arXiv:1611.00957v2 [math.CO] 9 Nov 2016 ata uso the of sums partial ae f( of cases sa riaygnrtn ucinta nmrtstegene the enumerates that function the generating that ordinary interpretation an important as particular the notice We ec rncnetfunction transcendent Lerch h series the 1.1. codn otececetidentity coefficient the to according hr h ento fte“ordinary” the of definition the where sums the as sequences generalized these define we is, That Date EAE OGNRLZDSILN UBR N PARTIAL AND NUMBERS STIRLING GENERALIZED TO RELATED Definitions. EASRE EEAIGFNTO TRANSFORMATIONS FUNCTION GENERATING SERIES ZETA 2016.06.15-v1. : Abstract. imn eafnto osat ttepstv integers positive seri the particular at constants and var function denominators, sum zeta quadratic Riemann Euler of exotic powers and fu with alternating chi constants, Legendre the in special function, prove for we beta Dirichlet results the new of the pansions num of enumerate Applications Stirling to series. the employed function for coefficients properties transformation known known the to su partial analogous the are or which withi sequences, number define harmonic we expa generalized coefficients series by transformation and new function The zeta Hurwitz function. the of sums partial the 2 where ) H n ( H USO H UWT EAFUNCTION ZETA HURWITZ THE OF SUMS r edfieagnrlzdcaso oie easre transfor series zeta modified of class generalized a define We The ) oie uwt eafunction zeta Hurwitz modified n ( ( r ,β α, ) ≡ generalized [ = ) H n , ( r Φ( ) z (1 n H ,s ,β α, s, z, Φ( ] , Φ( n ζ ( 0) r ,s ,β α, s, z, ( ) [email protected] r ,α β α, s, ,s ,β α, s, z, ( [ odrhroi numbers harmonic –order AI .SCHMIDT D. MAXIE 8 ,β α, 1. r 63.Adtoal,w en h nlgu “ analogous the define we Additionally, §6.3]. , odrhroi numbers harmonic –order 1 ) Introduction = ) − = ) ≡ = ) z ) α 1 − X k − ≤ X 1 ≥ n X s k β ≥ 1 ≤ · − 0 ( Φ( n , αk s ( ( , αn ζ ,s β/α s, z, αk ( | + 1 ,α β α, s, z z + | + n 1 β s < cin B–yesre identities series BBP–type nction, ) β ≥ β s h ril nld e eisex- series new include article the at,atraigzt functions zeta alternating iants, so h uwt eafunction, zeta Hurwitz the of ms soso h ec transcendent Lerch the of nsions ) 1 sdfiigseilcsso the of cases special defining es ) . s h ril aif expansions satisfy article the n ) 3 r ec rncnetfnto acts function transcendent Lerch ) ∨ endb h series the by defined , . . ainso h polylogarithm the of variants eso h rtkn n for and kind first the of bers aie amncnmesi ( in numbers harmonic ralized , for , , z H , = n ( H r | − z ) n ( | ( r 1 ,β α, ) ain generating mations < n , sgvnb h special the by given is , 1 ) ≥ rwhen or r enda the as defined are , 0 . z modified − ≡ 1 (3) (1) (2) by 2 ” ) 2 MAXIED.SCHMIDT [email protected]
1.2. Approach to Generating the Modified Zeta Function Series. The approach to enumerating the harmonic number sequences and series for special constants within this article begins in Section 2 with a brief overview of the properties of the harmonic number expansions in (2) obtained through the definition of a generalized Stirling number triangle extending the results in [13]. We can also employ transformations of the generating functions of many sequences, though primarily of the geometric series, to enumerate and approximate the gener- alized harmonic number sequences in (2) which form the partial sums of the modified Hurwitz zeta function in (1). For example, given the ordinary generating function, A(t), of the sequence an n 0, we can employ a known integral transformation involving A(t) termwise to enumerateh thei ≥ following modified forms of a when r 2 is integer-valued [3]: n ≥ r 1 1 an n ( 1) − r 1 z = − log − (t)A(tz)dt. (n + 1)r (r 1)! n 0 0 X≥ − Z For integers α 2 and 0 β < α, we can similarly transform the ordinary generating function, A(t), through≥ the previous≤ integral transformation and the αth primitive root of unity, ωα = exp(2πı/α), to reach an integral transformation for the modified Lerch transcendent function in (3) in the form of the next equation [15].
