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A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions. (English) Zbl 1131.33005

Using the extended Sturm-Liouville theorem a basic class of symmetric orthogonal polynomials with four free parameters is introduced. All the standard properties are obtained, including the explicit form, a generic second order differential equation, a generic orthogonality relation and a generic three term recurrence relation. Next it is shown that four classes of symmetric polynomials (generalized ultraspherical polynomials, generalized Hermite polynomials, and two finite classes of symmetric polynomials which are finitely orthogonal on \((-\infty, \infty)\) with respect to special weight functions) can be obtained from the introduced class and consequently all its standard properties can be generated. Two kinds of half-trigonometric orthogonal polynomials appears that the author calls fifth and sixth kind Chebyshev polynomials because they are generated using the first and second kind Chebyshev polynomials and have the half-trigonometric forms.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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