symbol [15, § § § Exponential Generating Functions.
The examples cited so far in the introduction have in-volved special cases of the geometric square series expansions, or the sequences over Fourier-type square series, which are readily expanded through the new integral representation for-mulas for these geometric-series-based sequence results. Before concluding this subsection, wesuggest additional, characteristically distinct applications that are derived from exponential-series-based sequences and generating functions. The resulting integral representations de-rived from Theorem 3.3 for these exponential square series types represent a much differentstylistic nature for these expansions than for the analogous non–exponential square series gen-erating functions demonstrated so far in the examples above ( cf . the characteristic expansionscited in Section 5.1).In particular, we consider the particular applications of two–variable sequence generatingfunctions over the binomial powers of a fixed series parameter, q ( n ) ≡ q n ( n − / , defined bythe series e E sq ( q, z ) := ∞ X n =0 q ( n ) z n n ! = F sq (cid:16)D q − n/ /n ! E ; q / , z (cid:17) (13)= Z ∞ e − t / √ π " X b = ± exp (cid:18) e bt √ Log( q ) · z √ q (cid:19) dt. For m, n ∈ N , let the function ℓ m ( n ) denote the number of labeled graphs with m edges on n nodes . The sequence is generated as the coefficients of the power series expansion formed bythe special case of the two–variable, double generating function in (13) given by [9, § b G ℓ ( w, z ) := X n,m ≥ ℓ m ( n ) w m z n n ! = ∞ X n =0 (1 + w )( n ) z n n ! (14)= Z ∞ e − t / √ π (cid:20) e e √ Log(1+ w ) t z √ w + e e − √ Log(1+ w ) t z √ w (cid:21) dt. Other examples of series related to the definition in (13) arise in the combinatorial interpreta-tions and generating functions over sequences with applications to graph theoretic contexts.The results proved in Section 5 provide further applications of the new integral representationsfor these variants of the exponential square series.
MAXIE D. SCHMIDT
Other Applications of the New Results in the Article.
There are a number of other applicationsof the new results and square series integral representations proved within the article. Thenext few identities provide additional examples and specific applications of these new results.A pair of q -series expansions related to Zagier’s identities and the second of the infiniteproducts defined in (6) are stated by the next equations for | q | < §
3; Thm. 1]. ∞ X n =0 ( z ; q ) n +1 z n = 1 + ∞ X n =1 ( − n h q n (3 n − / z n − + q n (3 n +1) / z n i (15)= 1 − qz Z ∞ e − t / √ π (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q z + 2 q / z cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt − q z Z ∞ e − t / √ π (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q z + 2 q / z cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt ∞ X n =0 [( q ) ∞ − ( q ) n ] = ( q ) ∞ ∞ X n =1 q n − q n + ∞ X n =1 ( − n h (3 n − q n (3 n − / + (3 n ) q n (3 n +1) / i = ( q ) ∞ ∞ X n =1 q n − q n − Z ∞ e − t / √ π q (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt + Z ∞ q / e − t / √ π (cid:16) q / + (1 + q ) cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt − Z ∞ e − t / √ π q (cid:16) q / cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt + Z ∞ q / e − t / √ π (cid:16) q / + (1 + q ) cosh (cid:16)p q ) t (cid:17)(cid:17)(cid:16) q + 2 q / cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt Other identities connecting sums involving products of terms in q -series are found in a numberof series from Ramanujan’s “lost” notebook, including many expansions that are adaptedeasily from the mock theta function series from Watson’s article [21, 2] [8, cf . § § n th prime number , p n , or alternately, a newfunctional equation for the ordinary generating function of the primes. In particular, if we letthe constant K be defined by the infinite series K = X k ≥ − k p k ≈ . , we have the following identities for p n in terms of its ordinary generating function, e P ( z ),defined such that e P (0) = 0 whenever | z | < / p n = j n K mod 10 n k QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 9 = $ [ z n ] R ∞ e − t / √ π − z cosh (cid:16) √ Log(100) t (cid:17) dtz − z cosh (cid:16) √ Log(100) t (cid:17) +1 R ∞ e − s / √ π X b = ± e P (cid:16) − e bı √ Log(100) t (cid:17) ds mod 10 n % . Unilateral Series Expansions of Bilateral Square Series.
For fixed parameters a, b, r , r , r ∈ Q and q ∈ C such that | q | <
1, let the function, B a,b ( r , r , r ; q ), be defined by the next formof the bilateral series in (16). The first series for this function is then expanded in the formof a second unilateral square series as B a,b ( r , r , r ; q ) := ∞ X n = −∞ ( − n ( an + b ) q ( r n + r n + r ) / (16)= q r / " ∞ X n =0 ( − n ( an + b ) q n ( r n + r ) / − ∞ X n =1 ( − n ( an − b ) q n ( r n − r ) / . An application of Proposition 4.3 requires that the second series term in the previous equationbe shifted to obtain the following unilateral series expansion (see Section 4): B a,b ( r , r , r ; q ) = q r / × ∞ X n =0 ( − n ( an + b ) q n ( r n + r ) / (17)+ q ( r − r + r ) / × ∞ X n =0 ( − n ( an + a − b ) q n ( r n +2 r − r ) / . Particular special cases of this series include the expansions corresponding to the tuples T ≡ ( a, b, r , r , r ) in the series forms of (16) and (17) for the following functions: the series forthe Dedekind eta function , η ( τ ), expanded in the bilateral series form of T = (0 , , , ,
0) [15, § Rogers–Ramanujan continued fraction , R ( q ), expanded by the series where T = (10 , , , ,
0) and T = (10 , , , ,
0) [2, cf . §
5] [6, cf . § T = (0 , , , − ,
0) for the coefficients of the q –series ( q ) ∞ modulo 5,and the cases where T = (0 , , , ± ,
0) related to the
Euler function , φ ( q ) ≡ ( q ) ∞ , and the pentagonal number theorem [13, Thm. 353] [7, Cor. 1.3.5] [6, cf . Entry 22(iii)]. Even furtherexamples of the series expansions phrased in terms of (16) are compared in the references [7, cf . Cors. 1.3.21 and 1.3.22] [13, cf . Thms. 355 and 356; (19.9.3)] [15, cf . § Statement and Constructions of the Main Theorem
The
Stirling numbers of the second kind , (cid:8) nk (cid:9) , are related to a specific key transformationresult of the OGF, F ( z ), of an arbitrary sequence stated in the next proposition. In particular,the triangular recurrence relation defining these numbers gives rise to the finite expression in(18) below for the generating function of the modified sequence, h n k f n i , when k ∈ Z + givenonly in terms of a sum over the Stirling numbers and the higher-order j th derivatives of theoriginal sequence generating function when these derivatives exist for all j ≤ k . The nextresult is an important ingredient to the proof of Theorem 3.3 given in this section. Proposition 3.1 (Generating Function Transformations Involving Stirling Numbers) . Forany fixed k ∈ Z + , let the ordinary generating function, F f ( z ) , of an arbitrary sequence, h f n i ,be defined such that its j th -order derivatives, F ( j ) ( z ) , exist for all j ≤ k . Then for this fixed k ∈ Z + , we have a finite expression for the OGF of the modified sequence, h n m f n i , given bythe following transformation identity: ∞ X n =0 n k f n z n = k X j =0 (cid:26) kj (cid:27) z j F ( j ) ( z ) . (18) The triangle is also commonly denoted by S ( n, k ) as in [15, § (cid:8) nk (cid:9) , for thesecoefficients is consistent with the notation employed by the reference [11, cf . § Proof.
