Email this article 
ESTIMATION OF THE REMAINDER TERM OF THE APPEL HYPERGEOMETRIC SERIES F2
- Authors: Bezrodnykh S.I1, Dunin-Barkovskaya O.V1,2
-
Affiliations:
- Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences
- Sternberg Astronomical Institute, Moscow State University
- Issue: Vol 65, No 12 (2025)
- Pages: 1973-1994
- Section: General numerical methods
- URL: https://journal-vniispk.ru/0044-4669/article/view/369547
- DOI: https://doi.org/10.7868/S3034533225120015
- ID: 369547
Cite item
Open Access
Access granted
Subscription Access
Abstract
Integral representations and estimates for the remainder term for the summation of the double hypergeometric Appell series F2 are constructed. The resulting formulas have applications in developing algorithms for computing the Appell functions F1 and F3 in C2 using analytic continuation formulas. The results have applications to problems in mathematical physics and computational function theory, including the construction of conformal mappings of complicated polygons based on the Christoffel-Schwarz integral.
About the authors
S. I Bezrodnykh
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences
Email: sbezrodnykh@mail.ru
Moscow, Russia
O. V Dunin-Barkovskaya
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences; Sternberg Astronomical Institute, Moscow State UniversityMoscow, Russia; Moscow, Russia
References
- Appel P. Sur les series hypergeometriques de deux variables et sur des equations diff’erentielles lin’eaires aux d’eriv’ees partielles // Comptes Rendus. 1880. V. 90. P. 296–298.
- Appel P., Kampe de Feriet J. Fonctions hypergeometriques et hyperspherique. Paris: Gauthier-Villars, 1926.
- Бейтмен Г., Эрдейи А. Высшие трансцендентные функции. Гипергеометрическая функция. Функции Лежандра. М.: Наука, 1973.
- Bezrodnykh S.I. Analytic continuation of Lauricella’s function FD(N) for large in modulo variables near hyperplanes {zj = zl} // Integral Transforms and Special Functions. 2022. V. 33. №4. P. 276–291.
- Bezrodnykh S.I. Analytic continuation of Lauricella’s functions FA(N), FB(N) andFD(N) // Integral Transforms and Special Functions. 2020. V. 31. №11. P. 921–940.
- Exton H. Multiple hypergeometric functions and application. New York: J. Willey & Sons inc, 1976.
- Erdelyi A. Hypergeometric functions of two variables // Acta Mat. 1950. V. 83. Iss. 131. P. 131–164.
- Безродных С.И., Дунин-Барковская О.В. Оценка остаточного члена при суммировании некоторых гипергеометрических рядов Горна // Ж. вычисл. матем. и матем. физ. 2024. Т. 64. №12. С. 2229–2242.
- Bezrodnykh S.I., Dunin-Barkovskaya O.V. Estimation of the remainder term of the Lauricella series FD(N) // Math. Notes. 2024. V. 116. №5. P. 905-919.
- Colavecchia F.D., Gasaneo G. fl: a code to compute Appell's F1 hypergeometric function // Comput. Phys. Communicat. 2004. V. 157. P. 32-38.
- Ananthanarayan B., Bera S., Friot S., Marichev O., Pathak T. On the evaluation of the Appell F2 double hypergeometric function // Comput. Phys. Communicat. 2023. V. 284. 108589.
- Bera S., Pathak T. Analytic continuations and numerical evaluation of the Appell F1, F3, Lauricella FD(3) and Lauricella-Saran FS(3) and their Application to Feynman Integrals // Comput. Phys. Communicat. 2025. V. 306. 109386589.
- Bezuglov M.A., Kniehl B.A., Onishchenko A.I., Veretin O.L. High-precision numerical evaluation of Lauricella functions // Nuclear Phys. B. 2025. 116994.
- Тарасов О.В. Применение функциональных уравнений для вычисления фейнмановских интегралов // Теор. и матем. физ. 2019. T. 200. №2. C. 324–342.
- Власов В.И., Скороходов С.Л. Аналитическое решение задачи о кавитационном обтекании клина. 1 // Ж. вычисл. матем. и матем. физ. 2020. T. 60. №12. C. 2098–2121.
- Власов В.И., Скороходов С.Л. Аналитическое решение задачи о кавитационном обтекании клина. II // Ж. вычисл. матем. и матем. физ. 2021. V. 61. №11. C. 1873–1893.
- Kalmykov M., Bytev V., Kniehl B., Moch S.-O., Ward B., Yost S. Hypergeometric functions and Feynman diagrams. In: Blumlein J., Schneider C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria). Springer, Cham, 2021.
- Karp D., Zhang Y. Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity // Math. Comp. 2023. V. 92. №344. P. 2769.
- Шилин И. А., Чой Дж. Алгебры Ли и специальные функции, связанные с изотропным конусом // Итоги науки и техн. Соврем. мат. и ее прил. Темат. обз., 222, ВИНИТИ РАН, М., 2023. C. 141–152.
- Claude Duhr, Franziska Porkert Feynman integrals in two dimensions and single-valued hypergeometric functions // J. High Energ. Phys. 2024. V. 2.
- Pantig R.C. Apparent and emergent dark matter around a Schwarzschild black hole // Physics of the Dark Universe. 2024. V. 45. 101550.
- Wei Fan. Celestial conformal blocks of massless scalars and analytic continuation of the Appell function F1 // J. High Energ. Phys. 2024. V. 2024. article number 145.
- Takahashi D.A. Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of Z(u|m)sc(u|m) // J. Phys. Soc. Japan. 2025. V. 94. №5. https://doi.org/10.7566/JPSJ.94.053001
- Wong R. Asymptotic approximations of integrals // Soc. Industr. and Appl. Math. 2001.
Supplementary files
