Expansions of Kampe de Feriet hypergeometric functions
DOI:
https://doi.org/10.26577/JMMCS2024-122-02-b5Keywords:
Hypergeometric function in two variables, Kamp´ e de F´ eriet function, generalized hypergeometric function, expansion formula, Burchnall-Chaundy method, symbolic H-operator, Appell and Humbert hypergeometric functionsAbstract
When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called $H$-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these $H$-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kamp\'{e} de F\'{e}riet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kamp\'{e} de F\'{e}riet functions of the superior order. Some Kamp\'{e} de F\'{e}riet functions are expanded in terms of the Appell and Humbert functions as illustrative examples
References
Appell P., Kamp´e de F´ eriet J., Fonctions Hyp´ erg´ eom´ etriques et Hyp´ erspheriques; Polynomes d’Hermite. (Paris: Gauthier Villars, 1926): 448.
Erd´ elyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, (New York; Toronto; London McGraw-Hill Book Company, 1953): 302.
Srivastava H.M., Karlsson P.W., Multiple Gaussian Hypergeometric Series. (New York, Chichester, Brisbane and Toronto: Halste Press, 1985): 428.
Barnes E.W., "The asymptotic expansions of integral functions defined by generalized hypergeometric series", Proc. London Math. Soc. V. 5, No 1 (1907): 59-116.
Kamp´ e de F´eriet J. "Les fonctions hyperg´eom´etriques d’ordre sup´ erieur а deux variables", C.R.Acad. Sci. Paris. V. 173, (1921): 401 404.
Burchnall J.L., Chaundy T.W., "Expansions of Appell’s double hypergeometric functions(II)", The Quarterly J. of Mathematics, Oxford. No 12. (1941): 112-128.
Srivastava H.M., Panda R., "An integral representation for the product of two Jacobi polynomials", J. London Math. Soc. V. 12, No 4 (1976): 419-425.
Srivastava H.M., Daoust M.C., "A note on the convergence of Kamp´ e de F´ eriet’s double hypergeometric series", Math. Nachr. V. 53, (1972): 151-159.
Kim Y.S., "On certain reducibility of Kamp´ e de F´ eriet function", Honam Math. J. 31(2) V. 31, No 2 (2009): 167-176.
Kim I., Paris R.B., Rathie A. K., "Some new results for the Kamp´ e de F´ eriet function with an application", Symmetry. V. 14, No 12 (2022).
Choi J., Milovanovi´ c G.V., Rathie A. K., "Generalized summation formulas for the Kamp´ e de F´ eriet function", Axioms. V. 10, No 4 (2021): 318.
Liu H., Wang W. "Transformation and summation formulae for Kamp´e de F´eriet series", Journal of Math. Anal. and Appl. V. 409, No 1 (2014): 100-110.
Ergashev T.G., Hasanov A., Yuldashev T.K., "Multiple Euler type integral representations for the Kamp´ e de F´ eriet functions", Chelyabinsk Physical and Mathematical journal. V. 8, No 4, (2023): 553-567.
Burchnall J.L., Chaundy T.W., "Expansions of Appell’s double hypergeometric functions", The Quarterly Journal of
Mathematics.Oxford. No 11 (1940): 249-270.
Hasanov A., Srivastava H.M., "Some decomposition formulas associated with the Lauricella function F(r) A and other multiple hypergeometric functions", Applied Mathematic Letters V. 19, No 2 (2006): 113-121.
Hasanov A., Srivastava H.M., "Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions", Computers and Mathematics with Applications V. 53, No 7 (2007): 1119-1128.
Hasanov A., Turaev M., Choi J., "Decomposition formulas for the generalized hypergeometric 4F3 function", Honam Mathematical J. V. 32, No 1, (2010): 1-16.
Choi J., Hasanov A., "Applications of the operator H to the Humbert double hypergeometric functions", Computers and Mathematics with Applications V. 61, No 3 (2011): 663-671.
Choi J., Kim S.Y., Hasanov A., "Applications of the operator H to the Saran function FE and some results" , Honam Mathematical J. V. 33, No 4 (2011): 441-452.
Bin-Saad M.G., Hasanov A., Turaev M., "Decomposition formulas of Kamp´ e de F´eriet double hypergeometric functions",
Analysis in Theory and Applications V. 34, No 3,(2018): 275-292.
Hasanov A., Bin-Saad M.G., Seilkhanova R.B., "Applications of symbolic operators to the Kamp´ e de F´eriet double hypergeometric series", Palestine Journal of Mathematics V. 7, No 1 (2018): 191-201.
Bin-Saad M.G., Ergashev T.G., Ergasheva D.A., Hasanov A., "Confluent Kamp´ e de F´eriet series arising in the solutions of Cauchy problem for degenerate hyperbolic equation of the second kind with the spectral parameter", Mathematica Pannonica, New Series. V. 29, No 2 (2023): 153-168.
Ergashev T.G., Komilova N.J., "The Kamp´ e de F´ eriet series and the regular solution of the Cauchy problem for degenerating hyperbolic equation of the second kind", Lobachevskii Journal of Mathematics V. 43, No 11 (2022): 3616-3625.
Appell P., "Sur les s´ eries hyperg´eom´ etriques de deux variables, et sur des ´ equations differentielles lin´ eaires aux d´ eriv´ ee partielles" , C.R. Acad. Sci.Paris V. 90 (1880): 296-298.
Humbert P., "The confluent hypergeometric functions of two variables", Proc. Roy. Soc. Edinburgh V. 41 (1922): 73-96.
Srivastava H.M., Hasanov A., Choi J., "Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation", Sohag J.Math. V. 2, No 1 (2015): 1-10.
Ergashev T.G., "Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation", Ufa Mathematical Journal V. 10, No 4 (2018): 111-121.
Hasanov A., Ergashev T.G., "Infinite summation formulas for triple Lauricella hypergeometric functions", Journal of Mathematical Sciences V. 274, No 2 (2023): 215-227.
HasanovA., Ryskan A., Choi J., "Decomposition formulas for second-order quadruple Gaussian hypergeometric series by means of operators H and H ", Montes Taurus Journal of Pure and Apllied Mathematics V. 4, No 3 (2022) 41-60.
Poole E. G., Introduction to the theory of linear differential equations. (Oxford, Clarendon, Oxford University, Press, 1936): 202.
Srivastava H.M., Manocha H.L., A treatise on generating functions. (New York, Chichester, Brisbane and Toronto: Halsted Press, Ellis Horwood, Chichester. John Wiley and Sons 1984): 570.
RyskanА.R., Arzikulov Z.O., Ergashev T.G. "Particular solutions of multidimensional generalized Euler-Poisson-Darboux equations of various (elliptic or hyperbolic) types", Journal of Mathematics, Mechanics and Computer Science No 1(121) (2024): 76-88
