What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$? The classical Euler's gamma function $\Gamma(z)$ can be defined by
\begin{equation}
\Gamma(z)=\lim_{n\to\infty}\frac{n!n^z}{\prod_{k=0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}.
\end{equation}
The extended binomial coefficient $\binom{z}{w}$ for $z,w\in\mathbb{C}$ is defined by
\begin{equation}
\binom{z}{w}=
\begin{cases}
\dfrac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}, & z\not\in\mathbb{N}_-,\quad w,z-w\not\in\mathbb{N}_-;\\
0, & z\not\in\mathbb{N}_-,\quad w\in\mathbb{N}_- \text{ or } z-w\in\mathbb{N}_-;\\
\dfrac{\langle z\rangle_w}{w!},& z\in\mathbb{N}_-, \quad w\in\mathbb{N}_0;\\
\dfrac{\langle z\rangle_{z-w}}{(z-w)!}, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_0;\\
0, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_-;\\
\infty, & z\in\mathbb{N}_-, \quad w\not\in\mathbb{Z},
\end{cases}
\end{equation}
where
\begin{align}
\mathbb{Z}&=\{0,\pm1,\pm2,\dotsc\}, & \mathbb{N}&=\{1,2,\dotsc\},\\
\mathbb{N}_0&=\{0,1,2,\dotsc\}, & \mathbb{N}_-&=\{-1,-2,\dotsc\}
\end{align}
and
\begin{align}
\langle\alpha\rangle_n&=\prod_{k=0}^{n-1}(\alpha-k)\\
&=
\begin{cases}
\alpha(\alpha-1)\dotsm(\alpha-n+1), & n\ge1\\
1, & n=0
\end{cases}
\end{align}
is called the falling factorial.
My question is: what are the general formulas of the finite sums
$$
\sum_{k=0}^{n}(\pm1)^{k} \binom{n}{k}\binom{k/2}{m}
$$
for $n\ge0$ and any suitable number $m$?

*

*When $m\ne0$ and $n\ge0$, it is known that the identity
$$
\sum_{k=0}^{n}\binom{n}{k}\binom{k/2}{m}=\frac{n}{m}\binom{n-m-1}{m-1}2^{n-2m}
$$
is valid.

*When $2m\ge n+1\ge1$, it is known that the identity
\begin{align}
\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}
&=(-1)^m\biggl[\binom{2m-n-1}{m-1}-\binom{2m-n-1}{m}\biggr]2^{n-2m}\\
&=(-1)^m\frac{n}{m}\binom{2m-n-1}{m-1}2^{n-2m}
\end{align}
is valid.

These two answers are slightly modifications in form of Items (3.163) and (3.164) on pages 91--92 in the monograph:
R. Sprugnoli, Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006. (Has this monograph been formally published somewhere?)


*The identity
\begin{equation}
\sum_{k=0}^{r}(-1)^k\binom{r}{k}\binom{k/2}{q}
=\frac{(-1)^q}{2^{2q-r}}\frac{r}{2q-r}\binom{2q-r}{q}, \quad 0\le r\le q
\end{equation}
at Finite summation with binomial coefficients, $\sum (-1)^k\binom{r}{k} \binom{k/2}{q}$ has a more strict restriction $0\le r\le q$ than the above restriction $2m\ge n+1\ge1$.

Finally and essentially speaking, my question is: if without the restriction $2m\ge n+1\ge1$, what is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m\in\mathbb{N}$ and $n\ge0$? even for $m\in\mathbb{C}$ and $n\ge0$?
 A: Let $\mathbb{N}_0=\{0,1,2,\dotsc\}$.

*

*For $n,\ell\in\mathbb{N}_0$, we have
\begin{equation}\tag{1}
\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{\ell}
=
\begin{cases}
0, & n>\ell\in\mathbb{N}_0;\\ \displaystyle
(-1)^{\ell}n!\frac{[2(\ell-n)-1]!!}{(2\ell)!!}\binom{2\ell-n-1}{2(\ell-n)}, & \ell\ge n\in\mathbb{N}_0.
\end{cases}
\end{equation}

*For $n,\ell\in\mathbb{N}_0$, we have
\begin{equation}\tag{2}
\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\binom{2k}{\ell}
=
\begin{cases}
0, & n>\ell\in\mathbb{N}_0;\\\displaystyle
(-1)^n\binom{n}{\ell-n}2^{2n-\ell}, & \ell\ge n\in\mathbb{N}_0.
\end{cases}
\end{equation}

*For $\ell\ge n\in\mathbb{N}_0$, we have
\begin{equation}\tag{3}
\sum_{n=0}^\ell\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{\ell}
=(-1)^\ell\frac{(2\ell-1)!!}{(2\ell)!!}.
\end{equation}
Reference

*

*Siqintuya Jin, Bai-Ni Guo, and Feng Qi, Partial Bell polynomials, falling and rising factorials, Stirling numbers, and combinatorial identities, Computer Modeling in Engineering & Sciences (2022), in press; accepted on 24 January 2022; available online at https://dx.doi.org/10.32604/cmes.2022.019941 or https://www.researchgate.net/publication/358050501.