r 1 1 aαn+β αn+β ( 1) − r 1 mβ m z = − log − (t) ω− A (ω tz) dt (αn + β + 1)r α (r 1)! α α n 0 Z0 0 m<α X≥ · − ≤X The main focus of this article is on the applications and expansions of generating function transformations introduced in Section 3 that generalize the forms of the coefficients defined in [14]. In many respects, the generalizations we employ here to transform geometric–series– based generating functions into series in the form of (3) are corollaries to the results in the first article. The next examples illustrate the new series we are able to obtain using these generalized forms of the generating function transformations proved in the reference. 1.3. Examples of the New Results. 1.3.1. BBP-Type Formulas and Identities. Many special constants such as those given as ex- amples in the next sections satisfy series expansions given by BBP–type formulas of the form [2] k aj P (s, b, m, A)= b− (mk + j)s k 0 1 j m X≥ ≤X≤ + where A = (a1, a2, . . . , am) is a vector of constants and m, b, s Z . A pair of particular examples of first-order BBP-type formulas which we list in this∈section to demonstrate the new forms of the generalized coefficients listed in Table 3 and Table 4 of the article below provide series representations for a real–valued multiple of π and a special expansion of the natural logarithm function [2, §11] [10, §3]. 4√3π 1 k 2 1 = + 9 −8 (3k + 1) (3k + 2) k 0 X≥ 1 1 8 j + 1 − 1 j + 2 − = 2 3 + 3 9j+1 1 2 2 j 0 3 3 ! X≥ n2 n + 1 1 k+1 n2 n 2 Log − = n2 −n3 3k + 1 − 3k + 2 − 3k + 3 k 0 X≥ 1 1 1 j + 1 − n j + 2 − 2 = n2 3 3 − (n3 + 1)j+1 1 − 2 2 − 3(j + 1) j 0 3 3 ! X≥ GENERATINGFUNCTIONTRANSFORMATIONS 3
1.3.2. New Series for Special Zeta Functions. The next two representative examples of special zeta functions serve to demonstrate the style of the new series representations we are able to obtain from the generalized generating function transformations established by this article. The Dirichlet beta function, β(s), is defined for (s) > 0 by the series ℜ n ( 1) s β(s)= − = 2− Φ( 1, s, 1/2). (2n + 1)s − n 0 X≥ The series in the previous equation is expanded through the generalized coefficients when (α, β) = (2, 1) as in the listings in Table 2. The first few special cases of s over the positive integers are expanded by the following new series [5, cf. §8]: 1 1 j + 1 − β(1) = 2 2j+1 1 j 0 2 X≥ 1 1 j + 1 − β(2) = 2 1+ H(1) (2, 1) 2j+1 1 · j j 0 2 X≥ 1 1 j + 1 − 1 β(3) = 2 1+ H(1) (2, 1) + H(1) (2, 1)2 + H(2) (2, 1) . 2j+1 1 · j 2 j j j 0 2 X≥ For z < 1, the Legendre chi function, χ (z), is defined by the series | | ν 2k+1 z ν 2 χ (z)= = 2− z Φ(z , ν, 1/2). ν (2k + 1)ν × k 0 X≥ The first few positive integer cases of ν 1 are similarly expanded by the forms of the next series given by ≥ 1 j + 1 − z ( z2)j χ (z)= 2 · − 1 1 (1 z2)j+1 j 0 2 X≥ − 1 j + 1 − z ( z2)j χ (z)= 2 1+ H(1) (2, 1) · − 2 1 · j (1 z2)j+1 j 0 2 X≥ − 1 j + 1 − 1 z ( z2)j χ (z)= 2 1+ H(1) (2, 1) + H(1) (2, 1)2 + H(2) (2, 1) · − . 3 1 · j 2 j j (1 z2)j+1 j 0 2 X≥ − Other special function series and motivating examples of these generalized generating function transformations we consider within the examples in Section 4 of the article include special cases of the Hurwitz zeta and Lerch transcendent function series in (1) and (3). In particular, we consider concrete new series expansions of BBP-like series for special constants, alternating and exotic Euler sums with cubic denominators, alternating zeta function sums with quadratic denominators, polygamma functions, and several particular series defining the Riemann zeta function, ζ(2k + 1), over the odd positive integers.
2. Generalized Stirling Numbers of the First Kind 2.1. Definition and Generating Functions. We first define a generalized set of coefficients in the symbolic polynomial expansions of the next products over x as an extension of the results first given in [13]1.
