Characterization of Kummer hypergeometric Bernoulli polynomials and applications

Abstract

In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials Ba,b,n(x) and Kummer hypergeometric polynomials of the second kind Ka,b,n(x), which are respectively defined by the exponential generating functions:extM(a,a+b;t)=∑n=0∞Ba,b,n(x)tnn! and extU(a,a+b;t)=∑n=0∞Ka,b,n(x)tnn! with M(a,b;t)=∑n=0∞(a)n(b)ntnn!, where U(a,a+b;t) is the Kummer hypergeometric function of the second kind.First we construct Gauss–Weierstrass-type convolution operators Twa,b with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions.