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Mathematics of Computation

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Volterra-type convolution of classical polynomials


Authors: Ana F. Loureiro and Kuan Xu
Journal: Math. Comp. 88 (2019), 2351-2381
MSC (2010): Primary 42A85, 44A35, 33C05, 33C20, 33C45; Secondary 42A10, 41A10, 41A58
DOI: https://doi.org/10.1090/mcom/3427
Published electronically: April 2, 2019
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Abstract: We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $ \{P_k(x)\}_{k \geqslant 0}$ with $ \deg P_k(x) = k$. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.


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Additional Information

Ana F. Loureiro
Affiliation: School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Sibson Building, Canterbury, CT2 7FS, United Kingdom
Email: A.Loureiro@kent.ac.uk

Kuan Xu
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui, 230026, People’s Republic of China
Email: kuanxu@ustc.edu.cn

DOI: https://doi.org/10.1090/mcom/3427
Keywords: Convolution, Volterra convolution integral, orthogonal polynomials, Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials, Laguerre polynomials
Received by editor(s): April 26, 2018
Received by editor(s) in revised form: October 29, 2018
Published electronically: April 2, 2019
Additional Notes: The first author is the corresponding author
The second author was funded in part by Anhui Initiative in Quantum Information Technologies under grant AHY150200.
Article copyright: © Copyright 2019 American Mathematical Society