A triangular field of rational numbers related to Stirling numbers and Hyperbolic functions
aa r X i v : . [ m a t h . N T ] F e b A triangular field of rational numbers related to Stirlingnumbers and Hyperbolic functions
Andreas B. G. [email protected] 23, 2021
Abstract
A triangular field of rational numbers is characterized, with relations to Stirling num-bers 2 nd kind, Hyperbolic functions, and centered Binomial distribution. A Generatingfunction is given. Contents
Preliminary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2The triangular array A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Relation to Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . 4Relation to centered Binomial distribution . . . . . . . . . . . . . . . . . . . 4The condensed form B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5The triangular field ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Expansion in terms of falling factorials . . . . . . . . . . . . . . . . . . . . . 6Special solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Relation to Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Relation to Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . 7The adjoint form ˜ ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7The Generating functions F n,λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Frequency representation of integer partitions . . . . . . . . . . . . . . . . . 9Solution of (39a) / (39b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Proof that (43) solves (39a) / (39b) . . . . . . . . . . . . . . . . . . . . . . . . . . 10Summation over λ ∈ Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Summation over n ∈ Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Solution of (50a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 reliminary definitions Let f ( x ) be a smooth function over some domain D . Define the operator [ r ] as follows: f [ r ] ( x ) := (cid:2) x · ddx (cid:3) r f ( x ) r ∈ N (1)Please notice that this definition uses square brackets, differing from the familiar notation f ( r ) ( x ), which involves round brackets and indicates the r th derivative of f ( x ).Define the elementary functions g, h : C \ { } −→ C g ( x ) := ( x + x − ) h ( x ) := ( x − x − ) (2)From (1) it follows immediately g [1] ( x ) = h ( x ) h [1] ( x ) = g ( x ) (3)Zeros of g and h are g ( i ) = 0 h (1) = 0 g ( − i ) = 0 h ( −
1) = 0 (4)Here, i denotes the imaginary unit.Based on (2), define the functions G s,j : C \ { , , − , i, − i } −→ C G s,j := g s − j · h j s ≥ j ∈ Z (5)The domain of G s,j , in general, excludes { , , − , i, − i } , because, if j < h (4)cause poles [Wikf] of G s,j , and so do the zeros of g in case j > s . For each given s ≥
0, theinfinite set of functions G s = (cid:8) . . . , G s, − , G s, − , G s, , G s, , G s, , . . . (cid:9) s ≥ linearly independent . The reason is, that for each j ∈ Z , G s,j has unique orders ofzeros/poles. For example, by definition (5), x = 1 is a zero of order j if j >
0, and a poleof order j if j <
0. Therefore, no finite linear combination of other elements in G s canreproduce the same order j of the zero/pole at x = 1. The triangular array A Application of the [ r ] operator with r = 1 to G s,j as defined in (5) yields G [1] s,j = j · G s,j − + ( s − j ) · G s,j +1 s ≥ j ∈ Z (7)2his suggests the expansion G [ r ] s, = X j ∈ Z A s,r,j · G s,j s, r ≥ r = 1] to (8) leads to a recurrence relation for the coefficients:Here are the steps in detail: G [ r +1] s, = X j ∈ Z A s,r,j · G [1] s,j = X j ∈ Z A s,r,j · j · G s,j − + X j ∈ Z A s,r,j · ( s − j ) · G s,j +1 = X j ∈ Z A s,r,j +1 · ( j + 1) · G s,j + X j ∈ Z A s,r,j − · ( s − ( j − · G s,j (9)(10a) follows, if one compares this with (8) after having replaced r → r + 1, and whenobserving linear independence of (6). A s,r +1 ,j = A s,r,j − · ( s − ( j − A s,r,j +1 · ( j + 1) j ∈ Z s, r ≥ A s, , = 1 (10b) A s,r,j = 0 if not ≤ j ≤ r (10c)Initial or seed values (10b) / (10c) make A s,r,j a Triangular array [Wike]. We can thereforewrite (8) as a finite sum: G [ r ] s, = r X j =0 A s,r,j · G s,j s, r ≥ j + ( s − j ) = s . By induction, this is readily extended tothe identity r X j =0 A s,r,j = s r s, r ≥ A s,r,j , as computed from (10a). Here, the( s ) k denote Falling factorials [Wikb]. The elements of A are integer functions of the sizeparameter s ≥
