Про обчислення гіпергеометричної функції F4 (1,2;2,2; z1, z2 ) за допомогою гіллястого ланцюгового дробу спеціального вигляду: Fìz.-mat. model. ìnf. tehnol. 2021, 32:86-90

• Дата: 2021
• Автори: Hladun, Volodymyr, Hoyenko, Nataliya, Ventyk, Levko, Manziy, Oleksandra
• Формат: Стаття
• Мова: Українська
• Опубліковано: Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України 2021
• Теми: діаграми Фейнмана; гіпергеометрична функція; рекурентні формули; розвинення; гілляcтий ланцюговий дріб; підхідний дріб
• Онлайн доступ: https://www.fmmit.lviv.ua/index.php/fmmit/article/view/165
• Назва журналу: Physico-mathematical modeling and informational technologies

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2. References Exton, H. (1976). Multiple hypergeometric functions and applications. New York-Sydney-Toronto, Chichester, Ellis Horwood. Huber, T., Maitre, D. (2008). HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun., 178, 755–776. DOI doi.org/10.1016/j.cpc.2007.12.008 Bytev, V. V., Kalmykov, M. Yu., Kniehl, B. A. (2010). Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case, Nucl. Phys., 836, 129–170. DOI doi.org/10.1016/j.nuclphysb.2010.03.025 Tai-Fu Feng, Chao-Hsi Chang, Jian-Bin Chen, Zhi-Hua Gu, Hai-Bin Zhang. (2018). Evaluating Feynman integrals by the hypergeometry. Nuclear Physics, 927, 516-549. DOI doi.org/10.1016/j.nuclphysb.2018.01.001 Beytmen, G., Еrdeyi, A. (1973). Vysshiye transtsendentnyye funktsii. (T.1.) – M.: Nauka. Lyuk, Yu. (1980). Spetsial'nyye matematicheskiye funktsii i ikh approksimatsii: Per. s angl. G.Babenko. – M.: Izd-vo «Mir». Kalmykov, M. Yu., Kniehl, B. A. (2012). Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions, Phys. Lett., 714, 103–109. DOI doi.org/10.1016/j.physletb.2012.06.045 Dzhouns, U., Tron, V. (1985). Nepreryivnyie drobi. Analiticheskaya teoriya i prilozheniya: Per. S angl.., M.: Mir. Cuyt, A., Brevik Petersen, V., Vendonk, B., Waadeland, H., Jones, W. B. (2008). Handbook of Continued Fractions for Special Functions. – New York: Springer Science + Business Media B.V. DOI doi.org/10.1007/978-1-4020-6949-9_2 Lorentzen, L., Waadeland, H. (1992). Continued Fractions with Application, Amsterdam: North-Holland. Skorobogatko, V. Ya. (1983). Teoriya vetvyaschihsya tsepnyih drobey i ee primenenie v vyichislitelnoy matematike. – M.: Nauka. Hoienko, N. P., Hladun, V. R., Manzii, O. S. (2014). Pro neskinchenni zalyshky hilliastoho lantsiuhovoho drobu Nordunda dlia hiperheometrychnykh funktsii Appelia. Karpatski matematychni publikatsiiб 6(1), 11–25. Manziy, O. S., Hladun, V. R., Ventyk, L. S. (2017). The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions. Mathematical Modeling and Computing, 4(1), 48-58. DOI doi.org/10.23939/mmc2017.01.048 Hoyenko, N., Antonova, T., Rakintsev, S. (2011). Nablyzhennya vidnoshen funktsiy Laurichelly-Sarana F_s z diysnymy parametramy hillyastymy lantsyuhovymy drobamy. Matematychnyy visnyk Naukovoho tovarystva im. Shevchenka, 8, 28-42. Hladun, V. R. (2016). Some Sets of Relative Stability Under Perturbations of Branched Continued Fractions with Complex Elements and a Variable Number of Branches. Journal of Mathematical Sciences, 255, 11-25. DOI doi.org/10.1007/s10958-016-2818-x Bodnar, D.I. (1986). Vetvyashchye tsepnye droby. K. Nauk. dumka. Bodnar, D. I. (1996). Bahatovymirni S-droby. Mat. metody ta fiz.-mekh. polya, 39(2), 39-46.