Infinite series involving r-Stirling numbers and hyperharmonic numbers

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发表日期：

2026年6月10日

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Xudong Chen,
Weiping Wang,
Heng Zhang

摘要

In this paper, using the recurrence relations and identities satisfied by the r-Stirling numbers of the first kind, together with structural properties of generalized polylogarithms and alternating multiple zeta values, we establish recurrence relations and initial conditions for the r-Stirling series $\varPhi _r(j,k,l;q)=\sum _{n=1}^{\infty }\genfrac[]{0.0pt}1{n}{k}_r\frac{q^n}{n!(n+j)^l}$ $Φ_{r} (j, k, l; q) = \sum_{n = 1}^{\infty} {[\binom{n}{k}]}_{r} \frac{q^{n}}{n! {(n + j)}^{l}}$ as well as for the shifted Euler-type series $\varPsi _r(j,l;q)=\sum _{n=1}^{\infty }\frac{h_n^{(r)}q^n}{(n+j)^l}$ $Ψ_{r} (j, l; q) = \sum_{n = 1}^{\infty} \frac{h_{n}^{(r)} q^{n}}{{(n + j)}^{l}}$ and $\varPsi _r(-j,l;q)=\sum _{n=j+1}^{\infty }\frac{h_n^{(r)}q^n}{(n-j)^l}$ $Ψ_{r} (- j, l; q) = \sum_{n = j + 1}^{\infty} \frac{h_{n}^{(r)} q^{n}}{{(n - j)}^{l}}$ , where $\genfrac[]{0.0pt}1{n}{k}_r$ ${[\binom{n}{k}]}_{r}$ denote the r-Stirling numbers of the first kind, and $h_n^{(r)}$ $h_{n}^{(r)}$ are the hyperharmonic numbers. Consequently, a broad class of such series can be systematically evaluated in a unified symbolic framework. Furthermore, these series are shown to be expressible in terms of zeta values when $q=1$ $q = 1$ , and to be reducible to alternating multiple zeta values when $q=-1$ $q = - 1$ or $q=1/2$ $q = 1 / 2$ .