Evaluations of sums involving harmonic numbers and binomial coefficients

ABSTRACT

In this paper, by the Faà di Bruno formula, we establish the decompositions of two general fractions involving the reciprocals of products of binomial coefficients. Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as $S_{{\pi }_1,q}\left(\mathit{k}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{H_n^{\left({\pi }_1\right)}}{n^q\prod _{i=1}^p\left(\genfrac{}{}{0em}{}{n+k_i}{k_i}\right)},S_{{\pi }_1}^q\left(\mathit{k}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{n^q{H}_n^{\left({\pi }_1\right)}}{\prod _{i=1}^p\left(\genfrac{}{}{0em}{}{n+k_i}{k_i}\right)},$ and some other forms. We present some explicit evaluations as examples and provide the Maple package to compute the sums $S_{{\pi }_1,q}\left(\mathit{k}\right)$ and $S_{{\pi }_1}^q\left(\mathit{k}\right)$. It can be found that this work gives a unified approach to such sums and generalizes many known results in the literature.

KEYWORDS:

• Euler-type sums
• harmonic numbers
• binomial coefficients
• Riemann zeta function

AMS CLASSIFICATIONS:

• 40A05
• 33B99
• 33E20
• 11M06
• 11M32

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, and Professor Petitot for his kind help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (under Grant 11671360) and the Fundamental Research Funds of Zhejiang Sci-Tech University (under Grant 2019Q063). The second author is supported by the China Scholarship Council (No. 201806310063).