Generating Function Transformations Related to Polylogarithm ...

ZETA SERIES GENERATING FUNCTION TRANSFORMATIONS RELATED TO POLYLOGARITHM FUNCTIONS AND THE k–ORDER HARMONIC NUMBERS

arXiv:1610.09666v1 [math.CO] 30 Oct 2016

MAXIE D. SCHMIDT [email protected]

Abstract. We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the j th derivatives of a sequence generating function and sets of generalized coefficients satisfying a non–triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the k–order harmonic number sequences. Other applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new “almost” linear recurrence relations between the integer–order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, ζ(k), at integers k ≥ 2.

1. Introduction  The Stirling numbers of the second kind, nk , are defined for n, k ≥ 0 by the triangular recurrence relation [6, §6.1]1       n−1 n−1 n + [n = k = 0]δ . (1) + =k k−1 k k It is also known, or at least straightforward to prove by induction, that for any sequence, hgn i, whose formal ordinary power series (OGF) is denoted by G(z), and natural numbers m ≥ 1, we have a generating function transformation of the form [6, cf. §7.4]2 m   X X m j (j) m n z G (z). (2) n gn z = j n≥0 j=0 We seek to study the properties of a related set of coefficients that provide the corresponding semi ”inverse” generating function transformations of the form X gn X k + 2 n z j G(j) (z), (3) z = k j n ∗ n≥1 j≥1 1

The notation for Iverson’s convention, [n = k]δ = δn,k , is consistent with its usage in [6]. Variants of (2) can be found in [9, §26.8(v)]. A special case of the identity for fn ≡ 1 appears in [6, eq. (7.46); §7.4]. Other related expansions for converting between powers of the differential operator D and the operator ϑ := zD are known as sums involving the Stirling numbers
of the first and second kinds [6, Ex. 6.13; cf. §6.5]. The particular identity that [z n ] (zD)k F (z) = nk fn is stated in [13, §2.2]. 2

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MAXIE D. SCHMIDT [email protected]

for integers k > 0. We readily see that the generalized coefficients, a two–index, non–triangular recurrence relation of the form

k

j ∗

      1 k 1 k−1 k =− + + [k = j = 1]δ j ∗ j j j−1 ∗ j ∗ X  j  (−1)j−m , = m j!mk−2 1≤m≤j

, are defined by

(4)

which provides a number of new properties, identities, and sequence applications involving these numbers. We likewise obtain a number of new, interesting relations between the r–order harP P (r) (r) monic numbers, Hn = nk=1 k −r and Hn (t) = nk=1 tk /k r , by their corresponding ordinary generating functions, Lir (z)/(1 − z) and Lir (tz)/(1 − z), through our study of the generalized transformation coefficients, kj , in (4). Most of the series expansions ∗ for special functions we define through (3) are new, and moreover, provide rational partial series approximations to the infinite series in z. Section 2.1 provides the details to a combinatorial proof of the zeta series transformations defined by (3) and (4).

Examples. The Dirichlet–generating–function–like series defined formally by (3) can be approximated up to any finite order u ≥ 1 by the terms of typically rational truncated Taylor series. We cite a few notable examples of these truncated ordinary generating functions in the following equations where k ∈ N, a, b, r, t ∈ R, and ωa = exp(2πı/a) denotes the primitive ath root of unity: !  u  j X zn X k + 2 (wz) j! (5a) = [w u ] k j+1 (1 − w) n (1 − wz) j ∗ j=1 1≤n≤u !  u  j wz X zn X k + 2 (wz) e (5b) = [w u ] k n! n j ∗ (1 − w) 1≤n≤u j=1 !  u  j X X k + 2 (wz) j! (5c) Hn(k) z n = [w u ] j+2 (1 − w) j (1 − wz) ∗ 1≤n≤u j=1 ! !  n u  j X X tm X k + 2 (wtz) j! (5d) z n = [w u ] k j+1 (1 − wz)(1 − w) m j ∗ (1 − wtz) 1≤n≤u m=1 j=1 ! !  n u  j wrz X X X k + 2 (wrz) e rm z n = [w u ] (5e) k m! j m (1 − wz)(1 − w) ∗ 1≤n≤u m=1 j=1 !  u  (k) j wz X Hn X k + 2 (wz) e (j + 1 + wz) (5f) z n = [w u ] n! j (1 − w) ∗ (j + 1) j=1 1≤n≤u !
a−1 au+b −mb b/a 1/a j X X X s + 2 ω z ( tz j! [b > 0]δ zn a + = [tau+b ] . s s m 1/a )j+1 (1 − t) (an + b) b j ∗ a ((1 − ωa tz m=0 j=1 1≤n≤u (5g)

