Congruences modulo powers of 5 for the rank parity function
CCONGRUENCES MODULO POWERS OF FOR THE RANK PARITY FUNCTION
DANDAN CHEN, RONG CHEN, AND FRANK GARVAN
Abstract.
It is well known that Ramanujan conjectured congruences modulo powers of 5,7 and and 11 for the partition function. These were subsequently proved by Watson (1938)and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of5 for the crank parity function. The generating function for rank parity function is f ( q ),which is the first example of a mock theta function that Ramanujan mentioned in his lastletter to Hardy. We prove congruences modulo powers of 5 for the rank parity function. Introduction
Let p ( n ) be the number of unrestricted partitions of n . Ramanujan discovered and laterproved that p (5 n + 4) ≡ , (1.1) p (7 n + 5) ≡ , (1.2) p (11 n + 6) ≡ . (1.3)In 1944 Dyson [16] defined the rank of a partition as the largest part minus the number ofparts and conjectured that residue of the rank mod 5 (resp. mod 7) divides the partitionsof 5 n + 4 (resp. 7 n + 5) into 5 (resp. 7) equal classes thus giving combinatorial explanationsof Ramanujan’s partition congruences mod 5 and 7. Dyson’s rank conjectures were provedby Atkin and Swinnerton-Dyer [7]. Dyson also conjectured the existence of another statistiche called the crank which would likewise explain Ramanujan’s partition congruence mod11. The crank was found by Andrews and the third author [4] who defined the crank asthe largest part, if the partition has no ones, and otherwise as the difference between thenumber of parts larger than the number of ones and the number of ones.Let M e ( n ) (resp. M o ( n )) denote the number of partitions of n with even (resp. odd) crank.Choi, Kang and Lovejoy [12] proved congruences modulo powers of 5 for the difference, whichwe call the crank parity function . Date : December 24, 2020.2020
Mathematics Subject Classification.
Key words and phrases. partition congruences, Dyson’s rank, mock theta functions, modular functions.The first and second authors were supported in part by the National Natural Science Foundation of China(Grant No. 11971173) and an ECNU Short-term Overseas Research Scholarship for Graduate Students(Grant no. 201811280047). The third author was supported in part by a grant from the Simon’s Foundation( a r X i v : . [ m a t h . N T ] J a n DANDAN CHEN, RONG CHEN, AND FRANK GARVAN
Theorem 1.1 (Choi, Kang and Lovejoy [12, Theorem 1.1]) . For all α ≥ we have M e ( n ) − M o ( n ) ≡ α +1 ) , if n ≡ α +1 ) . This gave a weak refinement of Ramanujan’s partition congruence modulo powers of 5: p ( n ) ≡ a ) , if 24 n ≡ α ) . This was proved by Watson [30].In this paper we prove an analogue of Theorem 1.1 for the rank parity function. Analogousto M e ( n ) and M o ( n ) we let N e ( n ) (resp. N o ( n )) denote the number of partitions of n witheven (resp. odd) rank. It is well known that the difference is related to Ramanujan’s mocktheta function f ( q ). This is the first example of a mock theta function that Ramanujan gavein his last letter to Hardy. Let f ( q ) = ∞ (cid:88) n =0 a f ( n ) q n = 1 + ∞ (cid:88) n =1 q n (1 + q ) (1 + q ) · · · (1 + q n ) = 1 + q − q + 3 q − q + 3 q − q + 7 q − q + 6 q − q + 12 q − q + · · · . This function has been studied by many authors. Ramanujan conjectured an asymptotic for-mula for the coefficients a f ( n ). Dragonette [15] improved this result by finding a Rademacher-type asymptotic expansion for the coefficients. The error term was subsequently improvedby Andrews [3], Bringmann and Ono [10], and Ahlgren and Dunn [1]. We have a f ( n ) = N e ( n ) − N o ( n ) , for n ≥ Theorem 1.2.
For all α ≥ and all n ≥ we have (1.4) a f (5 α n + δ α ) + a f (5 α − n + δ α − ) ≡ (cid:106) α (cid:107) ) , where δ α satisfies < δ α < α and δ α ≡ α ) . Below in Section 3.1 we show that the generating function for a f (5 n −
1) + a f ( n/ , is a linear combination of two eta-products. See Theorem 3.1. This enables us to use thetheory of modular functions to obtain congruences. Our presentation and method is similarto that Paule and Radu [25], who solved a difficult conjecture of Sellers [28] for congruencesmodulo powers of 5 for Andrews’s two-colored generalized Frobenious partitions [2]. InSection 2 we include the necessary background and algorithms from the theory of modularfunctions for proving identities. In Section 3 we apply the theory of modular functions toprove our main theorem. In Section 4 we conclude the paper by discussing congruencesmodulo powers of 7 for both the rank and crank parity functions. ANK PARITY FUNCTION CONGRUENCES 3
Some Remarks and Notation.