The proof follows by induction on k ≥
0. For the base case of k = 0, the right–hand–side of (18) evaluates to the original OGF, F ( z ), as claimed, since (cid:8) (cid:9) = 1. The Stirlingnumbers of the second kind are defined in [11, § n, k ≥ (cid:26) nk (cid:27) = k (cid:26) n − k (cid:27) + (cid:26) n − k − (cid:27) + [ n = k = 0] δ (19)Suppose that (18) holds for some k ∈ N . For this fixed k , let F ( z ) denote the OGF of thesequence h f n i such that the j th derivatives of the function exist for all j ∈ { , , , . . . , k + 1 } .It follows from our assumption that ∞ X n =0 n · (cid:16) n k f n (cid:17) z n = z ddz " k X j =0 (cid:26) kj (cid:27) z j F ( j ) ( z ) = k X j =0 (cid:26) kj (cid:27) (cid:16) jz j F ( j ) ( z ) + z j +1 F ( j +1) ( z ) (cid:17) = k X j =0 j (cid:26) kj (cid:27) z j F ( j ) ( z ) + k +1 X j =1 (cid:26) kj − (cid:27) z j F ( j ) ( z )= k +1 X j =1 (cid:18) j (cid:26) kj (cid:27) + (cid:26) kj − (cid:27)(cid:19) z j F ( j ) ( z ) (20)where (cid:8) k (cid:9) ≡ k = 0. Finally, we equate the left-hand-side of (19) to theinner right-hand-side terms in (20) to obtain that k +1 X j =0 (cid:26) k + 1 j (cid:27) z j F ( j ) ( z ) = ∞ X n =0 n k +1 f n z n . (21)Thus the sum in (18) expanded in terms of the Stirling number triangle holds for all k ∈ N . (cid:3) Remark (Applications of the Stirling Number Transformations) . The series transforma-tions given by the result in (18) effectively generalize the following known identity for thenegative-order polylogarithm function corresponding to the special case sequences of f n ≡ c n that holds whenever | cz | < § − m ( cz ) ≡ ∞ X n =0 n m ( cz ) n = m X j =0 (cid:26) mj (cid:27) ( cz ) j · j !(1 − cz ) j +1 , m ∈ Z + . The Stirling number identity in the proposition is of particular utility in finding generatingfunctions or other closed–form expressions for the series P n g ( n ) f n z n where the function g ( n )is expanded as an infinite power series in the variable n . For example, for any q ∈ C and anynatural number n ≥ g ( n ) = q n , as q n ≡ exp (cid:0) n Log( q ) (cid:1) = ∞ X i =0 Log( q ) i n i i ! . (22)This particular expansion is key to the proof of Theorem 3.3 immediately below. Otherexamples where this approach applies include the lacunary sequence case where g ( n ) = q n for some | q | < Theorem 3.3 (Square Series Transformations) . Fix q ∈ C and suppose that h f n i denotesa prescribed sequence whose ordinary generating function, F f ( z ) , is analytic for z on a QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 11 corresponding non-trivial region of R sq ( f, q ) . Then the unilateral square series functions, F sq ( f ; q, z ) , defined by (5) satisfy the two formulas F sq ( f ; q, z ) = Z ∞ e − t / √ π X b = ± ∞ X j =0 z j · F ( j ) f ( z ) j ! × (cid:16) e bt √ q ) − (cid:17) j dt (23a)= Z ∞ e − t / √ π " X b = ± F f (cid:16) e bt √ q ) · z (cid:17) dt, (23b) for any q ∈ C such that | q | ≤ and z ∈ R sq ( f, q ) such that the square series expansion has anon-trivial radius of convergence.Proof. We begin by expanding out the series in (5) in terms of the j th derivatives of the OGF, F f ( z ), for the sequence, h f n i , and an infinite series over the Stirling numbers of the secondkind as ∞ X n =0 q n f n z n = ∞ X n =0 ∞ X k =0 Log( q ) k n k k ! ! f n z n = ∞ X k =0 Log( q ) k k ! ∞ X n =0 n k f n z n ! = ∞ X k =0 Log( q ) k k ! k X j =0 (cid:26) kj (cid:27) F ( j ) ( z ) z j = ∞ X j =0 ∞ X k =0 (cid:26) kj (cid:27) Log( q ) k k ! ! z j F ( j ) ( z ) . (24)where the terms (cid:8) kj (cid:9) are necessarily zero for all j > k .We next proceed to prove an integral transformation for the inner Stirling number seriesin (24). In particular, we claim thatLog( q ) i i ! = 1(2 i )! × "Z ∞ e − t / √ π (cid:16)p q ) · t (cid:17) i dt , (25)and then that X k ≥ (cid:26) kj (cid:27) Log( q ) k k ! = 1 √ π Z ∞ " X b = ± j ! (cid:16) e √ q ) t − (cid:17) j e − t / dt. (26)To prove these claims, first notice that the next identity follows from the duplication formula for the gamma function as [15, § n )! = 4 n n ! √ π Γ (cid:18) n + 12 (cid:19) , (27)where an integral representation for the gamma function, Γ( n + 1 / § (cid:18) n + 12 (cid:19) = 2 n − / Z ∞ t n e − t / dt. (28)A well-known exponential generating function for the Stirling numbers of the second kind, (cid:8) kj (cid:9) , at any fixed natural number j ≥ § § X k ≥ (cid:26) kj (cid:27) z k (2 k )! = 12 j ! (cid:2) ( e z − j + ( e − z − j (cid:3) . Then we have from (24) and (26) that ∞ X n =0 q n f n z n = ∞ X j =0 √ π Z ∞ " X b = ± j ! (cid:16) e √ q ) t − (cid:17) j e − t / dt. ! × z j F ( j ) ( z ) . It is not difficult to show by the Weierstrass M-test that for bounded Log( q ) ∈ C , the sumover the j th powers in the previous equation with respect to j ≥ (cid:3) Geometric–Series–Based Sequences
Eq. f n F f ( z ) F ( j ) f ( z ) Parameters (T.2.1) c n − cz ) c j j !(1 − cz ) j +1 | cz | < (T.2.2) nc n cz (1 − cz ) c j j !( cz + j )(1 − cz ) j +2 | cz | < (T.2.3) n c n cz ( cz +1)(1 − cz ) c j j ! ( ( cz ) +(3 j +1) cz + j ) (1 − cz ) j +3 | cz | < (T.2.4) ( an + b ) c n ( a − b ) cz + b (1 − cz ) c j j ! × (( a − b ) cz + aj + b )(1 − cz ) j +2 | cz | < a, b ∈ C (T.2.5) n ( n − c n cz ) (1 − cz ) c j j ! ( cz ) +4 jcz + j ( j − ) (1 − cz ) j +3 | cz | < Exponential–Series–Based Sequences
Eq. f n F f ( z ) F ( j ) f ( z ) Parameters (T.2.6) r n n ! e rz r j e rz r ∈ C (T.2.7) n ( n − r n n ! ( rz ) e rz r j e rz [ ( rz ) +2 jrz + j ( j − ] r ∈ C Table 2.
Special Case Sequences and Generating Function Formulas
Listings of the ordinary generating functions, F f ( z ) , j th derivative formulas, F ( j ) f ( z ) for j ∈ N , and corresponding series parameters for several special case sequences, h f n i . Remark (Admissible Forms of Sequence Generating Functions) . In practice, the as-sumption made in phrasing Proposition 3.1 and in the theorem statement requiring that thesequence OGF is analytic on some non-trivial R ( f ), i.e. , so that F f ( z ) ∈ C ∞ ( R ( f )), doesnot impose any significant or particularly restrictive conditions on the applications that maybe derived from these transformations. Rather, provided that suitable choices of the seriesparameters in the starting OGF series for many variations of the geometric-series-based andexponential-series-based sequence forms lead to convergent generating functions analytic onsome open disk, the formulas for all non-negative integer-order OGF derivatives exist, andmoreover, are each easily obtained in closed-form as functions over all j ≥ j th derivatives of such sequence OGFscorresponding to a few particular classes of these square series expansions that arise frequentlyin our applications. The particular variations of the geometric square series in the forms of thegeneralized functions defined in Table 1 on page 4 arise naturally in applications includingthe study of theta functions, modular forms, in infinite products that form the generating QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 13 functions for partition functions, and in other combinatorial sequences of interest within thearticle. The new integral representations for the generalized classes of series expansions forthe functions defined by Table 1 given in the next sections are derived from Theorem 3.3 andfrom the formulas for the particular special case sequence OGFs given separately in Table 2.The next sections provide applications extending the examples already given in Section 2.4.4.
