Generalized hyperharmonic number sums with reciprocal binomial coefficients
The classical harmonic numbers are the sums \[ H_n^{(p)} = \sum_{j=1}^n \frac{1}{j^p}, \] where \(n\) and \(p\) are positive integers. The generalized hyperharmonic numbers are defined by the induction \[ H_n^{(p,r)} = \sum_{j=1}^n H_j^{(p,r-1)} \] and \(H_n^{(p,1)}=H_n^{(p)}\), where \(n\), \(p\), and \(r\) are positive integers. The main goal of this article is to give exact formulas for the sums \[ \sum_{n=1}^{\infty} \frac{H_n{(p,s)}}{n^m \binom{n+k}{k}}, \qquad \sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n{(p,s)}}{n^m \binom{n+k}{k}}, \] and \[ \sum_{n=1}^{\infty} \frac{H_n{(p_1,s_1)}H_n{(p_2,s_2)}}{n^m \binom{n+k}{k}}, \qquad \sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n{(p_1,s_1)}H_n{(p_2,s_2)}}{n^m \binom{n+k}{k}}. \] These formulas are given in terms of auxiliary quantities \(a(s,\ell_1,\ell_2)\), \(S(p,m,t,r,x)\) and \(T(p_1,p_2,m,t,r,x)\). In the examples, it is shown that these formulas can be given in terms of zeta values, powers of \(\pi\), polylogarithms, etc.
scientific article; zbMATH DE number 7604032
scientific article