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POLYLOGARITHMIC CONNECTIONS with EULER SUMS 1. Introduction SARAJEVO JOURNAL OF MATHEMATICS DOI: 10.5644/SJM.12.1.02 Vol.12 (24), No.1, (2016), 17{32 POLYLOGARITHMIC CONNECTIONS WITH EULER SUMS ANTHONY SOFO Abstract. Polylogarithmic functions are intrinsically connected with sums of harmonic numbers. In this paper we explore many relations and explicitly derive closed form representations of integrals of polylogarith- mic functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers. 1. Introduction and preliminaries In this paper we will develop identities, new families of closed form rep- resentations of alternating harmonic numbers and reciprocal binomial coef- ficients, including integral representations, of the form: Z 1 2 x ln (x) 1; 1; 2 3F2 − x dx (1.1) 0 1 − x 2 + k; 2 + k ·; ·; · for k ≥ 1 and where 3F2 z is the classical generalized hypergeo- ·; · metric function, also for integrals of the form Z 1 x ln2 (x) Φ(−x; 2; 1 + r) dx 0 1 − x m where Φ (z; t; a) = P1 z is the Lerch transcendent defined for jzj < 1 m =0 (m+a)t and R (a) > 0 and satisfies the recurrence Φ(z; t; a) = z Φ(z; t; a + 1) + a−t: 2010 Mathematics Subject Classification. Primary: 05A10, 05A19, 33C20; Secondary: 11B65, 11B83, 11M06. Key words and phrases. Polylogarithm function, integral representation, Lerch tran- scendent function, alternating harmonic numbers, combinatorial series identities, summa- tion formulas, partial fraction approach, binomial coefficients. Copyright c 2016 by ANUBIH. 18 ANTHONY SOFO The Lerch transcendent generalizes the Hurwitz zeta function at z = 1; 1 X 1 Φ (1; t; a) = t m =0 (m + a) and the Polylogarithm, or de Jonqui`ere's function, when a = 1; 1 X zm Li (z) := ; t 2 when jzj < 1; (t) > 1 when jzj = 1: t mt C R m =1 Moreover Z 1 ( Lit (px) ζ (1 + t) ; for p = 1 dx = −r : 0 x (2 − 1) ζ (1 + t) ; for p = −1 2 ln2 x Li (−x)+x− x R 1 2 4 We also obtain identities for integrals of the type 0 x3(1−x) dx: Let R and C denote, respectively the sets of real and complex numbers and let N := f1; 2; 3;::: g be the set of positive integers, and N0 := N [ f0g : A λ generalized binomial coefficient µ (λ, µ 2 C) is defined, in terms of the familiar gamma function, by λ Γ(λ + 1) := ; (λ, µ 2 ); µ Γ(µ + 1) Γ (λ − µ + 1) C which, in the special case when µ = n; n 2 N0; yields λ λ λ (λ − 1) ··· (λ − n + 1) (−1)n (−λ) := 1 and := = n (n 2 ); 0 n n! n! N where (λ)ν (λ, ν 2 C) is the Pochhammer symbol. Let n X 1 H = = γ + (n + 1) ; (H := 0) (1.2) n r 0 r=1 be the nth harmonic number. Here, as usual, γ denotes the Euler-Mascheroni constant and (z) is the Psi (or Digamma) function defined by d Γ0(z) Z z (z) := flog Γ(z)g = or log Γ(z) = (t) dt: dz Γ(z) 1 (m) A generalized harmonic number Hn of order m is defined, for positive integers n and m, as follows: n X 1 (m) H(m) := ; (m; n 2 ) and H := 0 (m 2 ); n rm N 0 N r=1 ALTERNATING HARMONIC NUMBER SUMS 19 in terms of integral representations we have the result m Z 1 m n (m+1) (−1) (ln x) (1 − x ) Hn = dx: (1.3) m! 0 1 − x In the case of non-integer values of n such as (for example) a value ρ 2 R, (m+1) the generalized harmonic numbers Hρ may be defined, in terms of the Polygamma functions dn dn+1 (n)(z) := f (z)g = flog Γ(z)g (n 2 ); dzn dzn+1 N0 by (−1)m H(m+1) = ζ (m + 1) + (m) (ρ + 1) (1.4) ρ m! (ρ 2 R n {−1; −2; −3;::: g ; m 2 N) ; where ζ (z) is the Riemann zeta function. Whenever we encounter harmonic (m) numbers of the form Hρ at admissible real values of ρ, they may be evalu- ated by means of this known relation (1.4). In the exceptional case of (1.4) (1) when m = 0, we may define Hρ by (1) Hρ = Hρ = γ + (ρ + 1) (ρ 2 R n {−1; −2; −3;::: g): We assume (as above) that (m) H0 = 0 (m 2 N): r In the case of non integer values of the argument z = q ; we may write the (α+1) generalized harmonic numbers, Hz , in terms of polygamma functions α (α+1) (−1) (α) r r H r = ζ (α + 1) + + 1 ; 6= {−1; −2; −3; :::g ; q α! q q where ζ (z) is the zeta function. When we encounter harmonic numbers at (α) possible rational values of the argument, of the form H r they maybe eval- q uated by an available relation in terms of the polygamma function (α) (z) r or, for rational arguments z = q ; and we also define (1) r (α) H r = γ + + 1 , and H0 = 0: q q (α) r The evaluation of the polygamma function a at rational values of the argument can be explicitly done via a formula as given by K¨olbig[9], or Choi and Cvijovic [3] in terms of the Polylogarithmic or other special functions. 20 ANTHONY SOFO Some specific values are listed in the books[16], [20] and [21]. Let us define the alternating zeta function 1 n+1 − X (−1) ζ (z) = = 1 − 21−z ζ (z) nz n =1 − with ζ (1) = ln 2; 1 n+1 (p) X (−1) Hn S+− = : p;q nq n =1 For first order powers (p = 1) ; of harmonic numbers, Sitaramachandra Rao [12] gave, for 1 + q an odd integer, q −1 − 2 − +− X 2S1;q = (1 + q) ζ (1 + q) − ζ (1 + q) − 2 ζ (2j) ζ (1 + q − 2j) : j=1 For the positive terms, [1] gave 1 (p) p Z 1 p−1 X Hn (−1) ln (t) Liq (t) = ζ (p) ζ (q) + dt nq (p − 1)! 1 − t n =1 0 q 1 q−1 (−1) Z ln (t) Li (t) = ζ (p + q) − p dt, by symmetry. (q − 1)! 0 1 − t Some results for sums of alternating harmonic numbers may be seen in the works of [2], [4], [5], [6], [7], [8], [10], [11], [13], [14], [15], [17], [18], [22], [23], [24] and [25] and references therein. The following lemma will be useful in the development of the main theo- rems. Lemma 1. Let r be a positive integer and p 2 N: Then: r j X (−1) 1 (p) (p) (p) = H + H − H (1.5) jp 2p [ r ] [ r−1 ] 2[ r+1 ]−1 j =1 2 2 2 where [x] is the integer part of x: Proof. The proof is given in the paper [19]. Lemma 2. The following identities hold. For 0 < t ≤ 1 1 n+1 1 + t X (−t) Hn t ln2 = 2 (1.6) t n + 1 n=1 ALTERNATING HARMONIC NUMBER SUMS 21 and when t = 1; 1 n+1 X (−1) Hn 1 ln2 2 = 2 = ζ (2) − 2L =: S : n + 1 i2 2 1 n=1 1 n+1 1 + t X (−t) t ln = ; hence t n n=1 1 n+1 1 1 X (−1) X 1 1 X Hn ln 2 = = = : (1.7) n n2n 2 2n n=1 n=1 n=1 Also 1 n+1 (3) X (−1) Hn 19 3 M (0) := = ζ (4) − ln 2ζ (3) ; (1.8) n 16 4 n=1 1 n+1 (3) X (−1) Hn 3 5 M (1) := = ln 2ζ (3) − ζ (4) n + 1 4 16 n=1 and 1 n+1 (3) X (−1) Hn 3 21 X (0) := = ζ (2) ζ (3) − ζ (5) ; (1.9) n2 4 32 n=1 1 n+1 (3) X (−1) Hn 51 3 X (1) := = ζ (5) − ζ (2) ζ (3) : 2 32 4 n=1 (n + 1) Proof. Firstly (1.6) and (1.7) are standard known results. Next from the definition (1.3), 1 n+1 (3) Z 1 2 1 n+1 n X (−1) Hn 1 ln x X (−1) (1 − x ) M (0) = = dx n 2 1 − x n n=1 0 n=1 1 Z 1 ln2 x = (ln 2 − ln (1 + x)) dx 2 0 1 − x 19 3 = ζ (4) − ln 2ζ (3) : 16 4 Here we have used the integral result Z 1 ln2 x ln (1 + x) 7 19 dx = ln 2ζ (3) − ζ (4) : 0 1 − x 2 8 1 n+1 (3) X (−1) Hn 3 5 M (1) = = ln 2ζ (3) − ζ (4) n + 1 4 16 n=1 22 ANTHONY SOFO follows by a change of counter, also by the integral expression we deduce Z 1 ln2 x Li (−x) 1 51 2 dx = ζ (2) ζ (3) − ζ (5) ; (1.10) 0 x (1 − x) 2 16 and hence, (1.8) follows. In a similar fashion 1 n+1 (3) Z 1 2 1 n+1 n X (−1) Hn 1 ln x X (−1) (1 − x ) X (0) = = dx n2 2 1 − x n2 n=1 0 n=1 1 Z 1 ln2 x ζ (2) = − Li2 (−x) dx 2 0 1 − x 2 1 1 Z 1 ln2 x Li (−x) 3 21 = ζ (2) ζ (3) − 2 dx = ζ (2) ζ (3) − ζ (5) : 2 2 0 1 − x 4 32 1 n+1 (3) X (−1) Hn 51 3 X (1) = = ζ (5) − ζ (2) ζ (3) 2 32 4 n=1 (n + 1) follows by a change of counter, or by the integral expression (1.10).
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