Applications of the Geometric Square Series
Initial Results.
In this subsection, we state and prove initial results providing integralrepresentations for a few generalized classes of the geometric square series cases where f n := c n which follow from Theorem 3.3. Once we have these generalized propositions at ourdisposal, we move along to a number of more specific applications of these results in the nextsubsections which provide new integral representations for the Jacobi theta functions andother closely-related special function series and constants. The last subsection of this sectionproves corresponding integral representations for the polynomial multiples of the geometricsquare series where f n := ( αn + β ) m · c n which have immediate applications to the expansionsof higher-order derivatives of the Jacobi theta functions. Proposition 4.1 (Geometric Square Series) . Let q ∈ C be defined such that | q | < andsuppose that c, z ∈ C are defined such that | cz | < . For these choices of the series parameters, q, c, z , the ordinary geometric square series, G sq ( q, c, z ) , defined by (T.1.1) of Table 1 satisfiesan integral representation of the following form: G sq ( q, c, z ) = Z ∞ e − t / √ π − cz cosh (cid:16) t p q ) (cid:17) c z − cz cosh (cid:16) t p q ) (cid:17) + 1 dt. (29a) Proof Sketch.
The general method of proof employed within this section is given along thelines of the second phrasing of the transformation results in (23b) of Theorem 3.3. Alternately,we are able to arrive in these results as in the supplementary
Mathematica notebook [17] byapplying the formula involving the sequence OGFs in (23a) using the results given in Table2 where the series parameter z is taken over the ranges stated in the rightmost columns ofthe table. Thus we may use both forms of the transformation result stated in the theoreminterchangeably to arrive at proofs of our new results in subsequent sections of the article. (cid:26) Proof.
We prove the particular result for this formula in (29a) as a particular example case ofthe first similar argument method to be employed in the further results given in this sectionbelow. In particular, for | cz | <
1, the main theorem and the OGF for the special case of thegeometric-series-based sequence, f n ≡ c n , given in (T.2.1) of the table imply that G sq ( q, c, z ) = Z ∞ e − t / √ π " X b = ± (cid:16) − e bt √ q ) cz (cid:17) − dt. (29b)The separate integrand functions in the previous equation are combined as X b = ± (cid:16) − e bt √ q ) cz (cid:17) − = 2 − cz h e t √ q ) + e − t √ q ) i c z − h e t √ q ) + e − t √ q ) i + 1 , (i)Since the hyperbolic cosine function satisfies [15, § × cosh (cid:16) t p q ) (cid:17) = 2 × " X b = ± e bt √ q ) , (29c)the integral formula claimed in the proposition then follows by rewriting the intermediateexponential terms in the combined formula from (i) given above. (cid:3) Remark (Integrals for Shifted Forms of the Geometric Square Series) . For any d ∈ Z + ,we note that the slightly modified functions, ϑ d ( q, c, z ), defined as the “shifted” geometricsquare series from (T.1.2) in Table 1 satisfy expansions through G sq ( q, c, z ) of the form e ϑ d ( q, c, z ) = (cid:16) q d cz (cid:17) d × G sq (cid:16) q, q d c, z (cid:17) . Moreover, for any fixed choice of d ∈ Z + , we may similarly obtain modified forms of theintegral representations in Proposition 4.1 as follows: G sq ( q, c, z ) = d − X i =0 q i ( cz ) i + e ϑ d ( q, c, z ) , if | cz | < min (cid:16) , | q | − d (cid:17) . If we then specify that | cz | < min (cid:0) , | q | − d (cid:1) for any fixed integer d ≥
0, we may expandintegral representations for the shifted series variants of these functions through the formulasgiven in Proposition 4.1 as e ϑ d ( c, q, z ) = (cid:16) q d cz (cid:17) d Z ∞ e − t / √ π − q d cz cosh (cid:16) t p q ) (cid:17) q d c z − q d cz cosh (cid:16) t p q ) (cid:17) + 1 dt. (29d)The series for the special case of the shifted series when d := 1 arises in the unilateral seriesexpansions of bilateral square series of the form ∞ X n = −∞ ( ± n q n ( r n + r ) / = 1 + ∞ X n =1 ( ± n q n ( r n + r ) / + ∞ X n =1 ( ± n q n ( r n − r ) / . Whenever | cz | < min (cid:0) , | q | − (cid:1) , this shifted square series function satisfies the special caseintegral formula given by e ϑ ( c, q, z ) = Z ∞ e − t / √ π qcz − q c z cosh (cid:16) t p q ) (cid:17) q c z − q cz cosh (cid:16) t p q ) (cid:17) + 1 dt. (29e) Corollary 4.3 (A Special Case) . Suppose that a, b ∈ C are fixed scalars and that c, z ∈ C are chosen such that | cz | < . For these choices of the series parameters in the definition of (T.1.3) , the square series function, Q a,b ( q, c, z ) , has an integral representation of the form Q a,b ( q, c, z ) = Z ∞ acze − t / √ π (cid:0) c z + 1 (cid:1) cosh (cid:16) t p q ) (cid:17) − cz (cid:16) c z − cz cosh (cid:16) t p q ) (cid:17) + 1 (cid:17) dt (30)+ b · G sq ( q, c, z ) , where the second term in (30) given in terms of the function, G sq ( q, c, z ) , satisfies the integralrepresentation from Proposition 4.1 given above.Proof. The result follows by applying the theorem to the sequence OGF formulas in (T.2.1)and (T.2.2), respectively, and then combining the denominators in the second integrand using(29c) as in the first proof of Proposition 4.1. Equivalently, we notice that the integrand in(29a) of the previous proposition may be differentiated termwise with respect to z to formthe first term in the new integral representation formulated by (30). In this case, the function Q a,b ( q, c, z ) defined as the power series in (T.1.3) is related to the function, G sq ( q, c, z ), by Q a,b ( q, c, z ) = az · G ′ sq ( q, c, z ) + b · G sq ( q, c, z ) , where the right-hand-side derivative is taken with respect to z , and where G sq ( q, c, z ) may bedifferentiated termwise with respect to z . In particular, the first derivative of the function, QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 15 G sq ( q, c, z ), is obtained through Proposition 4.1 by differentiation as G ′ sq ( q, c, z ) = Z ∞ e − t / √ π × ∂∂z − cz cosh (cid:16)p q ) t (cid:17) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 dt = Z ∞ ce − t / √ π (cid:0) c z + 1 (cid:1) cosh (cid:16)p q ) t (cid:17) − cz (cid:16) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) dt. (cid:3) Jacobi Theta Functions and Related Special Function Series.