0. In particular, elements in the j = 0 column can be expressed in terms ofcosh (15). At the same time, for any given r ≥
0, the r th item in the j = 0 column equalsthe r th moment of a centered Binomial distribution of size s (20). Coefficients of ( s ) k in the j = 0 column match the triangle of numbers given in [Int], when selecting only even valuesof r . 3 r · · · · s · · · s s ) · · s ) + s s ) · s ) + s s ) + 4 ( s ) s ) s ) + 15 ( s ) + s s ) + 10 ( s )
06 15 ( s ) + 15 ( s ) + s s ) + 75 ( s ) + 16 ( s ) . . . s ) + 210 ( s ) + 63 ( s ) + s . . . s ) + 210 ( s ) + 63 ( s ) + s . . . Table 1: Apex of A s,r,j Relation to Hyperbolic functions
Exactly the same table is generated if one replaces g / h from (2) by cosh / sinh [Wikc], and,at the same time, replaces the [ r ] operator from (1) by the r th derivative ( r ): g −→ cosh h −→ sinh[ r ] −→ ( r ) (13)With these replacements (11) becomes (cid:16) cosh( t ) s (cid:17) ( r ) = r X j =0 A s,r,j · cosh( t ) s − j sinh( t ) j s, r ≥ t = 0: A s,r, = " (cid:16) cosh( t ) s (cid:17) ( r ) t =0 s, r ≥ Relation to centered Binomial distribution G s, as defined in (5), if written as the Laurent series G s, ( x ) = X j ∈ Z b s,j · x j s ≥ Binomial distribution [Wika]. For if we define the set of events V s := (cid:8) k − s | ≤ k ≤ s (cid:9) s ≥ v ∈ V s is associated with b s,v from (16), which can be written as b s,v = 12 s (cid:18) s v + s (cid:19) v ∈ V s , s ≥ Moment generating function M s ( t ) is defined by the Expectedvalue of e t · v M s ( t ) = s X k =0 s (cid:18) sk (cid:19) e t · (2 k − s ) = cosh( t ) s s ≥ µ s,r denotes the r th moment of (18) and M ( r ) s denotes the r th derivative of M s ( t )with respect to the formal parameter t , we have µ s,r = M ( r ) s (0) = " (cid:16) cosh( t ) s (cid:17) ( r ) t =0 s, r ≥ G [ r ] s, , we have the identity µ s,r = G [ r ] s, (1) = r X j =0 A s,r,j · G s,j (1) = A s,r, s, r ≥ The condensed form B In order to exclude the intermediate zero elements in table 1, we derive from A the “con-densed” form B s,r,n : = A s,r,r − n n ∈ Z s, r ≥ B s,r +1 ,n = B s,r, n · ( s − ( r − n ))+ B s,r, n − · ( r − n − s, r ≥ B s, , = 1 (23b) B s,r,n = 0 if not ≤ n ≤ (cid:4) r (cid:5) (23c) (cid:4) r (cid:5)X n =0 B s,r,n = s r s, r ≥ he triangular field ϕ Expansion in terms of falling factorials
Consider the transformation B s,r,n = X ≤ k ≤ j ≤ n ( s ) r − n − j · ( r ) n + k · ϕ n,j,k s, r ≥ n ∈ Z (25)If one inserts (25) into (23a), while applying appropriate identities to match factorial powers[Wikb] of s and r on both sides, one gets the recurrence relation(2 n + k ) · ϕ n,j, k = ( n − ( j − · ϕ n,j − , k − + ( k + 1) · ϕ n − ,j, k +1 + ϕ n − ,j, k ≤ k ≤ j ≤ n (26a) ϕ , , = 1 (26b) ϕ n,j,k = 0 if not ≤ k ≤ j ≤ n (26c) Seed values (26b) / (26c), which are counterparts of conditions (23b) / (23c), make ϕ atriangular array in 3 dimensions. For given n ≥ ϕ n,j,k can be regarded as a lower triangularmatrix of dimension n + 1. Indices j and k denote row and column indices respectively. Thematrix associated with n = 0 reduces to the scalar 1. Special solutions
Some special solutions of the triple (26b) / (26c) / (26a) are easily verified: ϕ n,k,k = 12 n ( n − k )! k ! 3 k n ≥ k ≥ ϕ n,n − , = 1(2 n )! ϕ n,n, = 1(2 n + 1)! n ≥ ϕ n,n, = δ n, n ∈ Z (27c)From (27a) we have in particular ϕ n, , = 12 n n ! ϕ n,n,n = 16 n n ! n ≥ elation to Stirling numbers Inserting (25) into (24) gives, after rearrangement r X i =0 ( s ) r − i X j ≥ ( r ) j " X n ≥ ϕ n,i − n,j − n = s r s, r ≥ nd kind [Wikd] r X i =0 ( s ) r − i (cid:26) rr − i (cid:27) = s r s, r ≥ X j ≥ ( r ) j " X n ≥ ϕ n,i − n,j − n = (cid:26) rr − i (cid:27) ≤ i ≤ r (31) Relation to Hyperbolic functions
