On Decomposition Formulas Related to the Gaussian Hypergeometric Functions in Three Variables

Hasanov, Anvar; Younis, Jihad; Aydi, Hassen

doi:https://doi.org/10.1155/2021/5538583

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Volume 2021 | Article ID 5538583 | https://doi.org/10.1155/2021/5538583

On Decomposition Formulas Related to the Gaussian Hypergeometric Functions in Three Variables

Anvar Hasanov,^1,2Jihad Younis,³and Hassen Aydi^4,5,6

Academic Editor: Gangadharan Murugusundaramoorthy

Received17 Feb 2021

Revised04 Mar 2021

Accepted06 Mar 2021

Published16 Mar 2021

Abstract

In this paper, by using certain inverse pairs of symbolic operators introduced by Choi and Hasanov in 2011, we establish several decomposition formulas associated with the Gaussian triple hypergeometric functions. Some transformation formulas for these functions have also been obtained.

1. Introduction

The use of many mathematical operations goes beyond the class of elementary functions. The calculation of integrals, the summation of series, and solution of algebraic, transcendental, difference, and differential equations and their systems require expanding the class of studied functions. The development of the concept of a function, going in parallel with the development of the concepts of number and space, led to the emergence of new hypergeometric functions of many complex variables.

The great success of the theory of hypergeometric functions in a single variable has stimulated the development of the theory of hypergeometric functions in several variables by the fact that the solutions of partial differential equations arising in many applied problems of mathematical physics are given in terms of such hypergeometric functions (see, e.g., [1–6]). Multiple hypergeometric functions occur in numerous problems in hydrodynamics, control theory, electrical current, heat conduction, and classical and quantum mechanics (see, for details, [7–10], and the references cited therein). In view of theory and applications, a large number of hypergeometric functions have been developed; for example, as many as 205 hypergeometric functions are recorded in the monograph [11]. In particular, we recall the Gaussian functions , and in three variables defined by (see [11])

Here, denotes the Pochhammer symbol given as

Burchnall and Chaundy presented the inverse pairs of symbolic operators and [12, 13] (also see [14]) by means of which they established several decomposition formulas for Appell’s double hypergeometric functions in terms of the Gaussian hypergeometric functions in one variable. Recently, Hasanov and Srivastava [15, 16] introduced multivariable analogues of Burchnall-Chaundy’s symbolic operators, and with the help of these operators, the authors obtained a number of decomposition formulas associated with multiple Lauricella hypergeometric functions , and . In [17, 18], the authors gave the following multivariable symbolic operators:

Based on the operators (12) and (13), we aim in this work to establish certain decomposition formulas for second-order Gaussian hypergeometric functions in three variables (1), (2), (3), (4), (5), (6), (7), (8), (9), and (10), which are used to derive some interesting transformation formulas for these functions.

2. Symbolic Form

Applying the symbolic operators in (12) and (13), we construct the following set of operator identities involving the classical Gauss hypergeometric function [19], the Appell functions [20], the Horn functions [21], and the Gaussian triple hypergeometric functions defined in (1), (2), (3), (4), (5), (6), (7), (8), (9), and (10):

Each of the operator identities (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33), and (34) can be proved by means of Mellin and inverse Mellin transformation (see, for example, [11, 20, 22]). The proofs of the operator identities are omitted here.

3. Decomposition Formulas

In [23] (p. 93), it is proved that, for every analytic function , the following formulas hold true: where

In view of formulas (35) and (36), and taking into account the differentiation formula for hypergeometric functions, from operator identities (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33), and (34), we have

4. Transformation Formulas

The transformation formulas defined below follow from the expansion formulas (38), (39), (40), (41), (42), (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), and (57):

5. Concluding Remarks

In this present paper, with the help of the inverse pairs of symbolic operators, we established a number of decomposition formulas for some Gaussian triple hypergeometric functions. Also, we investigated certain transformation formulas for these functions. We conclude that mutually inverse operators (12) and (13) can be applied to other multiple hypergeometric functions.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