The geometricsquare series , G sq ( q, c, z ), is related to the form of many special functions where the parameter z := p ( q ) is some fixed rational power of q . A number of such expansions of these seriescorrespond to special cases of this generalized function, denoted by G sq ( p, m ; q, ± c ), expandedthrough its integral representation below where the fixed p and m are considered to assumesome values chosen over the positive rationals. These integral representations, of course, donot lead to simple power series expansions in q of the integrand about zero, and cannot beintegrated termwise with respect to q in their immediate form following from Proposition 4.1.Notice that whenever p, m, q, c ∈ C are taken such that | q p | ∈ (0 ,
1) and | q m c | <
1, anintegral representation for the function, G sq ( p, m ; q, ± c ) ≡ Gi sq ( q p , c · q m , ± G sq ( p, m ; q, ± c ) = Z ∞ e − t / √ π (cid:16) ∓ cq m cosh (cid:16)p p Log( q ) t (cid:17)(cid:17) c q m ∓ cq m z cosh (cid:16)p p Log( q ) t (cid:17) + 1 dt. (31)The series for the Jacobi theta functions, ϑ i ( q ) ≡ ϑ i (0 , q ) when i = 2 , ,
4, and
Ramanujan’sfunctions , ϕ ( q ) and ψ ( q ), are examples of these special case series that are each cited asparticular applications of the result in (31) in the next subsections of the article. Integral Representations for the Jacobi Theta Functions, ϑ i ( q ) . The variants of the classicalJacobi theta functions, ϑ i ( u, q ) for i = 1 , , ,
4, defined by the functions, ϑ i ( q ) ≡ ϑ i (0 , q )where ϑ i ( q ) = 0 for i = 2 , , Proposition 4.4 (Integral Representations for Jacobi Theta Functions) . For any q ∈ C suchthat | q | ∈ (0 , / , the theta function, ϑ ( q ) , has the integral representation ϑ ( q ) = 4 q / Z ∞ e − t / √ π − q cosh (cid:16)p q ) t (cid:17) q − q cosh (cid:16)p q ) t (cid:17) + 1 dt. (32a) Moreover, for any q ∈ C such that | q | ∈ (0 , / , the remaining two theta function cases, ϑ i ( q ) , for i = 3 , , satisfy respective integral representations given by ϑ ( q ) = 1 + 4 q Z ∞ e − t / √ π − q cosh (cid:16)p q ) t (cid:17) q − q cosh (cid:16)p q ) t (cid:17) + 1 dt (32b) ϑ ( q ) = 1 − q Z ∞ e − t / √ π q cosh (cid:16)p q ) t (cid:17) q + 2 q cosh (cid:16)p q ) t (cid:17) + 1 dt. (32c) Proof.
After setting u := 0 in the Fourier series forms for the full series for the Jacobi thetafunctions, ϑ i ( u, q ), from the introduction, the series for the theta functions, ϑ i ( q ), are ex-panded for a fixed q as follows: ϑ ( q ) = 2 q / ∞ X n =0 q n q n ≡ q / G sq ( q, q,
1) (i) ϑ ( q ) = 1 + 2 q ∞ X n =0 q n q n ≡ qG sq (cid:0) q, q , (cid:1) (ii) ϑ ( q ) = 1 − q ∞ X n =0 q n ( − n q n ≡ − qG sq (cid:0) q, q , − (cid:1) . (iii)The integrals given on the right–hand–sides of (32) follow by applying (31) to each of theseseries first with | q | ∈ (0 , / | q | ∈ (0 , /
4) in the last two cases, respectively. (cid:3)
The next integral formulas for the
Mellin transform involving the Jacobi theta functions, ϑ i (0 , ıx ), with i = 2 , , gamma function , Γ( z ), and the Riemann zeta function , ζ ( s ), for Re( s ) > § Corollary 4.5 (Mellin Transforms of the Jacobi Theta Functions) . Let s ∈ C denote a fixedconstant such that Re( s ) > . Then we have that (2 s − π − s/ Γ( s/ ζ ( s ) = Z ∞ x s − ϑ (0 , ıx ) dx = Z ∞ Z ∞ e − t − πx √ π x s − (cid:16) − e − πx cos (cid:0) √ πtx (cid:1)(cid:17)(cid:0) e − πx − e − πx cos (cid:0) √ πtx (cid:1) + 1 (cid:1) dtdxπ − s/ Γ( s/ ζ ( s ) = Z ∞ x s − (cid:0) ϑ (0 , ıx ) − (cid:1) dx = Z ∞ Z ∞ e − t − πx √ π x s − (cid:16) − e − πx cos (cid:0) √ πtx (cid:1)(cid:17)(cid:0) e − πx − e − πx cos (cid:0) √ πtx (cid:1) + 1 (cid:1) dtdx (1 − − s ) π − s/ Γ( s/ ζ ( s ) = Z ∞ x s − (cid:0) − ϑ (0 , ıx ) (cid:1) dx = Z ∞ Z ∞ e − t − πx √ π x s − (cid:16) e − πx cos (cid:0) √ πtx (cid:1)(cid:17)(cid:0) e − πx + 2 e − πx cos (cid:0) √ πtx (cid:1) + 1 (cid:1) dtdx Proof.
The Mellin transform integrals given in terms of the Jacobi theta function variantsof the form ϑ i (0 | ıx ) = ϑ i (cid:16) e − πx (cid:17) are stated in the results of [15, § (cid:3) Explicit Values of Ramanujan’s ϕ –Function and ψ –Function. The series for
Ramanujan’sfunctions , ϕ ( q ) and ψ ( q ), are expanded similarly through the series for the classical thetafunctions as ϕ ( q ) ≡ ϑ (0 , q ) and ψ ( q ) ≡ (cid:0) q / (cid:1) − × ϑ (cid:0) , q / (cid:1) [12, cf . § XII] [6, cf . § Corollary 4.6 (Integral Representations) . Suppose that q ∈ C and | q | < . For these in-puts of q , Ramanujan’s functions, ϕ ( q ) and ψ ( q ) , respectively, have the following integral QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 17 representations: ϕ ( q ) = 1 + Z ∞ e − t / √ π q (cid:16) − q cosh (cid:16)p q ) t (cid:17)(cid:17) q − q cosh (cid:16)p q ) t (cid:17) + 1 dtψ ( q ) = Z ∞ e − t / √ π (cid:16) − √ q cosh (cid:16)p Log( q ) t (cid:17)(cid:17) q − √ q cosh (cid:16)p Log( q ) t (cid:17) + 1 dt. Proof.
On the stated region where | q | <
1, these two special functions have series expansionsin the form of (31) given by ϕ ( q ) = 1 + 2 q ∞ X n =0 q n q n ≡ qG sq (cid:0) q, q , (cid:1) ψ ( q ) = ∞ X n =0 ( √ q ) n ( √ q ) n ≡ G sq ( √ q, √ q, . The new integral representations for these functions then follow from Proposition 4.1 appliedto the series on the right-hand-side of each of the previous equations. (cid:3)
Corollary 4.7 (Special Values of Ramanujan’s ϕ -Function) . For any k ∈ R + , the variant ofthe Ramanujan ϕ -function, ϕ (cid:0) e − kπ (cid:1) ≡ ϑ (cid:0) e − kπ (cid:1) , has the integral representation ϕ (cid:16) e − kπ (cid:17) = 1 + Z ∞ e − t / √ π e kπ (cid:16) e kπ − cos (cid:16) √ πkt (cid:17)(cid:17) e kπ − e kπ cos (cid:16) √ πkt (cid:17) + 1 dt. (33) Moreover, the special values of this function corresponding to the particular cases of k ∈{ , , , } in (33) have the respective integral representations π / Γ (cid:0) (cid:1) = 1 + Z ∞ e − t / √ π " e π (cid:0) e π − cos (cid:0) √ πt (cid:1)(cid:1) e π − e π cos (cid:0) √ πt (cid:1) + 1 dt (34) π / Γ (cid:0) (cid:1) · p √ Z ∞ e − t / √ π " e π (cid:0) e π − cos (2 √ πt ) (cid:1) e π − e π cos (2 √ πt ) + 1 dtπ / Γ (cid:0) (cid:1) · p √ / / = 1 + Z ∞ e − t / √ π " e π (cid:0) e π − cos (cid:0) √ πt (cid:1)(cid:1) e π − e π cos (cid:0) √ πt (cid:1) + 1 dtπ / Γ (cid:0) (cid:1) · p √ / = 1 + Z ∞ e − t / √ π " e π (cid:0) e π − cos (cid:0) √ πt (cid:1)(cid:1) e π − e π cos (cid:0) √ πt (cid:1) + 1 dt. Proof.
