Combinatorial Identities and Hypergeometric Functions

We use properties of the Gaussian hypergeometric function to prove the following identities for combinatorial polynomials: ∑j=0n(n+αj)(n+βn−j)zj=(n+αn) ∑j=0n(nj)(n+j+α+βj)(j+αj)(z−1)n−j and m(m+nm)(1−z)n∑k=0n(nk)m+k(z1−z)k−∑k=0n(m+nk)(−z)k=∑k=0n(m+nk)(−z)n−k−∑k=0n(m+nn−k)(−z)n−k. These formulas extend two combinatorial identities published by Brereton et al. in 2011.