n k 1 = [x − ]x(x + α + β)(x + 2α + β) (x + (n 1)α + β) [n 1] (4) k · · · − ≥ δ (α,β) 1 The notation for Iverson’s convention, [n = k]δ = δn,k, is consistent with its usage in [8]. 4 MAXIED.SCHMIDT [email protected]
k 01 2 3 4 5678 j 0 10 0 0 0 0000 1 01 0 0 0 0000 2 03 1 0 0 0000 3 015 8 1 0 0 000 4 0 105 71 15 1 0 0 0 0 5 0 945 744 206 24 1 0 0 0 6 0 10395 9129 3010 470 35 1 0 0 7 0 135135 129072 48259 9120 925 48 1 0 8 0 2027025 2071215 852957 185059 22995 1645 63 1 Table 1. The Generalized Stirling Numbers of the First Kind, k j (2,1)
The polynomial coefficients of the powers of x in (4) are then defined by the following triangular recurrence for natural numbers n, k 0: ≥ n n 1 n 1 = (αn + β α) − + − + [n = k = 0] . (5) k − k k 1 δ (α,β) (α,β) − (α,β) We also easily arrive at generating functions for the column sequences and for the generalized (α) analogs to the Stirling convolution polynomials, σn(x) and σn (x), defined by x (x n 1)! σ(α,β)(x)= − − . n x n x! − (α,β) The series enumerating these coefficients are expanded as the following closed–form generating functions [8, §6.2] [9]2: n zn (1 αz) β/α = − − Log(1 αz)k (6) k n! k!αk − n 0 (α,β) X≥ αzeαz x xσ(α,β)(x)zn = eβz . n eαz 1 n 0 X≥ − When (α, β) = (1, 0), (α, 1 α) we arrive at the definitions of the respective triangular re- − n currences defining the Stirling numbers of the first kind, k , and the generalized α–factorial functions, n! , from the references [1, 8, 13]. Table 1 lists the first several rows of the triangle (α) in (5) corresponding to the special case of (α, β) = (2, 1) as considered in many special case expansions.
2 For fixed x, α, β, we have a known identity for the following exponential generating functions, which then k implies the first result in (6) by considering powers of x as functions of z where (x)n denotes the Pochhammer symbol [9] [11, §5.2(iii)]: x + β (αz)n = (1 − αz)−(x+β)/α. α n! ≥ n nX0 We can then apply a double integral transformation for the beta function, B(a,b), in the form of [11, §5.12]
−2 a + b (a + b)2 ∞ ∞ (ts)a−1 B a,b 2 dt ds, ( ) = · 2 2 = a+b a a b 0 0 [(1 + t)(1 + s)] ! Z Z for real numbers a,b > 0 such that b ∈ Q \ Z to these generating functions to obtain partially complete integral representations for the transformed series over the modified coefficients in (10) and (12)–(13) of Section 3 cited by the examples in Section 4. GENERATINGFUNCTIONTRANSFORMATIONS 5
2.2. Expansions by the Generalized Harmonic Number Sequences. We find, as in the references [1, 14], that this generalized form of a Stirling–number–like triangle satisfies a number of analogous harmonic number expansions to the Stirling numbers of the first kind (r) n r given in terms of the partial sums, Hn (α, β)= k=1(αk +β)− , of the modified Hurwitz zeta s function, ζ(s, α, β) = n 1 1/(αn + β) . For example, we may expand special case formulas ≥ P 3 for the triangle columns at k = 2, 3, 4 in the following forms : P n + 1 = n! H(1) (α, β) (7) 2 (α,β) × n (α,β) 2 n + 1 n!(α,β) (1) (2) = Hn (α, β) Hn (α, β) 3 (α,β) 2 × − 3 n + 1 n!(α,β) (1) (1) (2) (3) = Hn (α, β) 3Hn (α, β) Hn (α, β) + 2Hn (α, β) . 4 (α,β) 6 × − Similarly, we invert to expand the first few cases of the generalized r–order harmonic numbers through products of the coefficients in (5) as [13, cf. §4.3] 1 n + 1 2 n + 1 n + 1 H(2) (α, β)= 2 n 2 2 − 1 3 n!(α,β) (α,β) (α,β) (α,β)! 1 n + 1 3 n + 1 n + 1 n + 1 H(3) (α, β)= 3 n 3 2 − 1 2 3 n!(α,β) (α,β) (α,β) (α,β) (α,β) n + 1 2 n + 1 + 3 1 4 (α,β) (α,β)! 1 n + 1 4 n + 1 n + 1 2 n + 1 H(4) (α, β)= 4 n 4 2 − 1 2 3 n!(α,β) (α,β) (α,β) (α,β) (α,β) n + 1 2 n + 1 2 n + 1 3 n + 1 + 2 4 1 3 − 1 5 (α,β) (α,β) (α,β) (α,β) n + 1 2 n + 1 n + 1 + 4 . 1 2 4 (α,β) (α,β) (α,β)! In general, we can use the elementary symmetric polynomials implicit to the product-based definition of these generalized Stirling numbers in (4) to show that
