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An integral identity with applications in orthogonal polynomials


Author: Yuan Xu
Journal: Proc. Amer. Math. Soc. 143 (2015), 5253-5263
MSC (2010): Primary 33C45, 33C50, 42C10
DOI: https://doi.org/10.1090/proc/12635
Published electronically: April 14, 2015
MathSciNet review: 3411143
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Abstract | References | Similar Articles | Additional Information

Abstract: For $ \boldsymbol {\large {\lambda }} = (\lambda _1,\ldots ,\lambda _d)$ with $ \lambda _i > 0$, it is proved that

$\displaystyle \prod _{i=1}^d \frac { 1}{(1- r x_i)^{\lambda _i}} = \frac {\Gamm... ...ert\boldsymbol {\large {\lambda }}\vert}} \prod _{i=1}^d u_i^{\lambda _i-1} du,$    

where $ \mathcal {T}^d$ is the simplex in homogeneous coordinates of $ \mathbb{R}^d$, from which a new integral relation for Gegenbauer polynomials of different indexes is deduced. The latter result is used to derive closed formulas for reproducing kernels of orthogonal polynomials on the unit cube and on the unit ball.

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Additional Information

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: yuan@uoregon.edu

DOI: https://doi.org/10.1090/proc/12635
Keywords: Gegenbauer polynomials, orthogonal polynomials, several variables, reproducing kernel
Received by editor(s): April 3, 2014
Received by editor(s) in revised form: April 23, 2014, and September 11, 2014
Published electronically: April 14, 2015
Additional Notes: This work was supported in part by NSF Grant DMS-1106113
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society