**Title:** New Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-20-00049; **Volume 3 / Issue 3 / Year 2021 (Special Issue)**, Pages 317-326

**Document Type:** Research Paper

**Author(s):** Anvarjon Hasanov ^{a} , Tuhtasin Gulamjanovich Ergashev ^{b}

^{a}Institute of Mathematics, 81 Mirzo-Ulugbek Street, Tashkent 100170, Uzbekistan – Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39 Kari-Niyazi Street, Tashkent 100100, Uzbekistan Department of Mathematics, Analysis, Logic and Discrete Mathematics Ghent University, Belgium

^{b}Institute of Mathematics, 81 Mirzo-Ulugbek Street, Tashkent 100170, Uzbekistan – Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39 Kari-Niyazi Street, Tashkent 100100, Uzbekistan

Received: 18 December 2020, Accepted: 9 April 2021, Published: 25 April 2021.

**Corresponding Author:** Tuhtasin Gulamjanovich Ergashev (Email address: ergashev.tukhtasin@gmail.com)

**Full Text:** PDF

**Abstract**

Certain decomposition formulas associated with a number of simple and multiple Gaussian hypergeometric functions including the multiple Lauricella hypergeometric functions have been presented. In this paper, we aim to establish further new decomposition formulas involving the multiple Lauricella hypergeometric functions *F*_{A}^{(n)} and *F*_{B}^{(n)}.

**Keywords:** Lauricella functions, Multiple hypergeometric functions, Decomposition formula

**References:**

- P. Appell and J. Kampe de Feriet,
*Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite*, Gauthier – Villars, Paris, 1926. - J. L. Burchnall and T. W. Chaundy,
*Expansions of Appell’s double hypergeometric functions*, The Quarterly Journal of Mathematics, Oxford, Ser.**11**, 249-270, 1940. - J. L. Burchnall and T. W. Chaundy,
*Expansions of Appell’s double hypergeometric functions(II)*, The Quarterly Journal of Mathematics, Oxford, Ser.**12**, 112-128, 1941. - T. W. Chaundy,
*Expansions of hypergeometric functions*, Qiart. J. Math. Oxford Ser.**13**, 159-171, 1942. - J. Choi and A. Hasanov,
*Applications of the operator to the Humbert double hypergeometric functions*, Comput. Math. Appl.**61**, 663-671, 2011. - J. Choi and A. Hasanov,
*Certain decomposition formulas of generalized hypergeometric functions and some formulas of an analytic continuation of the Clausen function*, Commun. Korean Math. Soc.**27 (1)**, 107-116, 2012. - A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
*Higher Transcendental Functions*, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953. - T. G. Ergashev,
*On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients*, Comput. Math. Appl.**77 (1)**, 69-76, 2019. - 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 – 57, 2020. - 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. - T. G. Ergashev and A. Hasanov,
*Fundamental solutions of the bi-axially symmetric Helmholtz equation*, Uzbek Math. J.**1**, 55-64, 2018. - F. I. Frankl,
*Selected Works in Gas Dynamics*, Nauka, Moscow, 1973. - A. Hasanov,
*Fundamental solutions bi-axially symmetric Helmholtz equation*, Complex Var. Elliptic Equ.**52 (8)**, 673-683, 2007. - A. Hasanov and E. T. Karimov,
*Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients*, Appl. Math. Lett.**22**, 1828-1832, 2009. - A. Hasanov and H. M. Srivastava,
*Decomposition formulas associated with the Lauricella function*, Appl. Math. Lett.*F*_{A}^{(r)}and other multiple hypergeometric functions**19 (2)**, 113-121, 2006. - A. Hasanov and H. M. Srivastava,
*Decomposition formulas associated with the Lauricella multivariable hypergeometric functions*, Comput. Math. Appl.**53 (7)**, 1119-1128, 2007. - G. Lauricella,
*Sulle funzione ipergeometriche a più variabili*, Rend. Circ. Mat. Palermo**7**, 111-158, 1893. - G. Lohöfer,
*Theory of an electromagnetically deviated metal sphere*, I: Absorbed power, SIAM J. Appl. Nath.**49**, 567-581, 1989. - A. W. Niukkanen,
*Generalized hypergeometric series*, J. Phys. A: Math. Gen.^{N}*F*(*x*_{1},…,*x*_{N}) arising in physical and quantum chemical applications**16**, 1813-1825, 1983. - S. B. Opps, N. Saad and H. M. Srivastava,
*Some reduction and transformation formulas for the Appell hypergeometric functions*, J. Math. Anal. Appl.*F*_{2}**302**, 180-195, 2005. - P. A. Padmanabham and H. M. Srivastava,
*Summation formulas associated with the Lauricella function*, Appl. Math. Lett.*F*_{A}^{(r)}**13 (1)**, 65-70, 2000. - H. M. Srivastava,
*A class of generalised multiple hypergeometric series arising in physical and quantum chemical applications*, J. Phys. A: Math. Gen.**18**, L227-L234, 1985. - H. M. Srivastava,
*Reduction and summation formulae for certain classes of generalised multiple hypergeometric series arising in physical and quantum chemical applications*, J. Phys. A: Math. Gen.**18**, 3079-3085, 1985. - H. M. Srivastava,
*Neumann expansions for a certain class of generalised multiple hypergeometric series arising in physical and quantum chemical applications*, J. Phys. A: Math. Gen.**20**, 847-855, 1987. - H. M. Srivastava,
*Some Clebsch-Gordan type linearisation relations and other polynomial expansions associated with a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications*, J. Phys. A: Math. Gen.**21**, 4463-4470, 1988. - H. M. Srivastava and P. W. Karlsson,
*Multiple Gaussian Hypergeometric Series*, Halsted Press, New York, Chichester, Brisbane and Toronto, 1985. - H. M. Srivastava and A. W. Niukkanen,
*Some Clebsch-Gordan type linearization relations and associated families of Dirichlet integrals*, Math. and Comput. Model.**37**, 245-250, 2003. - A. K. Urinov and E. T. Karimov,
*On fundamental solutions for 3D singular elliptic equations with a parameter*, Appl. Math. Lett.**24**, 314-319, 2011. - R. J. Weinacht,
*Fundamental solutions for a class of singular equations*, Contrib. Differential Equations**3**, 43-55, 1964.