n + 1 k (1) k (k) = ( 1) n!(α,β) Yk Hn (α, β) ,..., ( 1) Hn (α, β) (k 1)! k (α,β) − · × − − · − n + 1 k+1 n + 1 n + 1 H(k) (α, β) = ( 1)k(k + 1) [tk+1] Log + tj , n − 1 1 j + 1 (α,β) (α,β) j 1 (α,β) X≥ where Yn(x1,x2,...,xn) denotes the exponential, or complete, Bell polynomial whose expo- j nential generating function is given by Φ(t, 1) exp j 1 xjt /j! [12, §4.1.8]. ≡ ≥ Additionally, the next recurrences are obtained for the generalized harmonic numbers in P terms of these coefficients corresponding to the partial sums in our definition of the “modified” Hurwitz zeta function, ζ(s, α, β)= α s ζ(s, β/α). − × p+1 j (j) p+1 n + 1 ( 1) Hn (α, β) n + 1 p( 1) H(p) (α, β)= − − + − n p + 1 j n! p + 1 n! 1 j
n + 1 ( 1)p+1 H(p) (α, β)= H(p) (α, β)+ − n+1 n p (n + 1)! (α,β) (α,β) n + 2 ( 1)p+1 j + − − p + 1 j (αn + α + β)j(n + 1)! 1 j
functional equation between our modified Hurwitz zeta function involving the generalized Stirling numbers of the first kind in (5). For (α, β) := (1, 0), the previous equation implies new functional equations relating the p-order and (p 1)-order polylogarithm functions, Li (z) Φ(z,s, 1, 0). − s ≡
3. Transformations of Ordinary Power Series by Generalized Stirling Numbers of the Second Kind
3.1. Definitions and Preliminary Examples. Another approach to the relations of the generalized harmonic number sequences to the forms of the triangles defined by (5) proceeds as in the reference [14]. In particular, the next definitions lead to new expansions of many series and BBP–type formulas for special functions and constants from the introduction and in Section 4 which are implied by the new identities we prove for the series expansions of the s modified Lerch transcendent function, Φ(z,s,α,β)= α− Φ(z,s,β/α). j m k 1 j ( 1) − = − k 2 (8) j ∗ j! m (αm + β) (α,β) 0 m j − ≤X≤ j s s + 2 j!z Φ(z,s,α,β)= β− + j+1 j ∗ (1 z) j 0 (α,β) X≥ − The definition of the generalized Stirling numbers of the second kind 4 provided by (8), and recursively by k k + 1 k + 1 = (αj + β) + α , j ∗ j ∗ · j 1 ∗ (α,β) (α,β) − (α,β) also implies the next new truncated partial power series identities for the “modified” Lerch transcendent function over some sequence, g , whose ordinary generating function, G(z), h ni has derivatives of all orders, for α 1 and 0 β < α, and for any fixed u 1, u0 0 (See [14]). ≥ ≤ ≥ ≥
u u+u0 j (j) gn n u k + 2 (wz) G (wz) k z = [w ] (9) (αn + β) j ∗ (1 w) n=1 (α,β) X Xj=1 − 4 See the conclusions in Section 5.1 for a short discussion of why we consider these transformation coefficients to be generalized Stirling numbers of the second kind. GENERATINGFUNCTIONTRANSFORMATIONS 7
k 01234 5 6 j 0 00000 0 0 1 1 1 1 1 9 3 1 3 9 27 81 7 41 223 1169 2 7 1 1 15 225 3375 50625 19 859 34739 1323019 3 1 1 1 35 3675 385875 40516875 187 27161 3451843 406586609 4 1 1 1 315 99225 31255875 9845600625 437 735197 1066933061 1418417467373 5 1 1 1 693 2401245 8320313925 28829887750125 1979 45087479 877474863971 15505503106933439 6 1 1 1 3003 135270135 6093243231075 274470141343773375 4387 103349119 2065307132299 1488524941286431 7 1 1 1 6435 289864575 13056949780875 23526012115180575 76627 31562623583 10971718559046811 683894055421671560539 8 1 1 1 109395 83770862175 64148794273438875 9824580289359984022875 Table 2. k j 1 A Table of the Generalized Coefficients ∗ j!( 1) − j (2,1) × −
j (k) n u k + 2 (wz) j! Hn (α, β) z = [w ] · j+2 j (α,β)∗ (1 w)(1 wz) 1 n u 1 j u+u0 ≤X≤ ≤ X≤ − − n j wz (k) z u k + 2 (wz) e (j +1+ wz) Hn (α, β) = [w ] · n! j (α,β)∗ (j + 1)(1 w) 1 n u 1 j u+u0 ≤X≤ ≤ X≤ − n k j t n u r + 2 (twz) j! r z = [w ] · j+1 (αk + β) j ∗ (1 w)(1 wz)(1 twz) 1 n u k=1 ! 1 j u (α,β) ≤X≤ X ≤X≤ − − − The generalized harmonic number expansions of the coefficients in (8) are considered next in Section 3.2. An even more general proof of the formal power series transformation suggested in the concluding remarks from [14] is given below in Section 3.3. Table 2, Table 3, and Table 4 each provide listings of useful particular special cases of the generalized transformation coefficients, or alternately, generalized Stirling numbers of the second kind within the context of this article, corresponding to (α, β) := (2, 1), (3, 1), (3, 2), respectively.