Throughout this paper we use the standard q -notation.For finite products we use( z ; q ) n = ( z ) n = n − (cid:89) j =0 (1 − zq j ) , n > , n = 0 . For infinite products we use( z ; q ) ∞ = ( z ) ∞ = lim n →∞ ( z ; q ) n = ∞ (cid:89) n =1 (1 − zq ( n − ) , ( z , z , . . . , z k ; q ) ∞ = ( z ; q ) ∞ ( z ; q ) ∞ · · · ( z k ; q ) ∞ , [ z ; q ] ∞ = ( z ; q ) ∞ ( z − q ; q ) ∞ = ∞ (cid:89) n =1 (1 − zq ( n − )(1 − z − q n ) , [ z , z , . . . , z k ; q ] ∞ = [ z ; q ] ∞ [ z ; q ] ∞ · · · [ z k ; q ] ∞ , for | q | < z , z , z ,. . . , z k (cid:54) = 0. For θ -products we use J a,b = ( q a , q b − a , q b ; q b ) ∞ , and J b = ( q b ; q b ) ∞ , and as usual(1.5) η ( τ ) = exp( πiτ / ∞ (cid:89) n =1 (1 − exp(2 πinτ ) = q / ∞ (cid:89) n =1 (1 − q n ) , where Im( τ ) > (cid:98) x (cid:99) denote the largest integer less and or equal to x , andlet (cid:100) x (cid:101) denote the smallest integer greater than or equal to x .We need some notation for formal Laurent series. See the remarks at the end of [25,Section 1, p.823]. Let R be a ring and q be an indeterminant. We let R (( q )) denote theformal Laurent series in q with coefficients in R . These are series of the form f = (cid:88) n ∈ Z a n q n , where a n (cid:54) = 0 for at most finitely many n <
0. For f (cid:54) = 0 we define the order of f (withrespect to q ) as the smallest integer N such that a N (cid:54) = 0 and write N = ord q ( f ). We notethat if f is a modular function this coincides with ord( f, ∞ ). See equation (2.1) below forthis other notation. Suppose t and f ∈ R (( q ) and the composition f ◦ t is well-defined as aformal Laurent series. This is the case if ord q ( t ) >
0. The t -order of F = f ◦ t = (cid:88) n ∈ Z a n t n , DANDAN CHEN, RONG CHEN, AND FRANK GARVAN where t = (cid:80) n ∈ Z b n q n , is defined to be the smallest integer N such that a N (cid:54) = 0 and write N = ord t ( F ). For example, if f = q − + 1 + 2 q + · · · , t = q + 3 q + 5 q + · · · , then F = f ◦ t = t − + 1 + 2 t + · · · , = q − − q − + 5 + · · · , so that ord q ( f ) = −
1, ord q ( t ) = 2, ord t ( F ) = − q ( F ) = − Modular Functions
In this section we present the needed theory of modular functions which we use to proveidentities. A general reference is Rankin’s book [26].2.1.
Background theory.
Our presentation is based on [8, pp.326-329]. Let H = { τ :Im( τ ) > } (the complex upper half-plane). For each M = (cid:18) a bc d (cid:19) ∈ M +2 ( Z ), the set ofinteger 2 × M ( τ ) is definedby M τ = M ( τ ) = aτ + bcτ + d . The stroke operator is defined by ( f | M ) ( τ ) = f ( M τ ) , and satisfies f | M S = f | M | S .
The modular group Γ(1) is defined byΓ(1) = (cid:26)(cid:18) a bc d (cid:19) ∈ M +2 ( Z ) : ad − bc = 1 (cid:27) . We consider the following subgroups Γ of the modular group with finite indexΓ ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ Γ(1) : c ≡ N ) (cid:27) , Γ ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ Γ(1) : (cid:18) a bc d (cid:19) ≡ (cid:18) ∗ (cid:19) (mod N ) (cid:27) , Such a group Γ acts on H ∪ Q ∪ ∞ by the transformation V ( τ ), for V ∈ Γ which induces anequivalence relation. We call a set F ⊆ H ∪ Q ∪ {∞} a fundamental set for Γ if it containsone element of each equivalence class. The finite set F ∩ ( Q ∪ {∞} ) is called the completeset of inequivalent cusps .A function f : H −→ C is a modular function on Γ if the following conditions hold:(i) f is holomorphic on H .(ii) f | V = f for all V ∈ Γ. ANK PARITY FUNCTION CONGRUENCES 5 (iii) For every A ∈ Γ(1) the function f | A − has an expansion( f A − )( τ ) = ∞ (cid:88) m = m b ( m ) exp(2 πiτ m/κ )on some half-plane { τ : Im τ > h ≥ } , where T = (cid:18) (cid:19) and κ = min (cid:8) k > ± A − T k A ∈ Γ (cid:9) . The positive integer κ = κ (Γ; ζ ) is called the fan width of Γ at the cusp ζ = A − ∞ . If b ( m ) (cid:54) = 0, then we write Ord( f, ζ, Γ) = m which is called the order of f at ζ with respect to Γ. We also write(2.1) ord( f ; ζ ) = m κ = m κ (Γ , ζ ) , which is called the invariant order of f at ζ . For each z ∈ H , ord( f ; z ) denotes the orderof f at z as an analytic function of z , and the order of f with respect to Γ is defined byOrd( f, z, Γ) = 1 (cid:96) ord( f ; z )where (cid:96) is the order of z as a fixed point of Γ. We note (cid:96) = 1, 2 or 3. Our main tool forproving modular function identities is the valence formula [26, Theorem 4.1.4, p.98]. If f (cid:54) = 0is a modular function on Γ and F is any fundamental set for Γ then(2.2) (cid:88) z ∈ F Ord( f, z,
Γ) = 0 . Eta-product identities.
We will consider eta-products of the form(2.3) f ( τ ) = (cid:89) d | N η ( dτ ) m d , where N is a positive integer, each d > m d ∈ Z . Modularity.
Newman [24] has found necessary and sufficient conditions under which an eta-product is a modular function on Γ ( N ). Theorem 2.1 ([24, Theorem 4.7]) . The function f ( τ ) (given in (2.3)) is a modular functionon Γ ( N ) if and only if (1) (cid:88) d | N m d = 0 , (2) (cid:88) d | N dm d ≡ , DANDAN CHEN, RONG CHEN, AND FRANK GARVAN (3) (cid:88) d | N N m d d ≡ , and (4) (cid:89) d | N d | m d | is a square.Orders at cusps. Ligozat [21] has computed the invariant order of an eta-product at thecusps of Γ ( N ). Theorem 2.2 ([21, Theorem 4.8]) . If the eta-product f ( τ ) (given in (2.3)) is a modularfunction on Γ ( N ) , then its order at the cusp ζ = bc (assuming ( b, c ) = 1 ) is (2.4) ord( f ( τ ); ζ ) = (cid:88) d | N ( d, c ) m d d . Chua and Lang [14] have found a set of inequivalent cusps for Γ ( N ). Theorem 2.3 ([14, p.354]) . Let N be a positive integer and for each positive divisor d of N let e d = ( d, N/d ) . Then the set ∆ = ∪ d | N S d is a complete set of inequivalent cusps of Γ ( N ) where S d = { x i /d : ( x i , d ) = 1 , ≤ x i ≤ d − , x i (cid:54)≡ x j (mod e d ) } . Biagioli [9] has found the fan width of the cusps of Γ ( N ). Lemma 2.4 ([9, Lemma 4.2]) . If ( r, s ) = 1 , then the fan width of Γ ( N ) at rs is κ (cid:16) Γ ( N ); rs (cid:17) = N ( N, s ) . An application of the valence formula.
