AIMSVSW 2018: Orthogonal Polynomials pp 69-83

# Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials

Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

## 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 pn(x) = xn 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 yk(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 yk by pk, 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 Ck(n), Ik(n) and Dk(n, a) exist and are unique since deg pn = n, deg yk = k and the polynomials {pk(x), k = 0, 1, …, n} or {yk(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 Ck(n), Ik(n) and Dk(n, a) for classical continuous orthogonal polynomials.

## Keywords

Orthogonal polynomials Inversion coefficients Multiplication coefficients Connection coefficients

## Mathematics Subject Classification (2000)

33C45 33D45 33D15 33F10 68W30

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