k 01234 5 6 j 0 00000 0 0 1 1 1 1 1 16 4 1 4 16 64 256 5 41 311 2273 2 17 1 1 14 392 10976 307328 − 59 2671 107369 4060291 3 1 1 1 140 19600 2744000 384160000 212 133849 73174943 37005870001 4 1 1 1 455 828100 1507142000 2742998440000 727 1936973 4393719979 9104269630637 5 1 1 1 1456 10599680 77165670400 561766080512000 7271 384155263 17071846526411 686298711281124727 6 1 1 1 13832 1913242240 264639666636800 36604958689202176000 23789 14322370919 7187615461845233 3237486239486747349191 7 1 1 1 43472 66143517440 100638684655308800 153123771476745445376000 76801 238206415289 611558324636496331 1400156984227714635455249 8 1 1 1 135850 1033492460000 7862397238696000000 59813973233103689600000000 Table 3. k j 1 A Table of the Generalized Coefficients ∗ j!( 1) − j (3,1) × − 8 MAXIED.SCHMIDT [email protected]
k 01234 5 6 j 0 00000 0 0 1 1 1 1 1 25 5 1 5 25 125 625 11 103 899 7567 2 14 2 1 40 1600 64000 2560000 − 139 15757 1609291 155016733 3 4 2 1 440 193600 85184000 37480960000 527 446837 334869917 233183599997 4 4 2 1 1540 4743200 14609056000 44995892480000 1889 28606807 378441183599 4602491925840703 5 4 2 1 5236 274156960 14354858425600 751620387164416000 19619 61764761 4214471373881 10491182677877357 6 4 2 1 52360 548313920 143548584256000 1503240774328832000 66337 4956449573 7985964568560547 466567679887167456041 7 4 2 1 172040 41436866240 249507946377536000 60095485932707810816000 110258 43971566839 4710810017671083829 3642461006944413986125043 8 4 2 1 279565 350141519728 137042239547861648000 429096793431016946178944000 Table 4. k j 1 A Table of the Generalized Coefficients ∗ j!( 1) − j (3,2) × −
3.2. Harmonic Number Expansions of the Generalized Transformation Coefficients. For integers α 1 and 0 β < α, let the first few special cases of the functions, Rk(α, β; j), defined by ≥ ≤ j e j + β j ( 1)m+1 αm R (α, β; j) := α − · [k 2] + [k = 1] + [k = 0] , k j m (αm + β)k ≥ δ δ δ mX=1 be expandede in explicit formulas by the next equations.
R1(α, β; j) = 1 (10) R (α, β; j)= H(1)(α, β) e2 j 1 R (α, β; j)= H(1)(α, β)2 + H(2)(α, β) e3 2 j j 1 (1) (1) (2) (3) Re (α, β; j)= H (α, β)3 + 3H (α, β)H (α, β) + 2H (α, β) 4 6 j j j j 1 (1) (1) (2) (2) Re (α, β; j)= H (α, β)4 + 6H (α, β)2H (α, β) + 3H (α, β)2 5 24 j j j j (1) (3) (4) e + 8Hj (α, β)Hj (α, β) + 6Hj (α, β) . For larger cases of k > 5, we employ the following heuristic to generate the harmonic num- ber expansions of these functions, which for concrete special cases are easily obtained from Mathematica’s Sigma package: m 2 − Rm 2 i(α, β; j) (i+1) R (α, β; j)= − − H (α, β) + [m = 1] . (11) m (m 1) · j δ i=0 X e − We then define ae vague analog to the harmonic number expansions in [14] through the expan- sions of these functions as 1 j k + 2 k + 1 1 j + β/α − ( 1) = + − Rk(α, β; j) (12) j ∗ j ∗ · β β/α β j! · (α,β) (α,β) · j 1 k 1 1 j ( 1) − j + β/α − ( 1) R e(α, β; j) = − − + − m+1 . (13) βk j! β/α j! · βk m m=0 − · X e GENERATINGFUNCTIONTRANSFORMATIONS 9
Notice that the heuristic we used to generate more involved cases of the expansions for the func- tions, Rk(α, β; j), imply recurrences for the k–order generalized harmonic number sequences in the following forms when k 3: ≥ e (k 1) n 1 j Hn − (α, β) n + 1 j + β/α − ( 1) H(k)(α, β)= + − R (α, β; j) n β j + 1 β/α β · k Xj=0 (k 2) e H − (α, β) H(k)(α, β)= n n β2 n 1 n + 1 j + β/α − j Rk(α, β; j) Rk 1(α, β; j) + ( 1) + − . j + 1 β/α − · β β2 j=0 ! X e e A pair of harmonic and Hurwitz zeta function related identities that follow from the generalized coefficient definitions in (8) are obtained by similar methods from the reference [14] for n 1 as follows: ≥ 1 n k + 2 k = j! (αn + β) j j ∗ · 0 j n (α,β) ≤X≤ j k + 2 = ( 1)j ( 1)mnm m j ∗ − − 1 m n m j n (α,β) ≤X≤ X≤ ≤ (k) n + 1 k + 2 Hn (α, β)= j! j + 1 j ∗ · 0 j n (α,β) ≤X≤ j + 1 k + 2 ( 1)j+1 = − ( 1)p (n + 1)p. p j ∗ (j + 1) − · 0 p n+1 0 j n (α,β) ≤X≤ ≤X≤ 3.3. Proofs of the Zeta Series Transformations of Formal Power Series.
