Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials

Abstract

AbstractOur main objective is to establish the so-called connection formula, $$\displaystyle \begin{aligned} p_n(x)=\sum_{k=0}^{n}C_{k}(n)y_k(x), \end{aligned} $$ (0.1) which for p n(x) = x n is known as the inversion formula $$\displaystyle \begin{aligned} x^n=\sum _{k=0}^{n}I_{k}(n)y_k(x), \end{aligned} $$ for the family y k(x), where \(\{p_n(x)\}_{n\in \mathbb {N}_0}\) and \(\{y_n(x)\}_{n\in \mathbb {N}_0}\) are two polynomial systems. If we substitute x by ax in the left hand side of (0.1) and y k by p k, we get the multiplication formula $$\displaystyle \begin{aligned} p_n(ax)=\sum _{k=0}^{n}D_{k}(n,a)p_k(x). \end{aligned} $$ The coefficients C k(n), I k(n) and D k(n, a) exist and are unique since deg p n = n, deg y k = k and the polynomials {p k(x), k = 0, 1, …, n} or {y k(x), k = 0, 1, …, n} are therefore linearly independent. In this session, we show how to use generating functions or the structure relations to compute the coefficients C k(n), I k(n) and D k(n, a) for classical continuous orthogonal polynomials.KeywordsOrthogonal polynomialsInversion coefficientsMultiplication coefficientsConnection coefficientsMathematics Subject Classification (2000)33C4533D4533D1533F1068W30