Orthogonalizing q −Bernoulli Polynomials

In this study, we utilize the Gram-Schmidt orthogonalization method to construct a new set of orthogonal polynomials called O B n ( x , q ) from the q −Bernoulli polynomials. We demonstrate the relationship between O B n ( x , q ) polynomials and the little q −Legendre polynomials, and derive a generalized formula for O B n ( x , q ) by leveraging the little q −Legendre polynomials. Furthermore, we present some properties of O B n ( x , q ) polynomials. Lastly, we introduce a hybrid of block-pulse function and orthogonal O B n ( x , q ) polynomials and examine various properties of these polynomials. 2000 Math. Subject. Classification: 11B68, 33C45