The first integral in (33) is obtained from Corollary 4.6 applied to the functions ϕ ( q ) ≡ ϑ ( q ) at q := e − kπ . The constants on the left-hand-side of the integral equations in (34)correspond to the known values of Ramanujan’s function, ϕ (cid:0) e − kπ (cid:1) , over the particular inputsof k ∈ { , , , } , in respective order. These explicit formulas for the values of ϕ (cid:0) e − kπ (cid:1) areestablished by Theorem 5.5 of [22]. (cid:3) The special cases of the right-hand-side integrals in the corollary are verified numericallyto match the constant values cited by each of the equations in (34) in [17]. Still other integralformulas for the values of the function, ϕ (cid:0) e − kπ (cid:1) , are known in terms of formulas involvingpowers of positive rational inputs to the gamma function, in the form of (33), for example,when k ∈ {√ , √ , √ } . Corollary 4.8 (Explicit Values of Ramanujan’s ψ -Function) . For any k ∈ R + , the forms ofthe Ramanujan ψ –function, ψ (cid:0) e − kπ (cid:1) ≡ e kπ/ ϑ (cid:0) e − kπ/ (cid:1) , have the integral representation ψ (cid:16) e − kπ (cid:17) = Z ∞ e − t / √ π cos (cid:16) √ kπt (cid:17) − e kπ/ cos (cid:16) √ kπt (cid:17) − cosh (cid:0) kπ (cid:1) dt. (35) The explicit values of this function corresponding to the choices of k ∈ { , , / } in (35) havethe following respective integral representations: π / Γ (cid:0) (cid:1) · e π/ / = Z ∞ e − t / √ π " cos ( √ πt ) − e π/ cos ( √ πt ) − cosh (cid:0) π (cid:1) dt (36) π / Γ (cid:0) (cid:1) · e π/ / = Z ∞ e − t / √ π " cos (cid:0) √ πt (cid:1) − e π cos (cid:0) √ πt (cid:1) − cosh ( π ) dtπ / Γ (cid:0) (cid:1) · (cid:0) √ (cid:1) / e π/ / = Z ∞ e − t / √ π " cos (cid:0)p π t (cid:1) − e π/ cos (cid:0)p π t (cid:1) − cosh (cid:0) π (cid:1) dt. Proof.
The integral representation in (35) is obtained from Corollary 4.6 at the inputs of q := e − kπ . The explicit formulas for constants on the left-hand-side of the integral equationsin (36) are the known values of Ramanujan’s function, ψ (cid:0) e − kπ (cid:1) , from Theorems 6.8 and 6.9of the reference [5] corresponding to these particular values of k ∈ Q + . (cid:3) Integral Formulas for Sequences Involving Powers of Linear Polynomials.
Theprimary applications motivating the next results proved in this section correspond to thehigher-order derivatives of the Jacobi theta functions defined by d ( j ) ϑ i ( u, q ) /du ( j ) | u = u forsome prescribed setting of u , including the special cases of the theta functions from the resultsfor these series from Section 4.2 where u = 0. The square series integral representations forpolynomial powers, and then more generally for any fixed linear polynomial multiple, ofthe geometric square series are easily obtained by applying Proposition 3.1 to the forms of G sq ( q, c, z ) with respect to the underlying series in z .4.3.1. Initial Results.
Proposition 4.9 (Integrals for Polynomial Powers) . Suppose that m ∈ N and q, c, z ∈ C aredefined such that | q | ∈ (0 , and where | cz | < . Then the square series function, ϑ ,m ( q, c, z ) ,defined by the series in (T.1.4) satisfies an integral formula given by ϑ ,m ( q, c, z ) = Z ∞ e − t / √ π m X k =0 (cid:26) mk (cid:27) ( cz ) k k ! × Num k (cid:16)p q ) t, cz (cid:17)(cid:16) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dt. (37) The numerator terms of the integrands in the previous equation denote the polynomials
Num k ( w, z ) ∈ C [ z ] of degree k + 1 in z defined by the following equation for non-negative integers k ≤ m : Num k ( w, cz ) := X b = ± e − bkw × (cid:16) − e bw cz (cid:17) k +1 . Proof.
By the proof of Theorem 3.3, the integral representation of the series in the first ofthe next equations in (i) can be differentiated termwise with respect to z to arrive at the j th derivative formulas given in (38). ϑ , ( q, c, z ) = Z ∞ e − t / √ π (cid:16) − e √ q ) t cz (cid:17) + 1 (cid:16) − e − √ q ) t cz (cid:17) dt (i) QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 19 ϑ ( j )0 ,m ( q, c, z ) = Z ∞ e − t / √ π e j √ q ) t c j · j ! (cid:16) − e √ q ) t cz (cid:17) j +1 + e − j √ q ) t c j · j ! (cid:16) − e − √ q ) t cz (cid:17) j +1 dt (38)= Z ∞ e − t / √ π c j j ! P b = ± e bj √ q ) t × (cid:16) − e − b √ q ) t cz (cid:17) j +1 (cid:16) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) j +1 dt. We may then apply Proposition 3.1 and the definition of the numerator terms, Num k ( w, cz ),to the previous equations to obtain the result stated in (37). (cid:3) Scaled Formulas for the Numerator Functions
Eq. k Scaled Numerator Function ( × Num k ( s, y )) (T.3.1) − y cosh( s ) (T.3.2) − y + ( y + 1) cosh( s ) (T.3.3) y − ( y + 3 y ) cosh( s ) + cosh(2 s ) (T.3.4) − y + ( y + 6 y ) cosh( s ) − y cosh(2 s ) + cosh(3 s ) (T.3.5) y − ( y + 10 y ) cosh( s ) + 10 y cosh(2 s ) − y cosh(3 s ) + cosh(4 s ) Table 3.
Several Formulas for the Numerator Functions, Num k ( s, y ) Explicit formulas for the numerator functions,
Num k ( s, y ) , from the definition given inProposition 4.9 over the first several special cases of k ∈ [0 , . Proposition 4.9 provides an integral representation of the polynomial multiples for anygeometric series sequences of the form f n := p ( n ) × c n corresponding to any fixed p ( n ) ∈ C [ n ].Notice that for polynomials defined as integral powers of the form p ( n ) := ( αn + β ) m for somefixed m ∈ Z + , for example, as in the series for the m th derivatives of the theta functions, ϑ ( m ) i ( q ), the repeated terms over the index k ∈ { , , . . . , m } in the inner sums from (37) maybe simplified even further to obtain the next simplified formulas in the slightly more generalresults for these series expansions. Table 3 lists several simplified expansions of the functions,Num k ( w, z ), defined in the statement of Proposition 4.9. Proposition 4.10.
Suppose that m ∈ N and α, β, q, c, z ∈ C are defined such that | q | ∈ (0 , and such that | cz | < . The modified square series from (T.1.5) satisfies an integralrepresentation of the form ϑ ,m ( α, β ; q, c, z ) (39)= Z ∞ e − t / √ π X ≤ i ≤ k ≤ m (cid:18) ki (cid:19) ( − k − i ( αi + β ) m ( cz ) k Num k (cid:16)p q ) t, cz (cid:17)(cid:16) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dt. Proof.
The proof is similar to the proof of Proposition 4.9. We employ the same notationand j th derivative expansions from the proof of the first proposition. To simplify notation,for j ≥ N j ( t, q, cz ) = ( cz ) j Num j (cid:16)p q ) t, cz (cid:17)(cid:16) c z − cz cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) j +1 . Since the Stirling numbers of the second kind are expanded by the finite sum formula [15, § (cid:26) ik (cid:27) = 1 k ! k X j =0 (cid:18) kj (cid:19) ( − k − j j i , by expanding out the polynomial powers of ( αn + β ) m according to the binomial theorem wearrive at the following expansions of Proposition 3.1: ϑ ,m ( α, β ; q, c, z ) = Z ∞ e − t / √ π × m X j =0 j X k =0 (cid:18) mj (cid:19)(cid:26) jk (cid:27) α j β m − j × k ! N k ( t, q, cz ) dt = Z ∞ e − t / √ π × m X j =0 j X k =0 k X i =0 (cid:18) mj (cid:19)(cid:18) ki (cid:19) ( αi ) j β m − j ( − k − i N k ( t, q, cz ) dt = Z ∞ e − t / √ π × j X k =0 k X i =0 (cid:18) ki (cid:19) ( αi + β ) m ( − k − i N k ( t, q, cz ) dt. (cid:3) Higher-Order Derivatives of the Jacobi Theta Functions.
