COMBINATORIAL IDENTITIES AND HYPERGEOMETRIC FUNCTIONS

Abstract

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

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

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.