In the paper [1] below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by \begin{equation*}%\label{S(n,k,x)-dfn} \frac{(\textrm{e}^z-1)^k}{k!}\textrm{e}^{\lambda z}=\sum_{n=k}^\infty R(n,k,\lambda)\frac{z^n}{n!} \end{equation*} and can be explicitly expressed by \begin{equation*}%\label{S(n,k,x)-satisfy-eq} R(n,k,r)=\frac1{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(r+j)^n \end{equation*} for $r\in\mathbb{R}$ and $n\ge k\ge0$. Specially, when $\lambda=0$, the quantity $R(n,k,0)$ becomes the Stirling numbers of the second kind $S(n,k)$.

I guess that the identities \begin{equation}\label{QGWSID1}\tag{SID1} \sum_{j=1}^k(-1)^j\binom{k}{j} \frac{R\bigl(2m+j-1,j,-\frac{j}2\bigr)}{\binom{2m+j-1}{j}}=0, \quad k,m\in\mathbb{N} \end{equation} and \begin{equation}\label{QGWSID2}\tag{SID2} \sum_{j=1}^k(-1)^j\binom{k}{j} \frac{R\bigl(2m+j,j,-\frac{j}2\bigr)}{\binom{2m+j}{j}}=0, \quad k>m\ge1 \end{equation} should be valid.

Stronger but simpler than \eqref{QGWSID1}, the identity \begin{equation}\label{QGWSID3}\tag{SID3} R\biggl(2m+j-1,j,-\frac{j}2\biggr) =\frac{(-1)^j}{j!}\sum_{\ell=0}^j(-1)^{\ell}\binom{j}{\ell}\biggl(\ell-\frac{j}{2}\biggr)^{2m+j-1} =0 \end{equation} for $k,m\in\mathbb{N}$ should be true.

These guesses \eqref{QGWSID1}, \eqref{QGWSID2}, and \eqref{QGWSID3} are related to series expansions at $x=0$ of the functions $$ \biggl(\frac{\sin x}{x}\biggr)^r \quad\text{and}\quad \biggl(\frac{\sinh x}{x}\biggr)^r $$ for real number $r\in\mathbb{R}$. For details, please read the paper [2] below.

Could you please confirm or deny these identities \eqref{QGWSID1}, \eqref{QGWSID2}, and \eqref{QGWSID3} involving the weighted Stirling numbers of the second kind $R(n,k,r)$?

References

- L. Carlitz,
*Weighted Stirling numbers of the first and second kind, I*, Fibonacci Quart.**18**(1980), no. 2, 147--162. - Feng Qi and Peter Taylor,
*Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials*, Applicable Analysis and Discrete Mathematics**18**(2024), no. 1, 92–115; available online at https://doi.org/10.2298/AADM230902020Q.

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