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Identities for the harmonic numbers and binomial coefficients. (English) Zbl 1234.11022

Leonhard Euler considered sums of the form \[ \sigma_h (s,t):= \sum_{n=1}^\infty \Biggl(1+ {1\over {2^s}}+ \dots+ {1\over {(n-1)^s}} \Biggr) n^{-t}, \] where \(s\) and \(t\) are positive integers. By a process of extrapolation from \(s+t\leq 13\), Euler discovered that \(\sigma_h (s,t)\) can be evaluated in terms of Riemann \(\zeta\)- functions when \(s+t\) is odd.
Both authors have separately explored elsewhere topics related to Euler sums like harmonic numbers, binomial coefficients and the Riemann zeta function, e.g., the first author in [Computational techniques for the summation of series. New York, NY: Kluwer Academic/Plenum Publishers (2003; Zbl 1059.65002)], in [Appl. Math. Comput. 207, No. 2, 365–372 (2009; Zbl 1175.11007)] and in [Integral Transforms Spec. Funct. 20, No. 11–12, 847–857 (2009; Zbl 1242.11014)] while the second author in [H. Alzer, D. Karayannakis and H. M. Srivastava, J. Math. Anal. Appl. 320, No. 1, 145–162 (2006; Zbl 1093.33010)], in [J. Choi and H. M. Srivastava, Ramanujan J. 10, No. 1, 51–70 (2005; Zbl 1115.11052)], in [J. Choi and H. M. Srivastava, Math. Nachr. 282, No. 12, 1709–1723 (2009; Zbl 1214.33002)], in [A. Petojević and H. M. Srivastava, Appl. Math. Lett. 22, No. 5, 796–801 (2009; Zbl 1228.11025)], in [T. M. Rassias and H. M. Srivastava, Appl. Math. Comput. 131, No. 2–3, 593–605 (2002; Zbl 1070.11038)] and in [H. M. Srivastava and J. Choi, Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001)].
In order to introduce their joint work the authors recall some interesting papers dealing with Euler sums, e.g., by A. Basu [Ramanujan J. 16, No. 1, 7–24 (2008; Zbl 1155.40002)], by A. Basu and T. M. Apostol [Ramanujan J. 4, No. 4, 397–419 (2000; Zbl 0971.40001)] and by P. Flajolet and B. Salvy [Exp. Math. 7, No. 1, 15–35 (1998; Zbl 0920.11061)].
The authors also mention recent works where the Beta function and other techniques are used to investigate Euler related sums, e.g., by X. Chen and W. Chu [Discrete Math. 310, No. 1, 83–91 (2010; Zbl 1250.33007)], by W. Chu and A. M. Fu [Ramanujan J. 18, No. 1, 11–31 (2009; Zbl 1175.05009)] and by C. Krattenthaler and K. Srinivasa Rao [J. Comput. Appl. Math. 160, No. 1–2, 159–173 (2003; Zbl 1038.33003)].
Hence the present paper extends various earlier works and it is perfectly complementary to a contemporary article by the first author [Adv. Appl. Math. 42, No. 1, 123–134 (2009; Zbl 1220.11025)] where he applies the method of integral representation for binomial sums and he extends the range of identities of finite sums with harmonic numbers \(H_n=\sum^{n}_{k=1}1/k\).
Here the integral and closed-form representations of sums with products of harmonic numbers and binomial coefficients are developed in terms of polygamma function whose evaluation at rational arguments is given through formulae supplied by K. S. Kölbig [J. Comput. Appl. Math. 75, No. 1, 43–46 (1996; Zbl 0860.33002)] and by J. Choi and D. Cvijović [J. Phys. A, Math. Theor. 40, No. 50, 15019–15028 (2007; Zbl 1127.33002)] in terms of polylogarithmic and other special functions.
In particular the authors establish several new representations involving Euler related results proved by D. Borwein, J. M. Borwein and R. Girgensohn [Proc. Edinb. Math. Soc., II. Ser. 38, No. 2, 277–294 (1995; Zbl 0819.40003)] and by B. Cloitre [http://mathworld.wolfram.com/\-HarmonicNumber.html].
Beyond the support of the computer algebra package Mathematica, in the proof the authors employ the Catalan constant, the Euler-Mascheroni constant, the Bernoulli polynomials, the Digamma function and the Clausen function.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
33B15 Gamma, beta and polygamma functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E05 Elliptic functions and integrals

Software:

Mathematica
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References:

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