Email this article Infinite elliptic hypergeometric series: convergence and diffrence equations
- Authors: Krotkov D.I.1, Spiridonov V.P.2,1
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Affiliations:
- HSE University
- Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics
- Issue: Vol 214, No 12 (2023)
- Pages: 106-134
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/147930
- DOI: https://doi.org/10.4213/sm9874
- ID: 147930
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Abstract
We derive finite difference equations of infinite order for theta-hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, and we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion of the convergence of q-hypergeometric series for |q|=1, qn≠1, to the elliptic level and prove the convergence of infinite very-well poised elliptic hypergeometric r+1Vr-series for restricted values of q.
About the authors
Danil Igorevich Krotkov
HSE University
Email: math-net2025_06@mi-ras.ru
Vyacheslav Pavlovich Spiridonov
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics; HSE University
Author for correspondence.
Email: math-net2025_06@mi-ras.ru
Doctor of physico-mathematical sciences, no status
References
- Р. Аски, Р. Рой, Дж. Эндрюс, Специальные функции, МЦНМО, М., 2013, 651 с.
- I. B. Frenkel, V. G. Turaev, “Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions”, The Arnold–Gelfand mathematical seminars, Birkhäuser Boston, Inc., Boston, MA, 1997, 171–204
- Дж. Гаспер, М. Рахман, Базисные гипергеометрические ряды, Мир, М., 1993, 349 с.
- S. Grepstad, L. Kaltenböck, M. Neumüller, “A positive lower bound for $lim inf_{Ntoinfty}prod_{r=1}^{N} |2sin pi rvarphi|$”, Proc. Amer. Math. Soc., 147:11 (2019), 4863–4876
- G. H. Hardy, J. E. Littlewood, “Notes on the theory of series (XXIV): a curious power-series”, Proc. Cambridge Philos. Soc., 42:2 (1946), 85–90
- D. S. Lubinsky, “The size of $(q; q)_n$ for $q$ on the unit circle”, J. Number Theory, 76:2 (1999), 217–247
- G. Petruska, “On the radius of convergence of $q$-series”, Indag. Math. (N.S.), 3:3 (1992), 353–364
- V. P. Spiridonov, “Theta hypergeometric series”, Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., 77, Kluwer Acad. Publ., Dordrecht, 2002, 307–327
- V. P. Spiridonov, “An elliptic incarnation of the Bailey chain”, Int. Math. Res. Not. IMRN, 2002:37 (2002), 1945–1977
- V. P. Spiridonov, “Theta hypergeometric integrals”, Алгебра и анализ, 15:6 (2003), 161–215
- В. П. Спиридонов, “Очерки теории эллиптических гипергеометрических функций”, УМН, 63:3(381) (2008), 3–72
- A. Zhedanov, “Elliptic polynomials orthogonal on the unit circle with a dense point spectrum”, Ramanujan J., 19:3 (2009), 351–384
- A. Zhedanov, “Umbral “classical” polynomials”, J. Math. Anal. Appl., 420:2 (2014), 1354–1375
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