Since eta-products have no zeros or poles in H thefollowing result follows easily from the valence formula (2.2). Theorem 2.5.
Let f ( τ ) , f ( τ ) , . . . , f n ( τ ) be eta-products that are modular functions on Γ ( N ) . Let S N be a set of inequivalent cusps for Γ ( N ) . Define the constant (2.5) B = (cid:88) ζ ∈S N ζ (cid:54) = ∞ min ( { Ord( f j , ζ, Γ ( N )) : 1 ≤ j ≤ n } ) , and consider (2.6) g ( τ ) := α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) , where each α j ∈ C . Then g ( τ ) ≡ if and only if (2.7) Ord( g ( τ ) , ∞ , Γ ( N )) > − B. ANK PARITY FUNCTION CONGRUENCES 7
An algorithm for proving eta-product identities.
STEP 0 . Write the identity in the following form:(2.8) α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) = 0 , where each α i ∈ C and each f i ( τ ) is an eta-product of level N . STEP 1 . Use Theorem 2.1 to check that f j ( τ ) is a modular function on Γ ( N ) for each1 ≤ j ≤ n . STEP 2 . Use Theorem 2.3 to find a set S N of inequivalent cusps for Γ ( N ) and the fanwidth of each cusp. STEP 3 . Use Theorem 2.2 to calculate the order of each eta-product f j ( τ ) at each cusp ofΓ ( N ). STEP 4 . Calculate B = (cid:88) ζ ∈S N ζ (cid:54) = ∞ min( { Ord( f j , ζ, Γ ( N )) : 1 ≤ j ≤ n } ) . STEP 5 . Show that Ord( g ( τ ) , ∞ , Γ ( N )) > − B where g ( τ ) = α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) . Theorem 2.5 then implies that g ( τ ) ≡ MAPLE package called
ETA which implements this algo-rithm. See http://qseries.org/fgarvan/qmaple/ETA/
A modular equation.
Define t := t ( τ ) := η ( τ ) η (10 τ ) η (2 τ ) η (5 τ ) (2.9)= q − q + 3 q − q + 11 q − q + 24 q − q + 57 q − q + 117 q + · · · . We note that t ( τ ) is a Hauptmodul for Γ (10) [22]. As an application of our algorithm weprove the following theorem which will be needed later. Theorem 2.6.
Let σ ( τ ) = − t, (2.10) σ ( τ ) = − t + 2 · t, (2.11) DANDAN CHEN, RONG CHEN, AND FRANK GARVAN σ ( τ ) = − t + 2 · t − · t, (2.12) σ ( τ ) = − t + 2 · t − · t + 12 · t, (2.13) σ ( τ ) = − t + 2 · t − · t + 12 · t − · t, (2.14) where t = t ( τ ) is defined in (2.9). Then (2.15) t ( τ ) + (cid:88) j =0 σ j (5 τ ) t ( τ ) j = 0 . Proof.
From Theorem 2.1 we find that t ( τ ) is a modular function on Γ (10) and t (5 τ ) is amodular function on Γ (50). Hence each term on the left side of (2.15) is a modular functionon Γ (50). For convenience we divide by t ( τ ) and let(2.16) g ( τ ) = 1 + (cid:88) j =0 σ j (5 τ ) t ( τ ) j − . From Theorem 2.3, Lemma 2.4 and Theorem 2.2 we have the following table of fan widthsfor the cusps of Γ (50), with the orders and invariant orders of both t ( τ ) and t (5 τ ). ζ / / / / / /
10 3 /
10 7 /
10 9 /
10 1 /
25 1 / κ (Γ (50) , ζ ) 50 25 2 2 2 2 1 1 1 1 2 1ord( t ( τ ) , ζ ) 0 − / t ( τ ) , ζ, Γ (50) 0 − t (5 τ ) 0 − /
25 0 0 0 0 − − − − t (5 τ ) , ζ, Γ (50) 0 − − − − − t (5 τ ) k t ( τ ) j − with 1 ≤ k ≤ j + 1where 0 ≤ j ≤
4, together with ( k, j ) = (0 , ζ of Γ (50), and thus giving lower bounds for Ord( g ( τ ) , ζ, Γ (
50) at each cusp in thefollowing. ζ / / / / / /
10 3 /
10 7 /
10 9 /
10 1 /
25 1 / g ( τ ) , ζ, Γ (50)) ≥ − − − − B in Theorem 2.5 is B = −
24. It suffices to show thatOrd( g ( τ ) , ∞ , Γ (50)) > . This is easily verified. Thus by Theorem 2.5 we have g ( τ ) ≡ (cid:3) The U p operator. Let p prime and f = ∞ (cid:88) m = m a ( m ) q m ANK PARITY FUNCTION CONGRUENCES 9 be a formal Laurent series. We define U p by(2.17) U p ( f ) := (cid:88) pm ≥ m a ( pm ) q m . If f is a modular function (with q = exp(2 πiτ )),(2.18) U p ( f ) = 1 p p (cid:88) j =0 f (cid:18) /p j/p (cid:19) = 1 p p (cid:88) j =0 f (cid:18) τ + jp (cid:19) . By [5, Lemma 7, p.138] we have
Theorem 2.7.
