On discrete q-ultraspherical polynomials and their duals |
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Authors: | NM Atakishiyev AU Klimyk |
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Institution: | a Instituto de Matemáticas, UNAM, CP 62210 Cuernavaca, Morelos, Mexico b Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine |
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Abstract: | A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q−1). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures. |
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Keywords: | Big q-Jacobi polynomials Little q-Jacobi polynomials Discrete q-ultraspherical polynomials Duality Orthogonality relation |
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