If one substitutes r = 2 n in (25) one gets B s, n,n = X ≤ k ≤ j ≤ n ( s ) n − j · (2 n ) n + k · ϕ n,j,k = (2 n ) n · X ≤ j ≤ n ( s ) n − j · ϕ n,j, s, n ≥ X ≤ j ≤ n ( s ) n − j · ϕ n,j, = 1(2 n )! " (cid:16) cosh( t ) s (cid:17) (2 n ) t =0 s, n ≥ The adjoint form ˜ ϕ It is useful to introduce the adjoint form˜ ϕ n,λ, k := ϕ n,n − λ + k, k λ, k ∈ Z n ≥ ϕ gives(2 n + k ) · ˜ ϕ n,λ, k = ( λ − ( k − · ˜ ϕ n,λ, k − + ( k + 1) · ˜ ϕ n − ,λ, k +1 + ˜ ϕ n − ,λ − , k ≤ k ≤ λ ≤ n (35a)7 ϕ , , = 1 (35b)˜ ϕ n,λ, k = 0 if not ≤ k ≤ λ ≤ n (35c)The ˜modifier in (34) can be viewed as a mapping which turns the lower triangular matrix ϕ n into the lower triangular matrix ˜ ϕ n . In that sense, it is an involution :˜˜ ϕ n = ϕ n n ≥ − · · · λ − λ · · · n n + 1 − n n ! ... ... . . . n − λ ˜ ϕ n,λ, n − λ + 1 ˜ ϕ n,λ, n − λ + 2 ˜ ϕ n,λ, ... . . . n − n )! · · · ˜ ϕ n,λ, λ − . . . n δ n, n +1)! · · · ˜ ϕ n,λ, λ · · · n n ! n + 1 Table 2: Triangular field ϕ n,j,k / ˜ ϕ n,λ, k Table 2 illustrates the triangular shape of ϕ n,j,k / ˜ ϕ n,λ, k . Row and column indices, j and k ,run over all integers Z . The blue area, which extends in all directions, marks zero elements.The diagonals of ϕ n,j,k form the rows of the adjoint field ˜ ϕ n,λ, k (34), and vice versa, byinvolution (36), the diagonals of ˜ ϕ n,λ, k form the rows of ϕ n,j,k . Special solutions (28), (27b),and (27c) have been inserted. 8 he Generating functions F n,λ Based upon the adjoint quantities (34), define the functions F n,λ ( z ) : = X k ∈ Z ˜ ϕ n,λ, k · z k n, λ ∈ Z (37)Here, z is a formal parameter. If F ′ n,λ denotes the derivative of (37) with respect to z , wehave F ′ n,λ ( z ) = X k ∈ Z k · ϕ n,n − λ + k, k · z k − n, λ ∈ Z (38)If one multiplies (35a) by z k , performs summation over all k ∈ Z , matches powers of z onboth sides, and replaces (37) and (38), one gets the recursive differential equation (2 n − λ z ) F n,λ ( z ) + z ( z + 1) F ′ n,λ ( z ) = F n − ,λ − ( z ) + F ′ n − ,λ ( z ) n ≥ λ ≥ F , = 1 (39b) F n,λ = 0 if not n ≥ λ ≥ ϕ . A distinctive special value is F n,λ (0) = ˜ ϕ n,λ, = ϕ n,n − λ, n, λ ∈ Z (40) Frequency representation of integer partitions
The solution (43) of (39a) involves integer partitions in frequency representation [And98].Let P n,λ denote the set of integer partitions of n ≥ λ ≥ π ∈ P n,λ be a partition of n with λ parts. Then π ( k ) denotes the frequency ofpart k ≥ π . This implies the identities n = n X k =1 π ( k ) · k (41a) λ = n X k =1 π ( k ) (41b)In the degenerate case λ = 0 we have P n, = (cid:8) empty partition (cid:9) if n = 0 ∅ if n > n = 0 has exactly one partition, namely the empty partition , without parts.And on the other hand, clearly, there is no partition of n > olution of (39a) / (39b)The recursive differential equation (39a), including boundary condition (39b), is solved by F n,λ ( z ) = X π ∈P n,λ n Y k =1 " π ( k )! · f k ( z ) π ( k ) n ≥ λ ≥ f k being defined as f k ( z ) : = 1(2 k )! (cid:18) z k + 1 (cid:19) k ≥ z = 0 in (43) while using (40) gives ϕ n,n − λ, = X π ∈P n,λ " n Y k =1 π ( k )! · (cid:0) (2 k )! (cid:1) π ( k ) − n ≥ λ ≥ λ = 0 this reduces to (27c), and for λ = n to the first part of (28). Proof that (43) solves (39a) / (39b) The proof is divided into three sections. First, get rid of the recursiveness in (39a) by formingpower series over the whole range of the indices λ and n . This leads to the linear partialdifferential equation (50a) including boundary condition (50b). Second, solve (50a) / (50b).And third, focus on specific powers of x and y in the representation of solution (51) as aninfinite product of infinite sums (52). Summation over λ ∈ Z Starting from (37), define the functions F n ( y, z ) : = X λ ∈ Z F n,λ ( z ) · y λ n ∈ Z (46)Here, y and z are formal parameters. If one multiplies (39a) by y λ , performs summationover all λ ∈ Z , matches powers of y and z on both sides, and replaces (46), or their partialderivatives, one gets the recursive partial differential equation2 n F n ( y, z ) + z ( z + 1) ∂ z F n ( y, z ) − z y ∂ y F n ( y, z ) = y F n − ( y, z ) + ∂ z F n − ( y, z ) n ≥ F = 1 (47b)10 n = 0 n < F n (0 ,