3.3.1. Proofs of the Geometric and Exponential Series Transformations. We claim that for any sequence, f(n) , which is not identically zero for n 0, any natural numbers k 1, and h i ≥ ≥ a sequence, gn , whose ordinary generating function, G(z), has derivatives of all orders, we have the (formal)h i series transformation
gn n k + 2 j (j) k z = z G (z), (14) f(n) j ∗ n 1 j 1 f X≥ X≥ where the coefficients implicit to the right–hand–side series are defined by j m k + 2 1 j ( 1) − = − k . (15) j ∗ j! m f(m) f 1 m j ≤X≤ As in [14], we primarily only work with these generalized series when the sequence generating function of gn is some variation of the geometric or exponential series. Therefore, for the content of our article, it suffices to prove the next two cases.
th Proof of the Geometric Series Case. Let gn 1 so that its corresponding j derivative is given by G(j)(z) = j!/(1 z)j+1. We proceed≡ to expand the right–hand–side of (14) as follows: −
k + 2 j!zj j ( 1)m ( z)j j+1 = − k − j+1 j ∗ (1 z) m f(m) (1 z) j 1 f j 1 1 m j X≥ − X≥ ≤X≤ − 10 MAXIED.SCHMIDT [email protected]
j ( z)j ( 1)m = − − . m (1 z)j+1 m!f(m)k m 1 j m X≥ X≥ − For a fixed c = 1, a known binomial sum identity gives that 6 j cm cj = , m (1 c)m+1 j m X≥ − which then implies that when c z/(1 z) we have 7→ − − j ( z)j ( 1)m ( 1)m − − = ( z)m − m (1 z)j+1 m!f(m)k − × m!f(m)k m 1 j m m 1 X≥ X≥ − X≥ Proof of the Exponential Series Case. Let g rn/n! so that its corresponding jth derivative n ≡ is given by G(j)(z) = rjerz for all j. In this case, we proceed to expand the right–hand–side of (14) as
m k + 2 j rz ( 1) j rz (rz) e = − k ( rz) e j ∗ m!(j m)!f(m) − j 1 f j 1 1 m j X≥ X≥ ≤X≤ − ( rz)j ( 1)m = − − erz (j m)! m!f(m)k m 1 j m X≥ X≥ − m rz m ( 1) rz = e− ( rz) − e − m!f(m)k m 1 X≥ rmzm = . m!f(m)k m 1 X≥ 3.3.2. Remarks on Symbolic Transformation Coefficient Identities. We observe that most of the identities formulated in [14, §3] are easily restated as symbolic identities for the coefficients in (15). If f(m) is polynomial in m, by expanding 1/f(m) in partial fractions over linear factors of m, we arrive at sums over the coefficient forms in (10) of Section 3.2 above. If 1/f(z) is a meromorphic function, we may alternately compute the symbolic coefficients in (15) by the next Nörlund–Rice integral over a suitable contour given by [6] j m k j ( 1) − j! := − k j ∗ · m f(m) f 1 m j ≤X≤ j! f(z) k = − dz. 2πı z(z 1)(z 2) (z j) I − − · · · − We provide a brief overview of examples of several finite sum expansions which are also easily proved along the lines given in the reference, and then quickly move on to the particular cases of the series transformations at hand in this article. For example, if we let the r–order f– (r) n r harmonic numbers, Fn (f) := k=1 f(k)− , be defined for a non-zero-valued function, f(n), we may expand the coefficients in (15) as P j 1 i k (j + 1)( 1) − − (k) = − Fi+1(f) j ∗ (j 1 i)!(i + 2)! × f 0 i j 1 i (k r) ( 1) − − F − (f) (i + 2) (j 1 i) = − i+1 + − − (j 1 i)!(i + 2)! f(i + 1)r f(i + 2)r 0 i 4. Examples of New Series Expansions for Modified Zeta Function Series 4.1. A Special Class of Alternating Euler Sums. A first pair of alternating Euler sums related to the Dirichlet beta function constants are expanded in the following forms [5, §8] 5: ( 1)n ( 1)n π − H = − Log(2) (2n + 1) n (2n + 1)2 − 2 n 0 n 0 X≥ X≥ 1 j + 1 − (H Log(2)) = 2 j − 1 2j+1 j 0 2 X≥ ( 1)n ( 1)n 7π π3 − H = 3 − ζ(3) Log(2) (2n + 1)3 n (2n + 1)4 − 16 − 16 n 0 n 0 X≥ X≥ 1 j + 1 − (H Log(2)) = 2 j − 1 2j+1 × j 0 2 X≥ 1 1+ H(1) (2, 1) + H(1) (2, 1)2 + H(2) (2, 1) . × j 2 j j The second and fourth series on the right–hand–side