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Journal of the London Mathematical Society 1978 s2-18(1):101-114; doi:10.1112/jlms/s2-18.1.101
© 1978 by London Mathematical Society
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© Oxford University Press

Positivity Proofs for Linearization and Connection Coefficients of Orthogonal Polynomials Satisfying an Addition Formula

Tom Koornwinder

Mathematisch Centrum 2e Boerhaavestraat 49, Amsterdam, Netherlands

Received 26 October 1977.

For orthogonal polynomials in one or several variables which satisfy an addition formula it is proved that the linearization coefficients are nonnegative. Furthermore, a sufficient condition for nonnegativity of the connection coefficients is given in the case of two orthogonal systems of polynomials satisfying related addition formulas. The results are applied to Jacobi, disk and Laguerre polynomials. Gasper's nonnegativity result for the linearization coefficients of Jacobi polynomials is proved in a less computational way. The linearization coefficients of disk polynomials are shown to be nonnegative. A positivity result due to Askey and Gasper for a certain integral of the product of three Laguerre polynomials of the same order and argument is extended to the case of four factors. Finally, Dunkl's expression of a certain Jacobi polynomial times a simple polynomial as a sum of Gegenbauer polynomials with nonnegative coefficients is generalized to all real {alpha} ≥ 0.


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