Confluent hypergeometric functions of many variables and their application to the finding of fundamental solutions of the generalized helmholtz equation with singular coefficients

A. K. Urinov ; T. G. Ergashev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 55 (2018), pp. 45-56

Résumé

An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions of one variable has stimulated the development of the corresponding theory for functions of two and more variables. In the theory of hypergeometric functions, an increase in the number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables in terms of an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitates the process of studying the properties of multidimensional functions. Confluent hypergeometric functions in all respects, including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper, we define a new class of confluent hypergeometric functions of several variables, study their properties, give integral representations, and establish decomposition formulas. An important application of confluent functions has been found. It turns out that all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas, the order of the singularity of the fundamental solutions of the above elliptic equation is determined.

Détail

Keywords: confluent hypergeometric function, Lauricella functions, fundamental solutions, generalized Helmholtz equation with several singular coefficients
Mots-clés : decomposition formula.

Bibliographie

[1] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher Transcendental Functions, v. 1, McGraw-Hill Book Company, New York–Toronto–London, 1953

[2] Srivastava H. M., Karlsson P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, New York–Chichester–Brisbane–Toronto, 1985, 428 pp. | MR | Zbl

[3] Jain R. N., “The confluent hypergeometric functions of three variables”, Proc. Nat. Acad. Sci. India Sect. A, 36 (1966), 395–408 | MR | Zbl

[4] Exton H., “On certain confluent hypergeometric of three variables”, Ganita, 21:2 (1970), 79–92 | MR | Zbl

[5] Urinov A. K., “Fundamental solutions for the some elliptic equations with singular coefficients”, Nauchnyj vestnik Ferganskogo gosudarstvennogo universiteta - Scientific Records of Fergana State University, 2006, no. 1, 5–11

[6] Hasanov A., “Fundamental solutions bi-axially symmetric Helmholtz equation”, Complex Variables and Elliptic Equations, 52:8 (2007), 673–683 | DOI | MR | Zbl

[7] Hasanov A., Karimov E. T., “Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients”, Applied Mathematic Letters, 22 (2009), 1828–1832 | DOI | MR | Zbl

[8] Urinov A. K., Karimov E. T., “On fundamental solutions for 3D singular elliptic equations with a parameter”, Applied Mathematic Letters, 24 (2011), 314–319 | DOI | MR | Zbl

[9] Karimov E. T., “On a boundary problem with Neumann's condition for 3D singular elliptic equations”, Applied Mathematics Letters, 23 (2010), 517–522 | DOI | MR | Zbl

[10] Karimov E. T., “A boundary-value problem for 3-D elliptic equation with singular coefficients”, Progress in Analysis and its Applications, 2010, 619–625 | DOI | MR | Zbl

[11] Lauricella G., “Sille funzioni ipergeometriche a piu variabili”, Rend. Circ. Mat. Palermo, 7 (1893), 111–158 | DOI

[12] Erdélyi A., “Integraldarstellungen fur Produkte Whittakerscher Funktionen”, Nieuw Arch. Wisk., 1939, no. 2(20), 1–34 | MR

[13] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions”, Quart. J. Math. (Oxford) Ser. 11, 1940, 249–270 | DOI | MR

[14] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions. II”, Quart. J. Math. (Oxford) Ser. 12, 1941, 112–128 | DOI | MR

[15] Hasanov A., Srivastava H. M., “Some decomposition formulas associated with the Lauricella function $F_A^{(r)}$ and other multiple hypergeometric functions”, Applied Mathematic Letters, 19:2 (2006), 113–121 | DOI | MR | Zbl

[16] Hasanov A., Srivastava H. M., “Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions”, Computers and Mathematics with Applications, 53:7 (2007), 1119–1128 | DOI | MR | Zbl

[17] Ergashev T. G., Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients, 9 pp., arXiv: 1805.03826