of the previous equations follow from an identity for the jth derivatives of the first–order harmonic number generating function given by Log(1 z) (H Log(1 z)) j! D(j) − = j − − , z − (1 z) (1 z)j+1 − − for all integers j 0. Since this identity is straightforward to prove by induction, we move quickly along to the≥ next example. We also observe the key difference between the generalized zeta series transform coefficients introduced in [14] and those defined by (10) and (13) of this article. In particular, the def- initions of the generalized coefficients given in this article imply functional equations for a number of special series. For example, the second series in the equations immediately above is given in terms of the first sum in the following form: ( 1)n ( 1)n − H = − H (2n + 1)3 n (2n + 1)2 n n 0 n 0 X≥ X≥ 5 These special cases of the generalized harmonic number sequences are expanded in terms of the ordinary (r) r-order harmonic numbers, Hn , considered in the expansions of [14] as (r) (r) −r (r) Hn (2, 1) = H2n+2 − 2 · Hn+1 − 1. 12 MAXIED.SCHMIDT [email protected] 1 j + 1 − (H Log(2)) + 2 j − H(1) (2, 1)2 + H(2) (2, 1) . 1 2j+2 j j j 0 2 X≥ We have similar relations for series defining rational multiples of the polygamma functions, ψs 1(z/2) ψs 1((z + 1)/2), for example, as in the next pair of related sums given by − − − k 1 ( 1) j + z − 1 1 1 − = + H(1) (1, z) (k + z)2 z z2 z j 2j+1 k 0 j 0 X≥ X≥ k k 1 ( 1) 1 ( 1) j + z − 1 1 − = − + H(1) (1, z)2 + H(2) (1, z) , (k + z)3 z (k + z)2 z z j j 2j+2 k 0 k 0 j 0 X≥ X≥ X≥ which then implies functional equations between the polygamma functions such as the follow- ing identity: z z z + 1 ψ(2) ψ(2) = −4 2 − 2 1 z z + 1 j + z − (1) 2 (2) 1 ψ′ ψ′ + 4 H (1, z) + H (1, z) . 2 − 2 × z j j 2j+2 j 0 X≥ h i 4.2. An Exotic Euler Sum with Powers of Cubic Denominators. Flajolet mentions a more “exotic” family of Euler sums in his article defined for positive integers q by [5] n 2 ( 1) Hn A∗ = − . q [(2n 1)(2n)(2n + 1)]q n 1 X≥ − It is not difficult to prove that we have the following two ordinary generating functions for the squares of the first–order harmonic numbers: 1 H2zn = Log(1 z)2 + Li (z) (16) n (1 z) − 2 n 0 − X≥ 1 z = 2 Li + Li (z) . −(1 z) 2 −1 z 2 − − A proof of these two series identities follows from the expansions of the polylogarithm function, (2) Li2(z)/(1 z), generating the second–order harmonic numbers, Hn , in [14, §4]. In particular, we expand− the polylogarithm series as (2) 2 j Li (z) Hj + Hj z 2 = , (1 z) − 2(1 z)2 −1 z j 0 − X≥ − − and then perform the change of variable z z/(1 z) to obtain these results. Since the derivatives of the polylogarithm7→ − functions− in each of the equations in (16) are tedious and messy to expand, we do not give any explicit series for these Euler sums, Aq∗. However, we do note that a sufficiently motivated reader may expand these sums by the generalized coefficients we defined in (8) by taking partial fractions of the denominators of Aq. For example, when q = 1, 2 we have the series n 2 ( 1) Hn 1 1 1 A∗ = − + 1 2 (2n + 1) − n (2n 1) n 1 X≥ − n 2 ( 1) Hn 3 3 1 1 1 A∗ = − + + + . 2 4 (2n + 1) − (2n 1) (2n + 1)2 n2 (2n 1)2 n 1 X≥ − − GENERATINGFUNCTIONTRANSFORMATIONS 13 To expand the more involved cases of the polylogarithm function derivatives, we first note that for integers s 2 and z 1 we have [7, §2.7] ≥ | |≤ 1 Dz [Lis(z)] = Lis 1(z), z − where for r Z+ we have the identity that [8, §7.4] ∈ j r z j! 1 r i+1 Li r(z)= = z , − j (1 z)j+1 (1 z)r+1 × i 0 j r 0 i r ≤X≤ − − ≤X≤ and where the composite derivatives in the second equation of (16) are expanded by Faá de Bruno’s formula [11, §1.4(iii)]. 