Functional Summation of Series
- Authors: Varin V.P.1
-
Affiliations:
- Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
- Issue: Vol 63, No 1 (2023)
- Pages: 16-30
- Section: ОБЩИЕ ЧИСЛЕННЫЕ МЕТОДЫ
- URL: https://journals.rcsi.science/0044-4669/article/view/134284
- DOI: https://doi.org/10.31857/S0044466923010131
- EDN: https://elibrary.ru/LDWDNY
- ID: 134284
Cite item
Abstract
We consider a summation technique which reduces summation of a series to the solution of some linear functional equations. Partial sums of a series satisfy an obvious difference equation. This equation is transformed to the functional equation on the interval [0,1] for the continuous argument. Then this equation is either solved explicitly (to within an arbitrary constant) or an asymptotic expansion of the solution is computed at the origin. The sum of the original series is determined uniquely as a constant needed for the matching of the asymptotic series with partial sums of the original series. The notion of a limit is not involved in this computational technique, which allows summation of divergent series as well.
About the authors
V. P. Varin
Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Author for correspondence.
Email: varin@keldysh.ru
125047, Moscow, Russia
References
- Харди Г. Расходящиеся ряды. М.: Ленанд, 1951.
- Варин В.П. Преобразование последовательностей в доказательствах иррациональности некоторых фундаментальных констант // Ж. вычисл. матем. и матем. физ. 2022. Т. 62. № 9. С. 3–30.
- Lanczos C. Applied Analysis. Dover Publications. New-York, 1956.
- Варин В.П. Об интерполяции некоторых рекуррентных последовательностей // Ж. вычисл. матем. и матем. физ. 2021. Т. 61. № 6. С. 913–925.
- Варин В.П. Инвариантные кривые некоторых дискретных динамических систем // Ж. вычисл. матем. и матем. физ. 2022. Т. 62. № 2. С. 199–216.
- Olver F.W.J. Asymptotics and special functions. Acad. Press, New-York, 1974.
- Candelpergher B. Ramanujan Summation of Divergent Series. Lecture Notes in Math., Springer. 2017.
- Borwein J.M., Calkin N.J., Manna D. Euler–Boole Summation Revisited // The American Mathematical Monthly. 2009. V. 116. № 5. P. 387–412.
- Arakawa T., Ibukiyama T., Kaneko M. Bernoulli Numbers and Zeta Functions. Springer, Japan. 2014.
- Варин В.П. Факториальное преобразование некоторых классических комбинаторных последовательностей // Ж. вычисл. матем. и матем. физ. 2018. Т. 59. № 6. С. 1747–1770.
- Borwein J.M., Bradley D.M., Crandall R.E. Computational strategies for the Riemann zeta function // J. of Computational and Applied Mathematics. 2000. V. 121. P. 247–296.
- Cohen H., Villegas F.R., Zagier D. Convergence Acceleration of Alternating Series // Experimental Mathematics. 2000. V. 9. № 1. P. 3–12.
- Sloane online encyclopedia of integer sequences. (http://oeis.org).
- Char B.W. On Stieltjes’ Continued Fraction for the Gamma Function // MATHEMATICS OF COMPUTATION. 1980. V. 34. № 150. P. 547–551.
- Hille E. Ordinary differential equations in the complex domain. New-York: John Wiley & Sons. (1976).
- Edgar G.A. Transseries for beginners // [arxiv:0801.4877v5], (2009). (http://arxiv.org/abs/0801.4877v5).