Let p be prime. If f is a modular function on Γ ( pN ) and p | N , then U p ( f ) is a modular function on Γ ( N ) . Gordon and Hughes [19, Theorem 4, p.336] have found lower bounds for the invariantorders of U p ( f ) at cusps. Let ν p ( n ) denote the p -adic order of an integer n ; i.e. the highestpower of p that divides n . Theorem 2.8 ([19, Theorem 4]) . Suppose f ( τ ) is a modular function on Γ ( pN ) , where p is prime and p | N . Let r = βδ be a cusp of Γ ( N ) , where δ | N and ( β, δ ) = 1 . Then Ord( U p ( f ) , r, Γ ( N )) ≥ p Ord( f, r/p, Γ ( pN )) if ν p ( δ ) ≥ ν p ( N )Ord( f, r/p, Γ ( pN )) if < ν p ( δ ) < ν p ( N )min ≤ k ≤ p − Ord( f, ( r + k ) /p, Γ ( pN )) if ν p ( δ ) = 0 . Theorems 2.5, 2.7 and 2.8 give the following algorithm.
An algorithm for proving U p eta-product identities. STEP 0 . Write the identity in the form(2.19) U p ( α g ( τ ) + α g ( τ ) + · · · + α n g k ( τ )) = α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) , where p is prime, p | N , each g j ( τ ) is an eta-product and a modular function on Γ ( pN ),and each f j ( τ ) is an eta-product and modular function on Γ ( N ). STEP 1 . Use Theorem 2.1 to check that f j ( τ ) is a modular function on Γ ( N ) for each1 ≤ j ≤ n , and g j ( τ ) is a modular function on Γ ( pN ) for each 1 ≤ j ≤ k . STEP 2 . Use Theorem 2.3 to find a set S N of inequivalent cusps for Γ ( N ) and the fanwidth of each cusp. STEP 3a . Compute Ord( f j , ζ, Γ ( N )) for each j at each cusp ζ of Γ ( N ) apart from ∞ . STEP 3b . Use Theorem 2.8 to find lower bounds L ( g j , ζ, N ) forOrd( U p ( g j ) , ζ, Γ ( N )) for each cusp ζ of Γ ( N ), and each 1 ≤ j ≤ k . STEP 4 . Calculate(2.20) B = (cid:88) ζ ∈S N ζ (cid:54) = ∞ min( { Ord( f j , ζ, Γ ( N )) : 1 ≤ j ≤ n } ∪ { L ( g j , ζ, N ) : 1 ≤ j ≤ k } ) . STEP 5 . Show that Ord( h ( τ ) , ∞ , Γ ( N )) > − B where h ( τ ) = U p ( α g ( τ ) + α g ( τ ) + · · · + α n g k ( τ )) − ( α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ )) . Theorem 2.5 then implies that h ( τ ) ≡ U p eta-product identity (2.19).The third author has included an implementation of this algorithm in his ETA
MAPLE package.As an application of our algorithm we sketch the proof of(2.21) U ( g ) = 5 f ( τ ) + 2 f ( τ ) , where g ( τ ) = η (50 τ ) η (5 τ ) η (4 τ ) η (2 τ ) η (100 τ ) η (25 τ ) η (10 τ ) η ( τ ) ,f ( τ ) = η (10 τ ) η ( τ ) η (5 τ ) η (2 τ ) , f ( τ ) = η (10 τ ) η ( τ ) η (20 τ ) η (5 τ ) η (4 τ ) η (2 τ ) . We use Theorem 2.1 to check that f j ( τ ) is a modular function on Γ (20) for each 1 ≤ j ≤
2, and g ( τ ) is a modular function on Γ (100). We use Theorem 2.3 to find a set S ofinequivalent cusps for Γ (20) and the fan width of each cusp. By Theorems 2.3, 2.2 andLemma 2.4 we have the following table of orders. ζ / / / /
10 1 / f ( τ ) , ζ, Γ (20)) 0 − − f ( τ ) , ζ, Γ (20)) 1 0 − − L ( g, ζ, ζ / / / /
10 1 / U ( g ) , ζ, Γ (20)) ≥ − − − / / − / B in (2.20) is B = − /
5. It suffices to show thatOrd( h ( τ ) , ∞ , Γ (20)) ≥ , where h ( τ ) = U ( g ) − (5 f ( τ ) + 2 f ( τ )) . This is easily verified. Thus by Theorem 2.5 we have h ( τ ) ≡ ANK PARITY FUNCTION CONGRUENCES 11
Generalized eta-functions.
The generalized Dedekind eta function is defined to be(2.22) η δ,g ( τ ) = q δ P ( g/δ ) (cid:89) m ≡± g (mod δ ) (1 − q m ) , where P ( t ) = { t } − { t } + is the second periodic Bernoulli polynomial, { t } = t − [ t ] is thefractional part of t , g, δ, m ∈ Z + and 0 < g < δ . The function η δ,g ( τ ) is a modular function onSL ( Z ) with a multiplier system. Let N be a fixed positive integer. A generalized Dedekindeta-product of level N has the form(2.23) f ( τ ) = (cid:89) δ | N 4) = 2 , N ( c,N ) , otherwise . In this theorem, it is understood as usual that the fraction ± corresponds to ∞ .Robins [27] has calculated the invariant order of η δ,g ( τ ) at any cusp. This gives a methodfor calculating the invariant order at any cusp of a generalized eta-product. Theorem 2.11 ([27]) . The order at the cusp ζ = ac (assuming ( a, c ) = 1 ) of the generalizedeta-function η δ,g ( τ ) (defined in (2.22) and assuming < g < δ ) is (2.25) ord( η δ,g ( τ ); ζ ) = ε δ P (cid:16) agε (cid:17) , where ε = ( δ, c ) .An algorithm for proving generalized eta-product identities. We note that the analog of Theorem 2.5 holds for generalized eta-products which are modularfunctions on Γ ( N ), and follows easily from the valence formula (2.2). Theorem 2.12 ([17, Cor.2.5]) . Let f ( τ ) , f ( τ ) , . . . , f n ( τ ) be generalized