Orthogonalizing Q-Bernoulli Polynomials

Abstract

In this study, we utilize the Gram-Schmidt orthogonalization method to construct a new set of orthogonal polynomials called OB n (x, q) {{\rm{OB}}}_{n}(x,q) from the q-Bernoulli polynomials. We demonstrate the relationship between polynomials OB n (x, q) {{\rm{OB}}}_{n}(x,q) and the little q-Legendre polynomials, and derive a generalized formula for OB n (x, q) {{\rm{OB}}}_{n}(x,q) by leveraging the little q-Legendre polynomials. Furthermore, we present some properties of polynomials OB n (x, q) {{\rm{OB}}}_{n}(x,q). Finally, we introduce a hybrid of block-pulse function and orthogonal polynomials OB n (x, q) {{\rm{OB}}}_{n}(x,q) and examine various properties of these polynomials.