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On the relation between Gegenbauer polynomials and the Ferrers function of the first kind. (English) Zbl 1513.33010

The authors use relations among the Gegenbauer, Jacobi, and Meixner polynomials to find properties and statements for the Ferrers function of the first and second kind that have been unknown so far. The paper includes statements about orthogonality on \([-1,1]\), some definite integral evaluations, Christoffel-Darboux formulas, expression for the Poisson kernel, and the so-called closure relations. (Closure relations are infinite series representations of the Dirac delta distribution.)

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
44A20 Integral transforms of special functions
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References:

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