Transformations of Hypergeometric Functions in Logarithmic Case with Applications to Study of Fundamental Solutions of Degenerate and Singular Elliptic Equations
Ключевые слова:
Gauss, Appell and Lauricella hypergeometric functions, reduction formulas, logarithmic behavior, logarithmic singularity, degenerate and singular elliptic equations, fundamental solutionsАннотация
Under some exceptional circumstances, such as when the sum of the upper numerical parameters of a hypergeometric function is equal to the sum of the lower parameters,
or when these sums differ from each other by integers, a logarithmic function may be used in
the expression of the hypergeometric function. In such cases we speak of logarithmic reducible
hypergeometric functions and of logarithmic reduction formulas. In this paper, using the wellknown logarithmic reduction formulas and statements about logarithmic behavior for the Gauss
hypergeometric function and the second Appell function, we establish similar reduction formulas and statements for the first Lauricella function of three or more variables. The established
theorems on logarithmic or power behavior for the single Gauss, double Appell and multiple
Lauricella hypergeometric functions are applied to the determination of logarithmic or power
singularities of fundamental solutions of degenerate and singular elliptic equations
Библиографические ссылки
P. Appell, ”Sur les s´eries hyperg´eom´etriques de deux variables, et sur des ´equations diff´erentielles
lin´eaires aux d´eriv´ees partielles”, C.R. Acad. Sci., Paris 90, 296–298 (1880).
G. Lauricella, ”Sulle funzione ipergeometriche a pi`u variabili”, Rend.Circ. Mat. Palermo 7, 111–158
(1893). https://doi.org/10.1007/BF03012437
P. Appell, J. Kamp´e de F´eriet, Fonctions Hyperg´eometriques et Hypersph´eriques: Polynˆomes
d’Hermite. Paris, Gauthier-Villars, 1926.
A. Erd´elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions. New
York, Toronto, London, McGraw-Hill, 1953.
H. M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series. New York, Chichester,
Brisbane and Toronto, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, 1985.
Sh. B. Opps, N. Saad, H. M. Srivastava, ”Some reduction and transformation formulas
for the Appell hypergeometric function F2”, J. Math. Anal. Appl. 302, 180–195 (2005).
https://doi.org/10.1016/j.jmaa.2004.07.052
S. I. Bezrodnykh, ”Analytic continuation of the Appell function F1 and integration of the associated
system of equations in the logarithmic case”, Computational Mathematics and Mathematical Physics
(4), 559–589 (2017). https://doi.org/10.1134/S0965542517040042
J. Murley, N. Saad, ”Tables of the Appell hypergeometric functions F2”, ArXiv:0809.5203v2, 1–31
(2008). https://doi.org/10.48550/arXiv.0809.5203
M. O. Abbasova, T. G. Ergashev, ”Tables of the Lauricella hypergeometric functions F
(3)
A ”, Vestnik
KRAUNC. Fiz.-Mat. Nauki 54 (1), 9–32 (2026). https://doi.org/10.26117/2079-6641-2026-54-1-9-32.
A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Volume 3. More Special
Functions, Amsterdam, Gordon and Breach Science Publishers, 1990.
E. T. Copson, ”On Hadamard’s elementary solution”, Proceedings of the Royal Society of Edinburgh
Section A: Mathematics 69 (1), 19–27 (1970). https://doi.org/10.1017/S0080454100008529
A. Hasanov, T. G. Ergashev, N. Djuraev, ”Lauricella hypergeometric function F
(n)
A with applications
to the solving Dirichlet problem for three-dimensional degenerate elliptic equation”, Uzbek Mathematical Journal 69 (3), 73–82 (2025). https://doi.org/10.29229/uzmj.2025-3-7
A. Hasanov, T. G. Ergashev, A. B. Okboev, ”Applications of Appell and Lauricella hypergeometric
functions to solving of Neumann problem for degenerate elliptic equation”, Bulletin of the Institute
of Mathematics 9 (1), 63 – 75 (2026).
T. G. Ergashev, A. B. Okboev, ”Boundary value problems with mixed Dirichlet and Neumann conditions for three-dimensional degenerate elliptic equation”, Journal of Osh University “Differential
Equations” 1 (1), 93–104 (2026).
T. G. Ergashev, M. O. Abbasova, ”Holmgren’s problem for the Laplace equation in the hyperoctant of a multidimensional ball”, Lobachevskii Journal of Mathematics 43 (6), 1303–1312 (2022).
https://doi.org/10.1134/S1995080222090062
M. O. Abbasova, T. G. Ergashev, T. K. Yuldashev, ”Dirichlet problem for the Laplace equation in
the hyperoctant of a multidimensional ball”, Lobachevskii Journal of Mathematics 44 (3), 1072–1079
(2023). https://doi.org/10.1134/S1995080223030022
A. Hasanov, T. G. Ergashev, ”New decomposition formulas associated with the Lauricella multivariable hypergeometric functions”, Montes Taurus Journal of Pure and Applied Mathematics 3 (3),
–326 (2021). https://doi.org/MTJPAM-D-20-00049
A. Ryskan, T. Ergashev, ”On some formulas for the Lauricella function”, Mathematics 11 (24), 4938,
–10 (2023). https://doi.org/10.3390/math11244978
T. G. Ergashev, A. Hasanov, T. K. Yuldashev, ”Some infinite expansions of the Lauricella functions and their application in the study of fundamental solutions of a singular elliptic equation”, Lobachevskii Journal of Mathematics 45 (3), 1072–1085 (2024).
https://doi.org/10.1134/S1995080224600742
T. G. Ergashev, A. R. Ryskan, N. N. Yuldashev, ”Recurrence free decomposition formulas for the
Lauricella special functions”, Bulletin of the Karaganda University. Mathematics Series 116 (4),
–106 (2024). https://doi.org/10.31489/2024M4/95-106
S. I. Bezrodnykh, ”Analytic continuation of the Lauricella function with arbitrary number of variables”, Integral Transforms and Special Functions 29 (1), 21–42 (2018).
https://doi.org/10.1080/10652469.2017.1402017
Z. O. Arzikulov, T. G. Ergashev, ”Some systems of PDE associated with the multiple confluent
hypergeometric functions and their applications”, Lobachevskii Journal of Mathematics 45 (2), 591–
(2024). https://doi.org/10.1134/S1995080224600250
E.W. Hobson, The theory of finctions of a real variable. Vol.II. Cambridge, Cambridge University
Press, 1926.
T. G. Ergashev, ”Fundamental solutions for a class of multidimensional elliptic equations with several
singular coefficients”, Journal of Siberian Federal University – Mathematics and Physics 13 (1), 48–
(2020). https://doi.org/10.17516/1997-1397-2020-13-1-48-57
T. G. Ergashev, Z. R. Tulakova, ”A problem with mixed boundary conditions for a singular elliptic equation in an infinite domain”, Russian Mathematics 66 (7), 51–63 (2022).
https://doi.org/10.3103/S1066369X22070039
A. Hasanov, E. T. Karimov, ”Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients”, Appl. Math. Lett. 22, 1828–1832 (2009).
https://doi.org/10.1016/j.aml.2009.07.006
A. H. Hasanov, A. S. Berdyshev, A. R. Ryskan, ”Fundamental solutions for a class of four-dimensional
degenerate elliptic equation”, Complex Variables and Elliptic Equations 65 (4), 632–647 (2020).
https://doi.org/10.1080/17476933.2019.1606803
T. G. Ergashev, ”Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables”, Lobachevskii Journal of
Mathematics 41 (1), 15–26 (2020). https://doi.org/10.1134/S1995080220010047
