Congruences for generalized Apéry numbers and Gaussian hypergeometric series

摘要

Abstract For positive integers $$f_1,f_2,m,l$$ f 1 , f 2 , m , l , we define a generalization of Apéry numbers $$A(f_1,f_2,m,l,lambda )$$ A ( f 1 , f 2 , m , l , λ ) given by $$egin{aligned} A(f_1,f_2,m,l,lambda ),{:=},sum _{j=0}^{f_2}{f_1+jtopwithdelims ()j}^m{f_2topwithdelims ()j}^{l}lambda ^j. end{aligned}$$ A ( f 1 , f 2 , m , l , λ ) : = ∑ j = 0 f 2 f 1 + j j m f 2 j l λ j . In this article, we deduce congruence relations satisfied by these generalized Apéry numbers extending results of (Coster in Supercongruences, Ph.D. thesis, Universiteit Leiden, 1988). We find expressions of $$A(f_1,f_2,m,l,lambda )$$ A ( f 1 , f 2 , m , l , λ ) in terms of Gaussian hypergeometric series and evaluate some new supercongruences similar to Beukers’ supercongruences.