Parametric binomial sums involving harmonic numbers

Abstract

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $$p=0,1,2$$ p = 0 , 1 , 2 and $$|t|\\le 1$$ | t | ≤ 1 . $$\\begin{aligned} \\sum _{k=1}^{\\infty }\\frac{H_{k-1}t^k}{k^p\\left( {\\begin{array}{c}n+k\\\\ k\\end{array}}\\right) } \\quad \\text{ and }\\quad \\sum _{k=1}^{\\infty }\\frac{t^k}{k^p\\left( {\\begin{array}{c}n+k\\\\ k\\end{array}}\\right) }. \\end{aligned}$$ ∑ k = 1 ∞ H k - 1 t k k p n + k k and ∑ k = 1 ∞ t k k p n + k k . We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$\\begin{aligned} \\zeta (n+1)=\\sum _{k=n}^{\\infty }\\frac{s(k,n)}{kk!}, \\quad n=1,2,3,\\ldots . \\end{aligned}$$ ζ ( n + 1 ) = ∑ k = n ∞ s ( k , n ) k k ! , n = 1 , 2 , 3 , … . As examples, $$\\begin{aligned} \\zeta (3)=\\frac{1}{7}\\sum _{k=1}^{\\infty }\\frac{H_{k-1}4^k}{k^2\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }, \\quad \\text{ and }\\quad \\zeta (3)=\\frac{8}{7}+\\frac{1}{7}\\sum _{k=1}^{\\infty } \\frac{H_{k-1}4^k}{k^2(2k+1)\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }, \\end{aligned}$$ ζ ( 3 ) = 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 2 k k , and ζ ( 3 ) = 8 7 + 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 ( 2 k + 1 ) 2 k k , which are new series representations for the Apéry constant $$\\zeta (3)$$ ζ ( 3 ) .