Example 4.11 (Special Case Derivative Series) . An integral representation for the specialcase of the first derivative of the theta function, ϑ ′ ( q ) ≡ q / · Q , ( q, q, − ϑ ′ ( q ) = 2 q / ∞ X n =0 q n (2 n + 1)( − q ) n , is a consequence of the related statement in Proposition 4.3 already cited by the results givenabove. This series arises in expanding the cube powers of the infinite q -Pochhammer symbol,( q ) ∞ , cited as an example in Section 2.4.The higher-order series for the third derivative of this theta function provides anotherexample of the new generalization of this first result stated by Proposition 4.10 expanded asin the following equations for | q | ∈ (0 , / ϑ ′′′ ( q ) = (cid:16) q / (cid:17) × ∞ X n =0 q n (2 n + 1) ( − q ) n = Z ∞ e − t / √ π X ≤ i ≤ k ≤ (cid:18) ki (cid:19) ( − i (2 i + 1) q k +1 / Num k (cid:16)p t, − q (cid:17)(cid:16) q + 2 q cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dt. Notice that the subsequent cases of the higher-order, j th derivatives of these theta func-tions are then formed from the Fourier series expansions of the classical functions, ϑ i ( u, q ),expanded by the series in (T.1.8) through (T.1.11). More precisely, if j ∈ Z + , the new integralrepresentations for the higher–order j th derivatives of the functions, ϑ ( j ) i ( q ) ≡ ϑ ( j ) i (0 , q ), withrespect to their first parameter are then obtained from these series through Proposition 4.10over odd-ordered positive integers j := 2 m + 1 when i = 1, and at even-ordered non-negativecases of j := 2 m when i = 2 , , m ∈ N . Lemma 4.12 and Corollary 4.13 given belowstate the exact series formulas and corresponding integral representations for the higher-orderderivatives of these particular variants of the Jacobi theta functions. Lemma 4.12 (Generalized Series for Higher-Order Derivatives) . For any non-negative j ∈ Z ,the higher-order j th derivatives, ϑ ( j ) i (0 , q ) , of the classical Jacobi theta functions, ϑ i ( u, q ) ,with respect to u are expanded through the unilateral power series for the functions defined by QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 21 (T.1.5) as ϑ ( j )1 ( q ) = (cid:16) q / (cid:17) × ϑ ,j (2 , q, q, −
1) [ j ≡ δ (40) ϑ ( j )2 ( q ) = (cid:16) q / (cid:17) × ϑ ,j (2 , q, q, −
1) [ j ≡ δ ϑ ( j )3 ( q ) = [ j = 0] δ + (2 q ) × ϑ ,j (cid:0) , q, q , (cid:1) [ j ≡ δ ϑ ( j )4 ( q ) = [ j = 0] δ − (2 q ) × ϑ ,j (cid:0) , q, q , − (cid:1) [ j ≡ δ , for q ∈ C satisfying some | q | ∈ (0 , R q ( ϑ i )) such that the upper bound, R q ( ϑ i ) , on the intervalis defined as R q ( ϑ ) ≡ / when i := 1 , , and as R q ( ϑ ) ≡ / when i := 3 , .Proof. To prove the results for the special case series expanded by (40), first observe that forany j ∈ N , the higher-order derivatives of the next Fourier series with respect to u satisfy ∂ ( j ) ∂u ( j ) " c ( q ) × ∞ X n =0 q n sc (( αn + β ) · u ) c n z n u =0 (i)= [ j = 0] δ + c ( q ) × ∞ X n =0 h ( αn + β ) j · sc ( j ) (0) i × q n ( cz ) n , for some function c ( q ) that does not depend on u , series parameters α, β ∈ R , and where thederivatives of the trigonometric functions, denoted in shorthand by sc ∈ { sin , cos } , correspondto the known formulas for these functions from calculus. The theta functions, ϑ i ( q ) and ϑ ( j ) i ( q ), on the left-hand-side of (40) form special cases of the Fourier series for the Jacobitheta functions, ϑ i ( u, q ), that are then expanded by (i) where cos(0) = 1 and sin(0) = 0. (cid:3) Corollary 4.13 (Integrals for Higher-Order Derivatives of the Jacobi Theta Functions) . Let q ∈ C be defined on some interval | q | ∈ (0 , R q ( ϑ i )) where R q ( ϑ ) ≡ / when i := 1 , , andas R q ( ϑ ) ≡ / when i := 3 , . Then for any such q and fixed m ∈ Z + , the higher-orderderivatives of the theta functions, ϑ i ( q ) , satisfy each of the following integral representations: ϑ (2 m +1)1 ( q ) = Z ∞ e − t / √ π " X ≤ i ≤ k ≤ m +1 (cid:18) ki (cid:19) ( − i (2 i + 1) m +1 q k +1 / × (41) × Num k (cid:16)p q ) t, − q (cid:17)(cid:16) q + 2 q cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dtϑ (2 m )2 ( q ) = Z ∞ e − t / √ π " X ≤ i ≤ k ≤ m (cid:18) ki (cid:19) ( − i (2 i + 1) m q k +1 / ×× Num k (cid:16)p q ) t, − q (cid:17)(cid:16) q + 2 q cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dtϑ (2 m )3 ( q ) = Z ∞ m +1 e − t / √ π " X ≤ i ≤ k ≤ m (cid:18) ki (cid:19) ( − k − i ( i + 1) m q k +1 ×× Num k (cid:16)p q ) t, q (cid:17)(cid:16) q − q cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dtϑ (2 m )4 ( q ) = Z ∞ m +1 e − t / √ π " X ≤ i ≤ k ≤ m (cid:18) ki (cid:19) ( − i +1 ( i + 1) m q k +1 × × Num k (cid:16)p q ) t, − q (cid:17)(cid:16) q + 2 q cosh (cid:16)p q ) t (cid:17) + 1 (cid:17) k +1 dt. Proof.
The formulas in (41) follow immediately as consequences of the integral representa-tions proved in Proposition 4.10 applied to each of the series expansions for the higher-orderderivative cases provided by Lemma 4.12. (cid:3) Applications of Exponential Series Generating Functions
A Comparison of Characteristic Expansions of the Square Series Integrals.Example 5.1 (The Number of Edges in Labeled Graphs) . The number of edges in a labeledgraph on n ≥ nodes , denoted by the sequence h e ( n ) i , is given in closed-form by the formula[18, A095351] e ( n ) = 14 n ( n − n ( n − / . The ordinary and exponential generating functions of this sequence, defined to be e sq ( z ) and b e sq ( z ), respectively, correspond to the special cases of the second derivatives with respect to z of the ordinary and exponential generating functions, G sq ( q, c, z ) and E sq ( q, r, z ), from Table1 which are expanded as e sq ( z ) := OGF z [[ e (0) , e (1) , e (2) , . . . ]] ≡ z × G ′′ sq (cid:16) / , − / , z (cid:17)b e sq ( z ) := EGF z [[ e (0) , e (1) , e (2) , . . . ]] ≡ z × E ′′ sq (cid:16) / , − / , z (cid:17) . The special cases of the OGF formulas cited in (T.2.5) and (T.2.7) of Table 2 then lead tothe next integral representations for each of these sequence generating functions given by e sq ( z ) = Z ∞ e − t / √ π z − √ z ( z + 6) cosh (cid:16)p Log(2) t (cid:17) + 4 z cosh (cid:16) p Log(2) t (cid:17)(cid:16) z − √ z cosh (cid:16)p Log(2) t (cid:17) + 2 (cid:17) dt b e sq ( z ) = z × Z ∞ e − t / √ π " X b = ± exp (cid:18) e bt √ Log(2) z √ bt p Log(2) (cid:19) dt.
