## Abstract

We use properties of the Gaussian hypergeometric function to prove the following identities for combinatorial polynomials:

$$\sum _{j=0}^{n}\left(\begin{array}{c}n+\alpha \\ j\end{array}\right)\left(\begin{array}{c}n+\beta \\ n-j\end{array}\right){z}^{j}=\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)}\text{\hspace{0.17em}}{\displaystyle \sum _{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\frac{\left(\begin{array}{c}n+j+\alpha +\beta \\ j\end{array}\right)}{\left(\begin{array}{c}j+\alpha \\ j\end{array}\right)}{(z-1)}^{n-j}$$

and

$$\begin{array}{c}m\left(\begin{array}{c}m+n\\ m\end{array}\right){(1-z)}^{n}{\displaystyle \sum _{k=0}^{n}\frac{\left(\begin{array}{c}n\\ k\end{array}\right)}{m+k}{\left(\frac{z}{1-z}\right)}^{k}}-{\displaystyle \sum _{k=0}^{n}\left(\begin{array}{c}m+n\\ k\end{array}\right){(-z)}^{k}}={\displaystyle \sum _{k=0}^{n}\left(\begin{array}{c}m+n\\ k\end{array}\right){(-z)}^{n-k}}-{\displaystyle \sum _{k=0}^{n}\left(\begin{array}{c}m+n\\ n-k\end{array}\right){(-z)}^{n-k}}.\end{array}$$

These formulas extend two combinatorial identities published by Brereton et al. in 2011.

## Citation

Horst Alzer. Kendall C. Richards. "COMBINATORIAL IDENTITIES AND HYPERGEOMETRIC FUNCTIONS." Rocky Mountain J. Math. 52 (6) 1921 - 1928, December 2022. https://doi.org/10.1216/rmj.2022.52.1921

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