Parametric binomial sums involving harmonic numbers

2021/05/09 by Necdet Batır, Batir, Necdet
Mathematics · #05A19 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 05A10 #Secondary 33C20

paper · pdf · doi:10.48550/arxiv.2105.03927

openalex publication_date 2021/05/09 · openalex created_date 2021/05/24 · openalex updated_date 2026/07/28

Abstract

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for p=0,1,2 and |t|≤1. ∑_k=1^∞\fracH_k-1t^kk^p\binomn+kk and ∑_k=1^∞\fract^kk^p\binomn+kk. 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. ζ(n+1)=∑_k=n^∞(s(k,n))/(kk!), n=1,2,3,... . As examples, ζ(3)=(1)/(7)∑_k=1^∞\fracH_k-14^kk²\binom2kk, and ζ(3)=(8)/(7)+(1)/(7)∑_k=1^∞\fracH_k-14^kk²(2k+1)\binom2kk, which are new series representations for the Apéry constant ζ(3).

Citations

• Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

Related

• Mneimneh-type Binomial Sums of Multiple Harmonic-type Sums
• Sums of series involving central binomial coefficients & harmonic numbers
• Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values
• Discovering and Proving Infinite Binomial Sums Identities
• Combinatorial identities involving harmonic numbers