Email this article RATIONAL COEFFICIENTS OF ORTHOGONAL DECOMPOSITIONS OF CERTAIN FUNCTIONS
- Authors: Varin V.P1
-
Affiliations:
- Keldysh Institute of Applied Mathematics, RAS
- Issue: Vol 65, No 7 (2025)
- Pages: 1211-1224
- Section: Ordinary differential equations
- URL: https://kazanmedjournal.ru/0044-4669/article/view/688557
- DOI: https://doi.org/10.31857/S0044466925070105
- EDN: https://elibrary.ru/JYDVCR
- ID: 688557
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Abstract
Decompositions of many elementary and special functions into series by orthogonal polynomials have coefficients known explicitly. However, these coefficients are almost always irrational. Therefore, any numerical method gives these coefficients approximately when calculating in any arithmetic. This also applies to spectral methods that provide efficient approximations of holonomic functions. However, in some exceptional cases, the expansion coefficients obtained by the spectral method turn out to be rational and are calculated exactly in rational arithmetic. We consider such decompositions with respect to some classical orthogonal polynomials. It is shown that in this way it is possible to obtain an infinite set of linear forms for some irrationalities, in particular, for Euler’s constant.
About the authors
V. P Varin
Keldysh Institute of Applied Mathematics, RAS
Email: varin@keldysh.ru
Moscow, Russia
References
- Pashkovskii S. Computational Application of Chebyshev Polynomials and Series (Nauka, Moscow, 1983) [in Russian].
- Варин В.П. Аптроксимация дифференциальных операторов с учетом граничных условий // Ж. вычисл. матем. и матем. физ. 2023. Т. 63. № 8. С. 1251–1271.
- Варин В.П. Spectral methods for solution of differential and functional equations // Comp. Math. and Math. Phys. 2024. V. 64. № 5. P. 888–904.
- Арнёклеч А.І. On linear forms containing the Euler constant // [arXiv:0902.1768v2] (2009). (http://arxiv.org/abs/0902.1768v2).
- Арнёклеч А.І., Тиуакюч D.N. Four-termed recurrence relations for γ-forms // Sovrem. Probl. Math. 2007. Iss. 9. P. 37–43.
- Belabas K., Cohen H. Numerical Algorithms for Number Theory. Mathematical Surveys and Monographs, V. 254. Amer. Math. Soc. (2021).
- Варин В.П. Преобразование последовательностей в доказательствах иррациональности некоторых фундаментальных констант // Ж. вычисл. матем. и матем. физ. 2022. Т. 62. № 10. С. 1587–1614.
- Huang H., Kauers M. D-finite numbers // Inter. J. Number Theor. 2016. (https://www.researchgate.net/publication/310595024).
- Tchebichef P.I. Sur le développement des fonctions a une seule variable // Bull. de l’Acad. Imperiale des Sci. de St. Petersbourg. 1859. V. 1. P. 193–200.
- Gantmacher F.R. Application of the Theory of Matrices (Chelsea Press, New-York, 1960).
- Gradsheyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products (7th ed. Acad. Press, Elsevier, 2007).
- Варин В.П. Функциональное суммирование рядов // Ж. вычисл. матем. и матем. физ. 2023. Т. 63. № 1. С. 3–17.
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