Orthogonalizing q-Bernoulli polynomials

Abstract

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

Keywords

• q-bernoulli polynomials
• orthonormal polynomials
• block-pulse functions
• 33c45
• 11b68