COMBINATORIAL IDENTITIES AND HYPERGEOMETRIC FUNCTIONS

We use properties of the Gaussian hypergeometric function to prove the following identities for combinatorial polynomials: Sigma(n) (j =0) (n+alpha j ) (n+beta n- j) z(j) = (n+alpha n) Sigma(n) (j =0) (n j) (n+j+alpha+ss j) / (J + alpha J) (Z -1)(n-j) and m(m+n m) (1-z)(n) Sigma(n) (K=0) (nk)/m+k (z/1-z)(k) - Sigma(n) (k=0) (m+n k) (-z)(k) = Sigma(n) (k=0) (m+n k) (-z)(n) (k) - Sigma(n) (k=0) (m+n n-k) (-z)(n k). These formulas extend two combinatorial identities published by Brereton et al. in 2011.