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Expansion of the Riemann Ξ Function in Meixner–Pollaczek Polynomials

Published online by Cambridge University Press:  20 November 2018

Alexey Kuznetsov
Alexey Kuznetsov*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3. e-mail: kuznetsov@mathstat.yorku.ca
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Abstract

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In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner– Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

Keywords

41A1011M2633C45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 561 - 569
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bump, D., Choi, K.-K., Kurlberg, P., and Vaaler, J., A local Riemann hypothesis. I. Math. Z. 233(2000), no. 1, 1–19.Google Scholar
[2] Day, D. and Romero, L., Roots of polynomials expressed in terms of orthogonal polynomials. SIAM J. Numer. Anal. 43(2005), no. 5, 1969–1987 (electronic).Google Scholar
[3] Eremin, A. Yu., Kaporin, I. E., and Kerimov, M. K., The calculation of the Riemann zeta-function in the complex domain. (Russian) Zh. Vychisl.Mat. i Mat. Fiz. 25(1985), no. 4, 500–511.Google Scholar
[4] Gradshteyn, I. S. and Ryzhik, I. M., Tables of integrals, series and products. Sixth edition, Academic Press, Inc., San Diego, CA, 2000.Google Scholar
[5] Iserles, A. and Norsett, S. P., Zeros of transformed polynomials. SIAM J. Math. Anal. 21(1990), no. 2, 483–509.Google Scholar
[6] Iserles, A. and Norsett, S. P., On the theory of bi-orthogonal polynomials. Trans. Amer. Math. Soc. 306(1988), no. 2, 455–474.Google Scholar
[7] Iserles, A. and Saff, E. B., Zeros of expansions in orthogonal polynomials. Math. Proc. Cambridge Philos. Soc. 105(1989), no. 3, 559–573.Google Scholar
[8] Koekoek, R. and Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analog. Delft University of Technology, Faculty of Information Technology and Systems, Dept. of Technical Mathematics and Informatics, Report no. 98-17, (1998).Google Scholar
[9] Kuznetsov, A., Integral representations for the Dirichlet L-functions and their expansions in Meixner–Pollaczek polynomials and rising factorials. Integral Transforms Spec. Funct. 18(2007), no. 11-12, 809–817.Google Scholar
[10] Levin, B. Ja., Distribution of zeros of entire functions. Translations of Mathematical Monographs 5, American Mathematical Society, Providence, RI, 1980.Google Scholar
[11] Titchmarsh, E. C., The theory of the Riemann zeta-function. Second Edition, Oxford University Press, New York, NY, 1986.Google Scholar
[12] Weisstein, E. W., CRC concise encyclopedia of mathematics. Second Edition. CRC Press, Boca Raton, FL, 2002.Google Scholar