An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials

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In this Subject: <a title="Mathematics and Statistics" href="/content/subcat?j_subject=195&j_availability=">Mathematics and Statistics
By this author: Xu Y.

Author: Xu Y.

Source: Advances in Applied Mathematics, Volume 29, Number 2, August 2002 , pp. 328-343(16)

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Abstract:

The generalized Gegenbauer polynomials are orthogonal polynomials with respect to the weight function|x|²(1-x²)^-1/2. An integral formula for these polynomials is proved, which serves as a transformation between h-harmonic polynomials associated with Zopf ² invariant weight functions on the plane. The formula also gives a new integral transform for the Jacobi polynomials, which contains several well-known formulae as special cases. The new formulae can be used to prove the positivity of certain sums of the generalized Gegenbauer and Jacobi polynomials.