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Classical orthogonal polynomials of a discrete variable continuous orthogonality relation

✍ Scribed by S. K. Suslov

Publisher
Springer
Year
1987
Tongue
English
Weight
413 KB
Volume
14
Category
Article
ISSN
0377-9017

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✦ Synopsis

Based on the general theory, we consider the continuous orthogonality property for classical polynomials of a discrete variable on nonuniform lattices.

I. Introduction. Preliminary Notions and Notations

Classical orthogonal polynomials (Jacobi, Laguerre and Hermite) are the simplest solutions of hypergeometric-type differential equations [ 1 ]: a(x)y" + z(x)y' + 2), = 0. (1.1) Recently it has become clear [1-8] that all the fundamentals of the theory of these polynomials remain valid if Equation (1.1) is replaced by a hypergeometric-type difference equation on a lattice x = x(z) with the nonuniform step Ax(z) = x(z + 1) -x(z): A [Vy(z)l+s[x(z)][Ay(z ) + Vy(z)] + 2v(z) = 0 (1.2) a[x(z)] )~x(z-89 LTx(z)_l 2 LAx(z) Vx(z)/ " ' * This Letter w a s written m 1985 a n d revised in late 1986.

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