Abstract
We study integrals of the form
where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).
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Acknowledgements
The authors are grateful to Peter Paule for providing them with his Mathematica package, which implements Zeilberger’s algorithm and allows for transforming differential equations into holonomic linear recurrences for the coefficients. They are also indebted to Helmut Prodinger for pointing out to them that this method could be applied. The authors are grateful to an anonymous referee for her/his valuable comments.
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This research was supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”)
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Brauchart, J.S., Grabner, P.J. Weighted \(L^2\)-norms of Gegenbauer polynomials. Aequat. Math. 96, 741–762 (2022). https://doi.org/10.1007/s00010-022-00871-9
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DOI: https://doi.org/10.1007/s00010-022-00871-9