Expansion of the Riemann Ξ Function in Meixner–Pollaczek Polynomials
Canadian mathematical bulletin, Tome 51 (2008) no. 4, pp. 561-569

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2008-055-0
Mots-clés : 41A10, 11M26, 33C45
Kuznetsov, Alexey. Expansion of the Riemann Ξ Function in Meixner–Pollaczek Polynomials. Canadian mathematical bulletin, Tome 51 (2008) no. 4, pp. 561-569. doi: 10.4153/CMB-2008-055-0
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[1] [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] [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] [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] [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] [5] Iserles, A. and Norsett, S. P., Zeros of transformed polynomials. SIAM J. Math. Anal. 21(1990), no. 2, 483–509. Google Scholar

[6] [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] [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] [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] [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] [10] Levin, B. Ja., Distribution of zeros of entire functions. Translations of Mathematical Monographs 5, American Mathematical Society, Providence, RI, 1980. Google Scholar

[11] [11] Titchmarsh, E. C., The theory of the Riemann zeta-function. Second Edition, Oxford University Press, New York, NY, 1986. Google Scholar

[12] [12] Weisstein, E. W., CRC concise encyclopedia of mathematics. Second Edition. CRC Press, Boca Raton, FL, 2002. Google Scholar

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