Parametric binomial sums involving harmonic numbers
نویسندگان
چکیده
We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers $$p=0,1,2$$ and $$|t|\le 1$$ . $$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_{k-1}t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) } \quad \text{ }\quad }\frac{t^k}{k^p\left( }. \end{aligned}$$ also generalize relation between Stirling first kind Riemann zeta function to polygamma give some applications. \zeta (n+1)=\sum _{k=n}^{\infty }\frac{s(k,n)}{kk!}, n=1,2,3,\ldots As examples, (3)=\frac{1}{7}\sum }\frac{H_{k-1}4^k}{k^2\left( {\begin{array}{c}2k\\ }, (3)=\frac{8}{7}+\frac{1}{7}\sum \frac{H_{k-1}4^k}{k^2(2k+1)\left( which are new series representations Apéry constant $$\zeta (3)$$
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ژورنال
عنوان ژورنال: Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas
سال: 2021
ISSN: ['1578-7303', '1579-1505']
DOI: https://doi.org/10.1007/s13398-021-01025-3