eta-products thatare modular functions on Γ ( N ) . Let S N be a set of inequivalent cusps for Γ ( N ) . Definethe constant (2.26) B = (cid:88) s ∈S N s (cid:54) = ∞ min ( { Ord( f j , s, Γ ( N )) : 1 ≤ j ≤ n } ∪ { } ) , and consider (2.27) g ( τ ) := α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) + 1 , where each α j ∈ C . Then g ( τ ) ≡ if and only if (2.28) Ord( g ( τ ) , ∞ , Γ ( N )) > − B. The algorithm for proving generalized eta-product identities is completely analogous tothe method for proving eta-product identities described in Section 2.2. To prove an identityin the form α f ( τ ) + α f ( τ ) + · · · + α n f n ( τ ) + 1 = 0 , ANK PARITY FUNCTION CONGRUENCES 13 the algorithm simply involves calculating the constant B in (2.26) and then calculatingenough coefficients to show that the inequality (2.28) holds. A more complete description isgiven in [17].The third author has written a MAPLE package called thetaids which implements thisalgorithm. See http://qseries.org/fgarvan/qmaple/thetaids/ The rank parity function modulo powers of A Generating Function. In this section we prove an identity for the generatingfunction of a f (5 n − 1) + a f ( n/ , where it is understood that a f ( n ) = 0 if n is not a non-negative integer. Our proof dependson some results of Mao [23] who found 5-dissection results for the rank modulo 10. Theorem 3.1. (3.1) ∞ (cid:88) n =0 ( a f (5 n − 1) + a f ( n/ q n = J J J J J − q J J J J J J . Proof. From Watson [29, p.64] we have f ( q ) = 2( q ; q ) ∞ ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / q n . (3.2)We find that ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / n q n = ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / q n , (3.3) ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / n q n = ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / n q n . By [23, Lemma 3.1] we have ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / q n = P ( q , − q ; q ) − P ( q , − q ; q ) q + ( q ; q ) ∞ J ∞ (cid:88) n = −∞ ( − n q n ( n +1) / q n +5 , (3.4) ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / n q n = P ( q , − q ; q ) − q P ( q , − q ; q ) − ( q ; q ) ∞ J ∞ (cid:88) n = −∞ ( − n q n ( n +1) / q n +10 , (3.5) ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / n q n = P ( q , − q ) q − P ( q , − q ) q − ( q ; q ) ∞ J ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / − q n , (3.6)where(3.7) P ( a, b ; q ) = [ a, a ; q ] ∞ ( q ; q ) ∞ [ b/a, ab, b ; q ] ∞ . From (3.2)-(3.6), and noting that P ( q , − q ; q ) = P ( q , − q ; q ) we have f ( q ) = 2( q ; q ) ∞ ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / q n (3.8) = 2( q ; q ) ∞ ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / (1 − q n + q n − q n + q n )1 + q n = 2( q ; q ) ∞ ∞ (cid:88) n = −∞ ( − n q n (3 n +1) / (2 − q n + q n )1 + q n = 2 J (cid:26) q P ( q , − q ; q ) − P ( q , − q ; q ) q + P ( q , − q ) q − P ( q , − q ) q (cid:27) + 4 J ∞ (cid:88) n = −∞ ( − n q n ( n +1) / q n +5 + 4 J ∞ (cid:88) n = −∞ ( − n q n ( n +1) / q n +10 − q f ( q ) . We let(3.9) g ( q ) = 2 J (cid:26) q P ( q , − q ; q ) − P ( q , − q ; q ) q + P ( q , − q ) q − P ( q , − q ) q (cid:27) , and write the 5-dissection of g ( q ) as g ( q ) = g ( q ) + q g ( q ) + q g ( q ) + q g ( q ) + q g ( q ) . (3.10)From (3.2), (3.9) and (3.10), replacing q by q , we have(3.11) ∞ (cid:88) n =0 a f (5 n + 4) q n = − q f ( q ) + g ( q ) , after dividing both sides by q and replacing q by q .The 5-dissection of J is well-known(3.12) J = J (cid:18) B ( q ) − q − q B ( q ) (cid:19) , ANK PARITY FUNCTION CONGRUENCES 15 where B ( q ) = J , J , . See for example [18, Lemma (3.18)].From (3.9), (3.10) and (3.12) J ( g ( q ) + q g ( q ) + q g ( q ) + q g ( q ) + q g ( q )) (cid:18) B ( q ) − q − q B ( q ) (cid:19) (3.13) = 4 q P ( q , − q ; q ) − P ( q , − q ; q ) q + 2 P ( q , − q ) q − P ( q , − q ) q . By expanding the left side of (3.13) and comparing both sides according to the residue ofthe exponent of q modulo 5, we obtain 5 equations: B ( q ) g − q g − q B ( q ) g = 0 , (3.14) B ( q ) g − g − qB ( q ) g = − P ( q , − q ) q J , (3.15) B ( q ) g − g − B ( q ) g = − P ( q , − q ; q ) qJ , (3.16) B ( q ) g − g − B ( q ) g = 4 P ( q, − q ; q ) J , (3.17) B ( q ) g − g − B ( q ) g = 2 P ( q, − q ) q J , (3.18)where g j = g j ( q ) for 0 ≤ j ≤ g ( q ) = 1 J ( B − q − q /B ) (cid:18) q X B − q X B − + 4 X B + 4 X B − (3.19) − q X B + 8 qX B − − q X B − q X B − (cid:19) , where B := B ( q ) and X = P ( q , − q ) = q J , J , J , J , J J , , X = P ( q, − q ) = qJ , J , J , J , J J , ,X = P ( q , − q ; q ) = qJ , J , J , J , J , X = P ( q, − q ; q ) = J , J , J , J , J . The following identity is also well-known(3.20) J J = B − q − q B . See for example [20, Lemma (2.5)].By (3.19) and (3.20), we have g ( q ) = J , J , J , J , J , J qJ , J , J J − qJ , J , J , J , J , J J , J , J J + 4 J , J , J , J , J , J J , J J (3.21) + 4 qJ , J , J , J , J , J J , J J − J , J , J , J , J , J J , J J + 8 qJ , J , J , J , J , J J , J J − J , J , J , J , J , J J , J , J J − J , J , J , J , J , J J , J , J J . We prove(3.22) g ( q ) = − J J J J J J + 1 q J J J J J , using the algorithm described in Section 2.4.We first use (3.21) to rewrite (3.22) as thefollowing modular function identity for generalized eta-products on Γ (20).0 =1 − η , η , η , η , + 4 η , η , η , η , + 4 η , η , η , η , η , − η , η , η , η , η , + 8 η , η , η , η , η , (3.23) − η , η , η , η , − η , η , + 4 η , η , η , η , η , η , η , η , − η , η , η , η , η , η , η , η , . We use Theorem 2.9 to check each that each generalized eta-product is a modular functionon Γ (20). We then use Theorems 2.10 and 2.11 to calculate the order of each generalizedeta-product at each cusp of Γ (20). We calculate the constant in equation (2.26) to find that B = 24. We let g ( τ ) be the right side of (3.23) and easily show that Ord( g ( τ ) , ∞ , Γ (20)) > 24. The required identity follows by Theorem 2.12.From (3.11) and (3.22) we have(3.24) ∞ (cid:88) n =0 a f (5 n − q n + f ( q ) = q g ( q ) = J J J J J − q J J J J J J , which is our result (3.1). (cid:3) A Fundamental Lemma. We need the following fundamental lemma, whose prooffollows easily from Theorem 2.6. Lemma 3.2 (A Fundamental Lemma) . Suppose u = u ( τ ) , and j is any integer. Then U ( u t j ) = − (cid:88) l =0 σ l ( τ ) U ( u t j + l − ) , where t = t ( τ ) is defined in (2.9) and the σ j ( τ ) are given in (2.10)–(2.14). ANK PARITY FUNCTION CONGRUENCES 17 Proof. The result follows easily from (2.15) by multiplying both sides by u t j − and applying U . (cid:3) Lemma 3.3. Let u = u ( τ ) , and l ∈ Z . Suppose for l ≤ k ≤ l + 4 there exist Laurentpolynomials p u,k ( t ) ∈ Z [ t, t − ] such that U ( u t k ) = v p u,k ( t ) , (3.25) and ord t ( p u,k ( t )) ≥ (cid:24) k + s (cid:25) , (3.26) for a fixed integer s , where t = t ( τ ) is defined in (2.9) and where v = v ( τ ) . Then there existsa sequence of Laurent polynomials p u,k ( t ) ∈ Z [ t, t − ] , k ∈ Z , such that (3.25) and (3.26) holdfor all k ∈ Z .Proof. We proceed by induction on k . Let N > l + 4 and assume the result holds for l ≤ k ≤ N − 1. Then by Lemma 3.2 we have U ( u t N ) = − (cid:88) j =0 σ j ( τ ) U ( u t N + j − ) = − v (cid:88) j =0 σ j ( τ ) p u,N + j − ( t ) = v p u,N ( t ) , where p u,N ( t ) = − (cid:88) j =0 σ j ( τ ) p u,N + j − ( t ) ∈ Z [ t, t − ] , and ord t ( p u,N ( t )) ≥ min ≤ j ≤ (ord t ( σ j ) + ord t ( p u,N + j − ( t ))) ≥ min ≤ j ≤ (cid:18) (cid:24) N + j + s − (cid:25)(cid:19) = (cid:24) N + s (cid:25) . The result for all k ≥ l follows. The induction proof for k < l is similar. (cid:3) Lemma 3.4. Let u = u ( τ ) , and l ∈ Z . Suppose for l ≤ k ≤ l + 4 there exist Laurentpolynomials p u,k ( t ) ∈ Z [ t, t − ] such that U ( u t k ) = v p u,k ( t ) , (3.27) where p u,k ( t ) = (cid:88) n c u ( k, n ) t n , ν ( c u ( k, n )) ≥ (cid:22) n − k + r (cid:23) for a fixed integer r , where t = t ( τ ) is defined in (2.9) and where v = v ( τ ) . Then thereexists a sequence of Laurent polynomials p u,k ( t ) ∈ Z [ t, t − ] , k ∈ Z , such that (3.27) holds for k > l + 4 , where p u,k ( t ) = (cid:88) n c u ( k, n ) t n , and ν ( c u ( k, n )) ≥ (cid:22) n − k + r + 24 (cid:23) . Remark . Recall that ν p ( n ) denotes the p -adic order of an integer n ; i.e. the highest powerof p that divides n . Proof. We proceed by induction on k . Let N > l +4 and assume (3.27) holds for l ≤ k ≤ N − p u,k ( t ) = (cid:88) n c u ( k, n ) t n , ν ( c u ( k, n )) = (cid:22) n − k + r (cid:23) . As in the proof of Lemma 3.3 we have U ( u t N ) = v p u,N ( t ) , where p u,N ( t ) = − (cid:88) j =0 σ j ( τ ) p u,N + j − ( t ) ∈ Z [ t, t − ] , From Lemma 3.2 we have σ j ( t ) = j +1 (cid:88) l =1 s ( j, l ) t l ∈ Z [ t ] , where ν ( s ( j, l )) ≥ (cid:22) l + j (cid:23) , for 1 ≤ l ≤ j + 1, 0 ≤ j ≤ 4. Therefore p u,N ( t ) = − (cid:88) j =0 j +1 (cid:88) l =1 s ( j, l ) (cid:88) m c u ( N + j − , m ) t m + l = (cid:88) n c u ( N, n ) t n , where c u ( N, n ) = − (cid:88) j =0 j +1 (cid:88) l =1 s ( j, l ) c u ( N + j − , n − l ) , and ν ( c u ( N, n )) ≥ min ≤ l ≤ j +10 ≤ j ≤ (cid:18) ν ( s ( j, l )) + ν ( c u ( N + j − , n − l ) (cid:19) ≥ min ≤ l ≤ j +10 ≤ j ≤ (cid:18) (cid:22) l + j (cid:23) + (cid:22) n − l ) − ( N + j − 5) + r (cid:23) (cid:19) ≥ min ≤ l ≤ j +10 ≤ j ≤ (cid:22) l + j + 3( n − l ) − ( N + j − 5) + r − (cid:23) ≥ (cid:22) n − N + r + 24 (cid:23) . The result follows. (cid:3) ANK PARITY FUNCTION CONGRUENCES 19 We define the following functions which will be needed in the proof of Theorem 1.2.