The comparison given in Example 5.1 clearly identifies these characteristic, or at leaststylistic, differences in the resulting square series integral representations derived from theforms of these separate “ ordinary ” and “ exponential ” sequence types. We also see the generalsimilarities in form of the first geometric-series-like integrals to the Fourier series for the
Pois-son kernel , where we point out the similarities of the second exponential-series-like integrals togenerating functions for the non-exponential, Stirling-number-related
Bell polynomials , B n ( x )[15, § § § Initial Results.Proposition 5.2 (Exponential Square Series Generating Functions) . For any fixed parame-ters q, z, r ∈ C , the exponential square series functions, defined respectively as (T.1.12) and (T.1.13) in Table 1, have the following integral representations: E sq ( q, r, z ) = Z ∞ e − t / √ π (cid:20) e e √ q ) t rz + e e − √ q ) t rz (cid:21) dt (42a) e E sq ( q, r, z ) = Z ∞ e − t / √ π (cid:20) e e √ Log( q ) t rz √ q + e e − √ Log( q ) t rz √ q (cid:21) dt. (42b) QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 23
Proof.
First, it is easy to see from Theorem 3.3 applied to the first form of the exponentialseries OGF given in (T.2.6), that E ( q, r, z ) = Z ∞ e − t / √ π " X b = ± e e b √ q ) t rz dt, which implies the first result given in (42a). Next, the series for the second function, e E sq ( q, r, z ),over the binomial powers of q ( n ) ≡ q n ( n − / is expanded through first function as e E sq ( q, r, z ) = ∞ X n =0 q n ( n − / r n z n n ! ≡ E sq (cid:16) q / , rq − / , z (cid:17) . This similarly leads to the form of the second result stated in (42b). (cid:3)
Examples of Chromatic Generating Functions.
A class of generating function ex-pansions resulting from the application of Proposition 5.2 to exponential-series-based OGFsdefines chromatic generating functions of the form [19, § b F f ( q, z ) = ∞ X n =0 f ( n ) z n q ( n ) n ! , for some prescribed sequence of terms, h f ( n ) i . If f ( n ) denotes the number of labeled acyclicdigraphs with n vertices , for example as considered in [20, Prop. 2.1] [19, cf . Ex. 3.15.1(e)],then the chromatic generating function, b F f ( q, z ), has the form b F f (2 , z ) := ∞ X n =0 f ( n ) z n n ) n ! = ∞ X n =0 ( − n z n n ) n ! ! − . Example 5.3 below cites other particular chromatic generating functions corresponding tospecial cases of the integral representations for the series involving binomial square powers of q established in Proposition 5.2. Example 5.3 (Powers of a Special Chromatic Generating Function) . Suppose that G n is afinite, simple graph with n vertices and that χ ( G n , λ ) denotes the chromatic polynomial of G n evaluated at some λ ∈ C [20, § k , a variant of the EGF, c M k ( z ), for the sequence of M n ( k ) = P G n χ ( G n , k ) is expanded in integer powers of a squareseries integrals as [20, § c M k ( z ) = ∞ X n =0 M n ( k ) z n n ) n ! = ∞ X n =0 z n n ) n ! ! k . (43)The right-hand-side of (43) corresponds to the k th powers of the generating function, E sq (1 / , , z ),which is expanded for k ∈ Z + through the integral given in Proposition 5.2 as c M k ( z ) = Z ∞ e − t / √ π (cid:20) e e ı √ Log(2) t √ z + e e − ı √ Log(2) t √ z (cid:21) dt ! k . (44)Notice that since the integrands in (42b) can be integrated termwise in z , the right-hand-sideof (44) is also expressed as the multiple integral c M k ( z ) = Z ∞ · · · Z ∞ e − ( t + ··· + t k ) / (2 π ) k/ " k Y i =1 e e ı √ Log(2) ti √ z + e e − ı √ Log(2) ti √ z dt · · · dt k . Another Application: A Generalized Form of the Binomial Theorem.Proposition 5.4 (A Square Series Analog to the Binomial Theorem) . For constants q, r, c, d ∈ C and n ∈ N , a generalized analog to the binomial theorem involving square powers of theparameters q and r has the following double integral representation: n X k =0 (cid:18) nk (cid:19) c k q k d n − k r ( n − k ) = Z ∞ Z ∞ e − ( t + s ) / π "(cid:16) ce √ q ) t + de √ r ) s (cid:17) n (45)+ (cid:16) ce √ q ) t + de − √ r ) s (cid:17) n + (cid:16) ce − √ q ) t + de √ r ) s (cid:17) n + (cid:16) ce − √ q ) t + de − √ r ) s (cid:17) n dtds. Proof.
Let c, q ∈ C and for variable t ∈ R define the function, E ( t ) q ( c, z ), by E ( t ) q ( c, z ) = e e √ q ) t cz + e e − √ q ) t cz . (46)It follows from the transformation result in Proposition 5.2 that ∞ X n =0 q n c n z n n ! = Z ∞ e − t / √ π E ( t ) q ( c, z ) dt. (47)Fix the constants c, q, d, r ∈ C and observe that the next double integral results for thecoefficients in the discrete convolution of two power series in the form of (47). n X k =0 c k q k k ! · d n − k r ( n − k ) ( n − k )! = Z ∞ Z ∞ e − ( t + s ) / π · [ z n ] (cid:16) E ( t ) q ( c, z ) E ( s ) r ( d, z ) (cid:17) dtds (48)The left-hand-side sum in (45) is obtained by multiplying the coefficient definition on the right-hand-side the previous equation by a factor of n !. This operation can be applied termwise tothe series for E ( t ) q ( c, z ) E ( s ) r ( d, z ) through the next integral for the single factorial function, orgamma function, where Γ( n + 1) = n ! whenever n ∈ N [15, § § n + 1) = Z ∞ u n e − u du, Re( n ) > − Z ∞ e e a buz e − u du = Z ∞ e − (1 − e a bz ) u du = 1(1 − e a bz ) (50)for constants a, b and z such that the right–hand–side of (50) satisfies Re( e a bz ) <
1. Next,let the coefficient terms, b ( t,s ) i,j ( c, q, d, r ), be defined in the form of the next equation. b ( t,s ) i,j ( c, q, d, r ) = c · exp (cid:16) ( − i p q ) t (cid:17) + d · exp (cid:16) ( − j p r ) s (cid:17) By combining the results in (49) and (50), it follows that the function e E ( t,s ) q,r ( c, d, z ) := Z ∞ e − ( t + s ) / π E ( t ) q ( c, uz ) E ( s ) r ( d, uz ) e − u du = e − ( t + s ) / π X ( i,j ) ∈{ , }×{ , } − b ( t,s ) i,j ( c, q, d, r ) z ) = ∞ X n =0 X ( i,j ) ∈{ , }×{ , } e − ( t + s ) / π b ( t,s ) i,j ( c, q, d, r ) n z n . QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 25
The complete result given in (45) then follows from the last equation by integrating overnon-negative s, t ∈ R as n X k =0 (cid:18) nk (cid:19) c k q k d n − k r ( n − k ) = Z ∞ Z ∞ [ z n ] (cid:16) e E ( t,s ) q,r ( c, d, z ) (cid:17) dtds. Direct Expansions of Fourier-Type Square Series
Initial Results.Corollary 6.1 (Integral Representations of Fourier–Type Series) . For α, β ∈ R , and c, z ∈ C with | cz | < , the generalized Fourier-type square series functions defined by the series in (T.1.7) have the following integral representations: F cos ( α, β ; q, c, z ) = Z ∞ e ıβ e − t / √ π − e ıα cz cosh (cid:16) t p q ) (cid:17) e ıα c z − e ıα cz cosh (cid:16) t p q ) (cid:17) + 1 dt (51a)+ Z ∞ e − ıβ e − t / √ π − e − ıα cz cosh (cid:16) t p q ) (cid:17) e − ıα c z − e − ıα cz cosh (cid:16) t p q ) (cid:17) + 1 dtF sin ( α, β ; q, c, z ) = Z ∞ e ıβ e − t / √ πı − e ıα cz cosh (cid:16) t p q ) (cid:17) e ıα c z − e ıα cz cosh (cid:16) t p q ) (cid:17) + 1 dt (51b) − Z ∞ e − ıβ e − t / √ πı − e − ıα cz cosh (cid:16) t p q ) (cid:17) e − ıα c z − e − ıα cz cosh (cid:16) t p q ) (cid:17) + 1 dt. Proof.
