Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials

Abstract

Our 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.