(3.28) P A := J J J J J J − qJ J J J J J , P B := J J J qJ J J + 4 qJ J J J J J A := J J J J , B := qJ J . For f = f ( τ ) we define(3.29) U A ( f ) := U ( A f ) , U B ( f ) := U ( B f ) . First we need some initial values of U A ( P A t k ) and U B ( P B t k ). Lemma 3.6. Group I U A ( P A ) = P B (5 t − · t + 14 · t − · t + t ) ,U A ( P A t − ) = − P B t,U A ( P A t − ) = − P B t ,U A ( P A t − ) = − P B t ,U A ( P A t − ) = − P B t . Group II U B ( P B ) = P A ,U B ( P B t − ) = P A ( − t + 2) ,U B ( P B t − ) = P A (5 t − · t + 8) ,U B ( P B t − ) = P A (5 t − · t + 34) ,U B ( P B t − ) = P A ( − t + 16 · t − · t − · t + 6 · ) . Proof. We use the algorithm described in Section 2.3 to prove each of these identities. Theidentities take the form U ( g ) = f, where f , g are linear combinations of eta-products. For each identity we check that f is alinear combination of eta-products which are modular functions on Γ (100) and that g is alinear combination of eta-products which are modular functions on Γ (20). For each of theidentities we follow the 5 steps in the algorithm given after Theorem 2.8. We note that thesmallest value of B encountered is B = − 14. These steps have been carried out with thehelp of MAPLE , including all necessary verifications so that the results are proved. (cid:3) Following [25] a map a : Z × Z −→ Z is called a discrete array if for each i the map a ( i, − ) : Z −→ Z , by j (cid:55)→ a ( i, j ) has finite support. Lemma 3.7. There exist discrete arrays a and b such that for k ≥ U A ( P A t k ) = P B (cid:88) n ≥(cid:100) ( k +5) / (cid:101) a ( k, n ) t n , where ν ( a ( k, n )) ≥ (cid:22) n − k (cid:23) , (3.30) U B ( P B t k ) = P A (cid:88) n ≥(cid:100) k/ (cid:101) b ( k, n ) t n , where ν ( b ( k, n )) ≥ (cid:22) n − k + 24 (cid:23) . (3.31) Proof. From Lemma 3.6, Group I we find there is a discrete array a such that U A ( P A t k ) = P B (cid:88) n ≥(cid:100) ( k +5) / (cid:101) a ( k, n ) t n , where ν ( a ( k, n )) ≥ (cid:22) n − k − (cid:23) , for − ≤ k ≤ 0. Lemma 3.3 (with s = 4) and Lemma 3.4 (with r = − 2) imply (3.30) for k ≥ 1. From Lemma 3.6, Group II we find there is a discrete array b such that U B ( P B t k ) = P A (cid:88) n ≥(cid:100) k/ (cid:101) b ( k, n ) t n , where ν ( b ( k, n )) ≥ (cid:22) n − k (cid:23) , for − ≤ k ≤ 0. Lemma 3.3 (with s = 0) and Lemma 3.4 (with r = 0) imply (3.31) for k ≥ (cid:3) Proof of Theorem 1.2. For α ≥ δ α by 0 < δ < α and 24 δ α ≡ α ).Then δ α = 23 × α + 124 , δ α +1 = 19 × α +1 + 124 . We let λ α = λ α +1 = 524 (1 − α ) . For n ≥ c f ( n ) := a f (5 n − 1) + a f ( n/ . We find that for α ≥ ∞ (cid:88) n =0 (cid:0) a f (5 α n + δ α ) + a f (5 α − n + δ α − ) (cid:1) q n +1 = ∞ (cid:88) n =1 c f (5 α − n + λ α − ) q n . We define the sequence of functions ( L α ) ∞ α =0 by L := P A and for α ≥ L α +1 := U A ( L α ) , and L α +2 := U B ( L α +1 ) . Lemma 3.8. For α ≥ , L α = J J J ∞ (cid:88) n =0 c f (5 α n + λ α ) q n , ANK PARITY FUNCTION CONGRUENCES 21 and L α +1 = J J J ∞ (cid:88) n =0 c f (5 α +1 n + λ α +1 ) q n . Proof. L = P A = J J J J J J − q J J J J J J = J J J (cid:18) J J J J J − q J J J J J J (cid:19) = J J J ∞ (cid:88) n =0 ( a f (5 n − 1) + a f ( n/ q n = ∞ (cid:88) n =0 c f ( n + λ ) q n . This is the first equation with α = 0. The general result follows by a routine inductionargument. (cid:3) Our main result for the rank parity function modulo powers of 5 is the following theorem. Theorem 3.9. There exists a discrete array (cid:96) such that for α ≥ L α = P A (cid:88) n ≥ (cid:96) (2 α, n ) t n , where ν ( (cid:96) (2 α, n )) ≥ α + (cid:22) n − (cid:23) , (3.34) L α +1 = P B (cid:88) n ≥ (cid:96) (2 α + 1 , n ) t n , where ν ( (cid:96) (2 α + 1 , n )) ≥ α + 1 + (cid:22) n − (cid:23) . (3.35) Proof. We define the discrete array (cid:96) recursively. Define (cid:96) (1 , 1) = 1 , (cid:96) (1 , 2) = − · , (cid:96) (1 , 3) = 14 · , (cid:96) (1 , 4) = − · , (cid:96) (1 , 5) = 