0) = F n, (0) = ˜ ϕ n, , = ϕ n,n, = δ n, n ∈ Z (48)Here, we have made use of (40) and (27c). Summation over n ∈ Z Next, in a quite analogous manner, define the function F ( x, y, z ) : = X n ∈ Z F n ( y, z ) · x n (49)where F n has been declared in (46). Again, x , y , and z are formal parameters. If onemultiplies (47a) by x n , performs summation over all n ∈ Z , matches powers of x , y , and z on both sides, and replaces (49), or their partial derivatives, one gets the linear partialdifferential equation2 x ∂ x F ( x, y, z ) + z ( z + 1) ∂ z F ( x, y, z ) − z y ∂ y F ( x, y, z ) = y x F ( x, y, z ) + x ∂ z F ( x, y, z ) (50a) F (0 , ,
0) = F (0 ,
0) = 1 (50b)with boundary condition (50b) derived from (48).
Solution of (50a)The linear partial differential equation (50a), including boundary condition (50b), is solvedby F ( x, y, z ) = exp " y · X k ≥ f k ( z ) · x k (51)where f k is from (44). This is verified by insertion. Result (43) becomes clear if one writesdown (51) as an infinite product of infinite sums, as illustrated in (52). Here, each rowrepresents an infinite sum.1 + y f ( z ) x · + y f ( z ) x · + y f ( z ) x · + · · · y f ( z ) x · + y f ( z ) x · + y f ( z ) x · + · · · y f ( z ) x · + y f ( z ) x · + y f ( z ) x · + · · · ... ... ... ... . . . (52)1143) follows for any given pair n, λ ≥
0, if, upon multiplication of all rows, one combines allcoefficients of x n , and at the same time, all coefficients of y λ . Summary
Starting from the pair of elementary functions (2), the triangular array A (see Table 1) hasbeen generated, whose elements are integer functions of the size parameter s ≥
0. The3-dimensional triangular field of rational numbers ϕ has been introduced, being a result ofthe transformation (25), which involves the condensed form B (22). ϕ satisfies the tripleof conditions (26a) / (26b) / (26c). Distinctive special solutions have been given in (27a),(27b), (27c), and (45). Relations to Stirling numbers (31) and Hyperbolic functions (33)have been described. A Generating function (37), based upon the adjoint form ˜ ϕ (34), hasbeen defined, and it was shown that it can be written as (43). List of Tables A s,r,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Triangular field ϕ n,j,k / ˜ ϕ n,λ, k . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References [AS64] M. Abramowitz and I. Stegun.
Handbook of Mathematical Functions With For-mulas, Graphs, and Mathematical Tables . National Bureau of Standards, 1964,p. 831.[Bul79] M. G. Bulmer.
Principles of Statistics . Dover Publications, 1979, p. 75.[And98] G. E. Andrews.
The Theory of Partitions . Cambridge University Press, 1998.[Int] Online Encyclopedia of Integer Sequences.
Number of end rhyme patterns of apoem of an even number of lines . url : https://oeis.org/A156289 .[Wika] Wikipedia. Binomial distribution . url : https://en.wikipedia.org/wiki/Binomial_distribution .[Wikb] Wikipedia. Falling and rising factorials . url : https://en.wikipedia.org/wiki/Falling_and_rising_factorials .[Wikc] Wikipedia. Hyperbolic functions . url : https://en.wikipedia.org/wiki/Hyperbolic_functions .[Wikd] Wikipedia. Stirling numbers of the second kind . url : https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind .[Wike] Wikipedia. Triangular array . url : https://en.wikipedia.org/wiki/Triangular_array .[Wikf] Wikipedia. Zeros and poles . url : https://en.wikipedia.org/wiki/Zeros_and_poleshttps://en.wikipedia.org/wiki/Zeros_and_poles