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Erschienen in:
2020 | OriginalPaper | Buchkapitel
Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials
verfasst von : Daniel Duviol Tcheutia
Erschienen in: Orthogonal Polynomials
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Abstract
Our main objective is to establish the so-called connection formula, which for p
n(x) = x
n is known as the inversion formula
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
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.
$$\displaystyle \begin{aligned} p_n(x)=\sum_{k=0}^{n}C_{k}(n)y_k(x), \end{aligned} $$
(0.1)
$$\displaystyle \begin{aligned} x^n=\sum _{k=0}^{n}I_{k}(n)y_k(x), \end{aligned} $$
$$\displaystyle \begin{aligned} p_n(ax)=\sum _{k=0}^{n}D_{k}(n,a)p_k(x). \end{aligned} $$