We prove the forms of these two integral representations by first expanding the trigono-metric function sequences as follows [15, § αn + β ) · c n = e ıβ · ( e ıα c ) n + e − ıβ · (cid:0) e − ıα c (cid:1) n sin ( αn + β ) · c n = e ıβ ı · ( e ıα c ) n − e − ıβ ı · (cid:0) e − ıα c (cid:1) n . Then since | e ± ıα | ≡ α ∈ R , whenever | e ± ıα cz | ≡ | cz | <
1, the generating functionsfor each of these sequences are expanded in terms of the geometric series sequence OGFs in(T.2.1) as F cos ( α, β ; 1 , c, z ) = e ıβ · G sq (1 , e ıα c, z ) + e − ıβ · G sq (cid:0) , e − ıα c, z (cid:1) F sin ( α, β ; 1 , c, z ) = e ıβ ı · G sq (1 , e ıα c, z ) − e − ıβ ı · G sq (cid:0) , e − ıα c, z (cid:1) . The pair of integral formulas stated as the results in (51a) and (51b) are then obtained asthe particular special cases of Proposition 4.1 corresponding to the expansions in the previousequations. (cid:3)
Remark
Notice that when α, β ∈ R , the forms of the Fourier series OGFs, F sc ( α, β ; 1 , c, z )for each of the functions sc ∈ { cos , sin } , may be expressed in a slightly more abbreviated formby F sc ( α, β ; 1 , c, z ) = e ıα sc( β − α ) − sc( β )( e ıα − · (1 − e − ıα cz ) + e ıα ( e ıα sc( β ) − sc( β − α ))( e ıα − · (1 − e ıα cz ) The result in the corollary may then be expanded through this alternate form of the ordinarygenerating function as F sc ( α, β ; q, c, z ) = Z ∞ e − t / √ π " X b = ± b · e ıα (cid:0) e bıα sc( β ) − sc( β − α ) (cid:1) ( e ıα − × (51c) × − e bıα cz cosh (cid:16) t p q ) (cid:17) e bıα c z − e bıα cz cosh (cid:16) t p q ) (cid:17) + 1 dt, for any α, β ∈ R and whenever c, z ∈ C are chosen such that | cz | < Fourier Series Expansions of the Jacobi Theta Functions.
Exact new integralrepresentations for the asymmetric, unilateral Fourier series expansions of the Jacobi thetafunctions defined in the introduction are expressed through the results in Corollary 6.1 andRemark 6.2 as follows [15, § ϑ ( u, q ) = 2 q / F sin (2 u, u ; q, q, − ϑ ( u, q ) = 2 q / F cos (2 u, u ; q, q, ϑ ( u, q ) = 1 + 2 F cos (2 u, q, , ϑ ( u, q ) = 1 − qF cos (cid:0) u, u ; q, q , − (cid:1) We do not provide the explicit integral representations for each of these theta function vari-ants in this section since these expansions follow immediately by substitution of the integralformulas from the last subsection above.7.
Conclusions
Summary.
We have proved the forms of several new integral representations for geometric-series-type square series transformations, exponential-series-type square series transforma-tions, and Fourier-type square series transformations. Specific applications of the new resultsin the article include special case integrals for special infinite products, theta functions, andmany examples of new single and double integral representations for other special constantvalues and identities. The particular forms of the explicit special case applications cited withinthe article are easily extended to enumerate many other series identities and special functionsthat arise in practice.One key aspect we have not discussed within the article is the relations of the geometricand Fourier-type transformation integrals to other Fourier series expansions. We remarked inSection 5 about the similarity of the geometric square series characteristic expansions to thePoisson kernel [15, § X k ≥ sin( kx ) r k = r sin( x ) r − r cos( x ) + 1 X k ≥ cos( kx ) r k = r cos( x ) − r − r cos( x ) + 1 . One possible topic of future work on these transformations is to consider the relations ofthe square series generating function transformations to known Fourier series expansions,identities, and transforms – even when the integral depends on a formal power series parameter z which is independent of the squared series parameter q . QUARE SERIES GENERATING FUNCTION TRANSFORMATIONS 27
Comparisons to the Weierstrass Elliptic Functions.
Perhaps the most strikingsimilarities to the integral representations offered by the results in Section 4 and Section6 of this article are found in the
NIST Handbook of Mathematical Functions citing a fewrelevant properties of the
Weierstrass elliptic functions , ℘ ( z ) and ζ ( z ) [15, § § f ( s, τ ) and f ( s, τ ), defined by (52) below. The particular integralrepresentations for the functions, ℘ ( z ) and ζ ( z ), are restated in (53) and are then comparedto the forms of the integrals established in Proposition 4.1 and Remark 4.2 of this article.Let the parameter τ := ω /ω for fixed ω , ω ∈ C \ { } , and define the functions, f ( s, τ )and f ( s, τ ), over non–negative s ∈ R as in the following equations [15, eq. (23.11.1); § f ( s, τ ) := cosh (cid:0) τ s (cid:1) e − s − e − s cosh ( τ s ) + 1 (52) f ( s, τ ) := cos (cid:0) s (cid:1) e ıτs − e ıτs cos ( s ) + 1 . Provided that both − < Re( z + τ ) < | Im( z ) | < Im( τ ), these elliptic functions havethe integral representations given in terms of the auxiliary functions from (52) stated in therespective forms of the next equations [15, § ℘ ( z ) = 1 z + 8 × Z ∞ s h e − s sinh (cid:16) zs (cid:17) f ( s, τ ) + e ıτs sin (cid:16) zs (cid:17) f ( s, τ ) i ds (53) ζ ( z ) = 1 z + Z ∞ (cid:2) e − s ( zs − sinh( zs )) f ( s, τ ) − e ıτs ( zs − sinh( zs )) f ( s, τ ) (cid:3) ds Notice that the first function, f ( s, τ ), defined in (52) is similar in form to many of theidentities and special function examples cited as applications in Section 4 and Section 6. Inthis case, the parameter q corresponds to an exponential function of τ . The second definitionof the function, f ( s, τ ), given in terms of the cosine function is also similar in form to theintegrands that result from the computations of the explicit special values of Ramanujan’sfunctions, ϕ ( q ) and ψ ( q ), derived as applications in Corollary 4.7 and Corollary 4.8 in Section4.2 [15, cf . § Some Limitations of the Geometric-Series-Based Transformations.
Supposethat q, z ( q ) ∈ C are selected such that | q | < z ( q ) = 0. Then the bilateral squareseries, B sq ( q, z ) and B a,b ( r , s ; q, z ) defined below, converge and have resulting unilateral seriesexpansions [13, cf . § B sq ( q, z ) := ∞ X n = −∞ q n z n (54)= 1 + ∞ X n =1 q n (cid:0) z n + z − n (cid:1) B a,b ( r, s ; q, z ) := ∞ X n = −∞ ( − n ( an + b ) q n ( rn + s ) z n (55)= ∞ X n =0 ( − n ( an + b ) q n ( rn + s ) z n − ∞ X n =1 ( − n ( an − b ) q n ( rn − s ) z − n . Notice that even though both of the series defined in (54) and (55) converge for all z ( q ) ∈ C whenever | q | <
1, the square series integrals derived in Section 4 cannot be applied directlyto expand the forms of these bilateral series. This results from the constructions of thegeometric-series-based transformations which are derived from the OGF, F ( z ) ≡ (1 − z ) − ,which only converges when | z | <
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