GENERATING FUNCTION TRANSFORMATIONS

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These expansions follow easily as consequences of a few generating function operations and transformation results. First, for any fixed scalar t, the j th derivative of the geometric series satisfies " # 1 tj j! d(j) = . (1 − tz)j+1 dz (j) (1 − tz) We also have an integral transform that converts the ordinary generating function, F (z), of any sequence into its corresponding exponential generating function, Fb(z), according to [6, p. 566] Z +π
ıϑ 1 b F ze−ıϑ ee dϑ. F (z) = 2π −π

This integral transformation together with induction and integration by parts shows that
j Z +π wze−ıϑ 1 (wz)j ewz eıϑ e dϑ = (j + 1 + wz) , 2π −π (1 − wze−ıϑ )j+2 (j + 1)! which implies the second to last expansion in (5f). Lastly, there is a known “series multisection” generating function transformation over arithmetic progressions of a sequence for integers a > 1, b ≥ 0 of the form [8, §1.2.9] X X ω −mr a F (ωam z) fan+b z an+b = a n≥0 0≤m 0 and fixed j ∈ Z+ [9, §26.8(ii)]: ∞   X zj k k z = (10) j (1 − z)(1 − 2z) · · · (1 − jz) k=0     ∞   X (−1)j+1 z 2 k z k z = [j ≥ 2]δ + [j = 1]δ . j ∗ (1 − z)(2 − z) · · · (j − z) (1 − z) k=0

We also compare the generalized coefficient formula in (9) and its generating function representation in (10) to the Nörlund–Rice integral of a meromorphic function f over a suitable contour given by [5] I X j f (z) j! j−m dz. (−1) f (m) = 2πı z(z − 1)(z − 2) · · · (z − j) m 1≤m≤j

Corollary 2.1 (Harmonic Number Formulas). For j ∈ N, the following formulas provide expansions of the coefficients from (4) and (9) at the fixed cases of k ∈ [2, 6] ⊆ N:   (−1)j−1 2 = j! j  ∗ (−1)j−1 3 = Hj j ∗ j!

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MAXIE D. SCHMIDT [email protected]

   (−1)j−1  2 4 (2) Hj + Hj = 2j! j  ∗  (−1)j−1  3 5 (2) (3) = Hj + 3Hj Hj + 2Hj j ∗ 6j!      2 (−1)j−1 6 (2) (3) (4) 4 2 (2) Hj + 6Hj Hj + 3 Hj + 8Hj Hj + 6Hj . = 24j! j ∗

(11)

Proof. Both the Mathematica software suite and the package Sigma3 are able to obtain these formulas for small special cases [11]. Larger special cases of k ≥ 7 are easiest to compute by first generating a recurrence corresponding to the sum in (9), and then solving the resulting non–linear recurrence relation with the Sigma package routines. A more general heuristic harmonic–number–based recurrence formula that generates these expansions for all k ≥ 2 is suggested along the lines of the analogous formulas for the Stirling numbers of the first kind in the references as [1, §2]     X Hj(m+1) k + 1 − m (−1)j−1 k+2 + [k = 0]δ . (12) = j k j! j ∗ ∗ 0≤m