Q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

Abstract

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].