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Unified (p, q)-Bernoulli-Hermite polynomials

Pathan M. A.

Fasciculi Mathematici

2018

Nr 61

125--141

The Concepts of p-Bernoulli numbers Bn,p and p-Bernoulli polynomials Bn,p(x) are generalized to (p,q)-Bernoulli numbers Bn p q and (p,q)-Bernoulli polynomials Bn p q(x), respec- tively. Some properties, generating functions and Laplace hy- pergeometric integral representations of (p, q)-Bernoulli numbers Bn,p,q and (p,q)-Bernoulli polynomials Bn,p,q(x), are established. Unified (p,q)-Bernoulli-Hermite polynomials are defined by a generating function which aid in proving the generalizations of the results of Khan et al [8], Kargin and Rahmani [7], Dattoli [4] and Pathan [9]. Some explicit summation formulas and some relationships between Appell’s function F1, Gauss hypergeomtric function, Hurwitz zeta function and Euler’s polynomials are also given.