4.3. Zeta Function Series with Powers of Quadratic Denominators. We next provide examples of generalized zeta function series over denominator powers of quadratic polynomials. The method employed to expand these particular series is to factor and then take partial fractions to apply the generalized transformation cases we study within this article. For example, we observe that 1 ı 1 1 = n2 + 1 2 n + ı − n ı − 1 1 ı ı 1 1 = + + , (n2 + 1)2 −4 (n ı) − (n + ı) (n + ı)2 (n ı)2 − − which immediately leads to the first two of the next series examples. ( 1)n 1 − = (1+ π csch(π)) (n2 + 1) 2 n 0 X≥ 1 1 1 j + ı − j ı − = + − 2j+2 ı ı j 0 ! X≥ − ( 1)n 1 − = (2+ π(1 + π coth(π)) csch(π)) (n2 + 1)2 4 n 0 X≥ 1 1 1 j + ı − j ı − = 2+ ıH(1) (1, ı) − 2+ ıH(1) (1, ı) 2j+3 ı j − ı − j − j 0 ! X≥ − ( 1)n 1 − = 16 + 6π(1 + π coth(π)) csch(π)+ π3(3 + cosh(2π)) csch(π)3 (n2 + 1)3 32 n 0 X≥ 1 1 j + ı − = 8 + 5ıH(1) (1, ı) H(1) (1, ı)2 H(2) (1, ı) 2j+5 ı j − j − j j 0 X≥ 1 1 j ı − + − 8 5ıH(1) (1, ı) H(1) (1, ı)2 H(2) (1, ı) 2j+4 ı − j − − j − − j − j 0 X≥ − Another quadruple of trigonometric function series providing additional examples of expanding sums with quadratic denominators by partial fractions is given as follows [7, §1.2]: π 1 2( 1)n+1 = + − sin(πx) x x2 (n + 1)2 n 0 X≥ − 14 MAXIED.SCHMIDT [email protected] 1 1 1 j + 1 x − 1 j +1+ x − 1 1 = + − x 1 x 1 x − 1+ x 1+ x 2j+1 j 0 " # X≥ − − 1 π2 7π4 31π6 127π8 = + x + x3 + x5 + x7 + O(x9) x 6 360 15120 604800 1+ x csc(x) 2x( 1)n = − x x2 π2n2 n 0 X≥ − x 1 x 1 j + − 1 j − 1 1 = π + − π x x x x 2j+1 j 0 " π π # X≥ − 2 x 7 31 127 = + + x3 + x5 + x7 + O(x9) x 6 360 15120 604800 ∞ ( 1)n π sec(πx)= − 1 n= n + x + 2 −∞ X 1 1 j + x + 3 − 1 = 2 x + 1 − x + 3 2j+1 x + 3 2 j 0 2 2 X≥ 1 j + 1 x − 1 + 2 − 1 x 2j+1 1 x j 0 2 2 X≥ − − π3 5π5 61π7 277 π9 = π + x2 + x4 + x6 + x8 + O(x10) 2 24 720 8064 2+ x(1 + x cot(x)) csc(x) ( 1)n = − 4x4 (x2 π2n2)2 n 0 X≥ − ( 1)n b 1 = − + 4 x3(πn + bx) x2(πn + bx)2 b= 1 n 0 X± X≥ 1 j + bx − 2 b 1 = π + H(1) (π, bx) j x4 x3 j 2j+3 b= 1 j 0 X± X≥ 1 7 31x2 127x4 73x6 = + O(x7). x4 − 720 − 15120 − 403200 − 1710720 Remark 4.1 (Expansions of General Zeta Series with Quadratic Denominators). For general quadratic zeta series of the form ( 1)nzn − , (17) (an2 + bn + c)s n 0 X≥ for integers s 1 and constants a, b, c R, we can apply the same procedure of factoring the denominators≥ into linear factors of z and∈ taking partial fractions to write the first sums as a finite sum over functions of the form Φ(z,s,α,β) for variable parameters, α, β C. We also note that a less standard definition of the Lerch transcendent function, Φ(z, s, a∈), is given by zn Φ∗(z, s, a)= , 2 s/2 n 0 [(z + a) ] X≥ which also suggests an approach to a reduction to non–integer order exponents s from the general quadratic zeta series in the forms of (17). GENERATINGFUNCTIONTRANSFORMATIONS 15 4.4. Special Series Identities for the Riemann Zeta Function. 4.4.1. Series Generating the Riemann Zeta Function at the Even Integers. We first consider the following series identity for the zeta function constant, ζ(3), given by [4, §7.10.2] 2π2 ζ(2k) ζ(3) = Log(2) + 2 9 × 22k(2k + 3) k 0 X≥ Since we know the next ordinary generating function as [11, §25.8] π√z ζ(2k)zk = cot(π√z), − 2 k 0 X≥ (n) n (k) (n k) 2 where Dz [f(z) g(z)] = f (z)g (z), D [cot(z)] = 1 cot (z), and we can · k k − z − − expand the nth derivatives of the cotangent function according to the known formula on the P Wolfram Functions website as D(n)[cot(z)] = cot(z) [n = 0] csc2(z) [n = 1] z · δ − · δ 2k n 1 ( 1)k2n 2k(k j)n 1 nπ n − − − − − sin + 2(k j)z , − × j k (k + 1) sin2k+2(z) 2 − 0 j