Jensen polynomials for holomorphic functions. (English) Zbl 1530.33008
Summary: Recent work of M. Griffin et al. [Proc. Natl. Acad. Sci. USA 116, No. 23, 11103–11110 (2019; Zbl 1431.11105)]
shows that the Jensen polynomials for the Riemann xi function converge to the Hermite polynomials under a suitable normalization. We generalize this result, proving that the normalized Jensen polynomials for a large class of genus zero or one entire functions converge either to the Hermite polynomials, or to a class of polynomials which can be written as a confluent hypergeometric function.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 30C10 | Polynomials and rational functions of one complex variable |
| 11B83 | Special sequences and polynomials |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 30B10 | Power series (including lacunary series) in one complex variable |
Citations:
Zbl 1431.11105References:
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| [7] | Griffin, M., Ono, K., Rolen, L. and Zagier, D., Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci., USA116(23) (2019) 11103-11110. · Zbl 1431.11105 |
| [8] | M. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp and I. Wagner, Jensen polynomials for the Riemann Xi function, preprint, https://arXiv.org/abs/1910.01227. |
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| [12] | Pólya, G., Uber die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd.7 (1927) 3-33. · JFM 53.0309.01 |
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