5 , and (cid:96) (1 , k ) = 0 , for k ≥ . For α ≥ (cid:96) (2 α, n ) = (cid:88) k ≥ (cid:96) (2 α − , k ) b ( k, n ) (for n ≥ , and(3.37) (cid:96) (2 α + 1 , n ) = (cid:88) k ≥ (cid:96) (2 α, k ) a ( k, n ) (for n ≥ a and b are the discrete arrays given in Lemma 3.7. From Lemma 3.6, Group I andby Lemma 3.7 and equation (3.36) we have L = U A ( L ) = U A ( P A ) = P B (cid:88) n =1 (cid:96) (1 , n ) t n , where ν ( (cid:96) (1 , n )) ≥ (cid:22) n − (cid:23) .L = U B ( L ) = (cid:88) n =1 (cid:96) (1 , n ) U B ( P B t n ) , = (cid:88) n =1 (cid:96) (1 , n ) P A (cid:88) k ≥ b ( n, k ) t k = P A (cid:88) n ≥ (cid:88) k =1 (cid:96) (1 , k ) b ( k, n ) t n = P A (cid:88) n ≥ (cid:96) (2 , n ) t n , where ν ( (cid:96) (2 , n )) ≥ min ≤ k ≤ (cid:18) ν ( (cid:96) (1 , k )) + ν ( b ( k, n ) (cid:19) ≥ min ≤ k ≤ (cid:18) (cid:22) k − (cid:23) + (cid:22) n − k + 24 (cid:23) (cid:19) = (cid:22) n + 14 (cid:23) , since when k = 1, (cid:4) k − (cid:5) + (cid:4) n − k +24 (cid:5) = (cid:4) n +14 (cid:5) , and for k ≥ (cid:22) k − (cid:23) + (cid:22) n − k + 24 (cid:23) ≥ (cid:22) n + 2 k − (cid:23) ≥ (cid:22) n + 14 (cid:23) . Thus the result holds for L α when α = 1. We proceed by induction. Suppose the resultholds for L α for a given α ≥ 1. Then by Lemma 3.7 and equation (3.37) we have L α +1 = U A ( L α ) = (cid:88) n ≥ (cid:96) (2 α, n ) U A ( P A t n ) , = (cid:88) n ≥ (cid:96) (2 α, n ) P B (cid:88) k ≥ a ( n, k ) t k = P B (cid:88) n ≥ (cid:88) k ≥ (cid:96) (2 α, k ) a ( k, n ) t n = P B (cid:88) n ≥ (cid:96) (2 α + 1 , n ) t n , where ν ( (cid:96) (2 α + 1 , n )) ≥ min ≤ k (cid:18) ν ( (cid:96) (2 α, k )) + ν ( a ( k, n ) (cid:19) ≥ min ≤ k (cid:18) α + (cid:22) k − (cid:23) + (cid:22) n − k (cid:23) (cid:19) ≥ α + 1 + (cid:22) n − (cid:23) , since when k = 1, (cid:4) k − (cid:5) + (cid:4) n − k (cid:5) = 1 + (cid:4) n − (cid:5) , and for k ≥ (cid:22) k − (cid:23) + (cid:22) n − k (cid:23) ≥ (cid:22) n + 2 k − (cid:23) ≥ (cid:22) n − (cid:23) . ANK PARITY FUNCTION CONGRUENCES 23 Thus the result holds for L α +1 . Suppose the result holds for L α +1 for a given α ≥ 1. Thenagain by Lemma 3.7 and equation (3.36) we have L α +2 = U B ( L α +1 ) = (cid:88) n ≥ (cid:96) (2 α + 1 , n ) U B ( P B t n ) , = (cid:88) n ≥ (cid:96) (2 α + 1 , n ) P A (cid:88) k ≥ b ( n, k ) t k = P A (cid:88) n ≥ (cid:88) k ≥ (cid:96) (2 α + 1 , k ) b ( k, n ) t n = P A (cid:88) n ≥ (cid:96) (2 α + 2 , n ) t n , where (cid:96) (2 α + 1 , 1) = 0. Here ν ( (cid:96) (2 α + 2 , n )) ≥ min ≤ k (cid:18) ν ( (cid:96) (2 α + 1 , k )) + ν ( b ( k, n ) (cid:19) ≥ min ≤ k (cid:18) α + 1 + (cid:22) k − (cid:23) + (cid:22) n − k + 24 (cid:23) (cid:19) ≥ min ≤ k (cid:18) α + 1 + (cid:22) n + 2 k − (cid:23) (cid:19) = α + 1 + (cid:22) n − (cid:23) . Thus the result holds for L α +2 , and the result holds in general by induction. (cid:3) Corollary 3.10. For α ≥ and all n ≥ we have c f (5 α n + λ α ) ≡ α ) , (3.38) c f (5 α +1 n + λ α +1 ) ≡ α +1 ) . (3.39) Proof. The congruences follow immediately from Lemma 3.8 and Theorem 3.9. (cid:3) In view of (3.33) and Corollary 3.10 we obtain (1.4). This completes the proof of Theorem1.2. 4. Further results The methods of this paper can be extended to study congruences mod powers of 7 forboth the rank and crank parity functions. We describe some of these results, which we willprove in a subsequent paper [11]. Analogous to (3.1) we find that(4.1) ∞ (cid:88) n =0 ( a f ( n/ − a f (7 n − q n = J J (cid:18) J J J J + 6 q J J J J (cid:19) , which leads to the following Theorem 4.1. For all α ≥ and all n ≥ we have (4.2) a f (7 α n + δ α ) − a f (7 α − n + δ α − ) ≡ (cid:106) 12 ( α − (cid:107) ) , where δ α satisfies < δ α < α and δ α ≡ α ) . It turns out that for the crank parity function congruences mod powers of 7 are moredifficult. Define the crank parity function (4.3) β ( n ) = M e ( n ) − M o ( n ) , for all n ≥ 0. The following is our analog of Choi, Kang and Lovejoy’s Theorem 1.1. Theorem 4.2. For each α ≥ there is an integral constant K α such that (4.4) β (49 n − ≡ K α β ( n ) (mod 7 α ) , if n ≡ α ) . 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MR1581588 School of Mathematical Sciences, East China Normal University, Shanghai, People’sRepublic of China Email address : [email protected] School of Mathematical Sciences, East China Normal University, Shanghai, People’sRepublic of China Email address : [email protected] Department of Mathematics, University of Florida, Gainesville, FL 32611-8105 Email address ::