Orthogonalizing <i>q</i> -Bernoulli polynomials

Abstract In this study, we utilize the Gram-Schmidt orthogonalization method to construct a new set of orthogonal polynomials called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">OB</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{\rm{OB}}}_{n}(x,q) from the q -Bernoulli polynomials. We demonstrate the relationship between polynomials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">OB</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{\rm{OB}}}_{n}(x,q) and the little q -Legendre polynomials, and derive a generalized formula for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">OB</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{\rm{OB}}}_{n}(x,q) by leveraging the little q -Legendre polynomials. Furthermore, we present some properties of polynomials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">OB</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{\rm{OB}}}_{n}(x,q) . Finally, we introduce a hybrid of block-pulse function and orthogonal polynomials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">OB</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{\rm{OB}}}_{n}(x,q) and examine various properties of these polynomials.