References

A. Hasanov, “Fundamental solutions of generalized Helmholtz equation,” Reports of Uzbek Academy of Sciences, vol. 1, pp. 13–15, 2006.
View at: Google Scholar
A. Hasanov, “Fundamental solutions of generalized bi-axially symmetric Helmholtz equation,” Complex Variables and Elliptic Equations, vol. 52, no. 8, pp. 673–683, 2007.
View at: Publisher Site | Google Scholar
A. Hasanov and E. T. Karimov, “Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1828–1832, 2009.
View at: Publisher Site | Google Scholar
S. B. Opps, N. Saad, and H. M. Srivastava, “Some reduction and transformation formulas for the Appell hypergeometric function F₂,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 180–195, 2005.
View at: Publisher Site | Google Scholar
P. A. Padmanabham and H. M. Srivastava, “Summation formulas associated with the Lauricella function ,” Applied Mathematics Letters, vol. 13, no. 1, pp. 65–70, 2000.
View at: Publisher Site | Google Scholar
A. Saboor, G. Rahman, Z. Anjum, K. S. Nisar, and S. Araci, “A new extension of Srivastava’s triple hypergeometric functions and their associated properties,” Analysis, vol. 41, no. 1, pp. 13–24, 2021.
View at: Publisher Site | Google Scholar
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, NY, USA, 1958.
F. I. Frankl, Selected Works in Gas Dynamics, Nauka, Moscow, 1973.
G. Lohöfer, “Theory of an electromagnetically levitated metal sphere I: absorbed power,” SIAM Journal on Applied Mathematics, vol. 49, no. 2, pp. 567–581, 1989.
View at: Publisher Site | Google Scholar
A. W. Niukkanen, “Generalised hypergeometric series ^NF (x₁,…,x_N) arising in physical and quantum chemical applications,” Journal of Physics A: Mathematical and General, vol. 16, no. 9, pp. 1813–1825, 1983.
View at: Publisher Site | Google Scholar
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Lt1., Chichester, 1984.
J. L. Burchnall and T. W. Chaundy, “Expansions of Appell's double hypergeometric functions,” The Quarterly Journal of Mathematics, vol. 11, pp. 249–270, 1940.
View at: Publisher Site | Google Scholar
J. L. Burchnall and T. W. Chaundy, “Expansions of Appell's double hypergeometric functions II,” The Quarterly Journal of Mathematics, vol. 12, pp. 112–128, 1941.
View at: Publisher Site | Google Scholar
T. W. Chaundy, “Expansions of hypergeometric functions,” The Quarterly Journal of Mathematics, vol. 13, pp. 159–171, 1942.
View at: Publisher Site | Google Scholar
A. Hasanov and H. M. Srivastava, “Some decomposition formulas associated with the Lauricella function and other multiple hypergeometric functions,” Applied Mathematics Letters, vol. 19, pp. 113–121, 2006.
View at: Publisher Site | Google Scholar
A. Hasanov and H. M. Srivastava, “Decomposition formulas associated with the Lauricella multivariable hypergeometric functions,” Computers & Mathematcs with Applications, vol. 53, no. 7, pp. 1119–1128, 2007.
View at: Publisher Site | Google Scholar
J. Choi and A. Hasanov, “Applications of the operator H(α, β) to the Humbert double hypergeometric functions,” Computers & Mathematcs with Applications, vol. 61, no. 3, pp. 663–671, 2011.
View at: Publisher Site | Google Scholar
J. S. Choi, Y. S. Kim, and A. Hasanov, “Application of the operator H (α, β) to the saran function feand some other results,” Honam Mathematical Journal, vol. 33, no. 4, pp. 441–452, 2011.
View at: Publisher Site | Google Scholar
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
P. Appell and J. Kampé de Fériet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier - Villars, Paris, 1926.
J. Horn, “Üeber die Convergenz der hypergeometrischen Reihen Zweier und Dreier Veränderlichen,” Mathematische Annalen, vol. 34, no. 4, pp. 544–600, 1889.
View at: Publisher Site | Google Scholar
O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane and Toronto, 1982.
E. G. Poole, Introduction to the Theory of Linear Differential Equations, Clarendon (Oxford University) Press, Oxford, 1936.

Copyright

Copyright © 2021 Anvar Hasanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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