Title: Decomposition Formulas for Second-Order Quadruple Gaussian Hypergeometric Series by Means of Operators H(α,β) and H(α,β)
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00019; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 41-60
Document Type: Research Paper
aInstitute of Mathematics, Uzbek Academy of Sciences, 81 Mirzo-Ulugbek street, Tashkent 700170, Uzbekistan
bNational Pedagogical University, 86 Tole bi street, Almaty 0500012, Kazakhstan
cDepartment of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
Received: 2 February 2021, Accepted: 15 July 2021, Published: 30 July 2021.
Corresponding Author: Junesang Choi (Email address: email@example.com)
Full Text: PDF
Numerous decomposition formulas for various hypergeometric functions of several variables have been offered. In this paper, we aim to establish symbolic operator identities and decomposition formulas for second-order quadruple Gaussian hypergeometric series associated with Appell functions and Saran hypergeometric functions by mainly using mutually inverse symbolic operators H(α,β) and H(α,β), which were introduced in an earlier work. Mellin-Barnes type contour integrals are employed for proofs of the operator identities. Also we determine the regions of convergence of the 14 quadruple Gaussian hypergeometric series.
Keywords: Hypergeometric functions, Multiple hypergeometric functions, Inverse pairs of symbolic operators, Decomposition formulas, Mellin-Barnes contour integralsReferences:
- P. App ell, J. Kampé de Fériet, Fonctions hypergétriques et hypersphriques, polynômes d’Hermite, Gauthier-villars, Paris, 1926.
- L. Bers, Mathematical aspects of subsonic and transonic gas dynamics, John Wiley and Sons, 176, New York, 1958.
- J. L. Burchnall, T. W. Chaundy, Expansions of Appell’s double hypergeometric functions, Quart. J. Math (Oxford) 11 (1), 249–270, 1940. https://doi.org/10.1093/qmath/os-11.1.249
- J. L. Burchnall, T. W. Chaundy, Expansions of Appell’s double hypergeometric functions (II), Quart. J. Math (Oxford) 12 (1), 112–128, 1941. https://doi.org/10.1093/qmath/os-12.1.112
- P. Candelas, X. C. De La Ossa, P. S. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1), 21–74, 1991. https://doi.org/10.1016/0550-3213(91)90292-6
- T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math (Oxford) 13 (1), 159–171, 1942. https://doi.org/10.1093/qmath/os-13.1.159
- J. Choi, A. Hasanov, Applications of the operator H(α, β) to the Humbert double hypergeometric functions, Comput. Math. Appl. 61, 663–-671, 2011. doi:10.1016/j.camwa.2010.12.012
- J. Choi, A. Hasanov, Certain decomposition formulas of generalized hypergeometric functions pFq and some formulas of an analytic continuation of the Clausen function 3F2, Commun. Korean Math. Soc. 27 (1), 107–116, 2012. http://dx.doi.org/10.4134/CKMS.2012.27.1.107
- J. Choi, Y. S. Kim, A. Hasanov, Relations between the hypergeometric function of Appell F3 and Kampé de Fériet functions, Miskolc Math. Notes 12 (2), 131–148, 2011.
- V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang-Baxter equations, Reports of the USSR Academy of Sciences 268 (2), 285–287, 1983.
- V. G. Drinfeld, On constant quasiclassical solutions of the quantum Yang-Baxter equation, Reports of the USSR Academy of Sciences 273 (3), 531–535, 1983.
- V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Reports of the USSR Academy of Sciences 283 (5), , 1060–1064, 1985.
- V. G. Drinfeld, Quantum groups, Zap. scientific Semin. Leningrad Dep. Mat., Institute of the USSR Academy of Sciences 155, 19–49, 1986.
- 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.
- T. G. Ergashev, A. Hasanov, Fundamental solutions of the bi-axially symmetric Helmholtz equation, Uzbek Math. J. 1, 55–64, 2018.
- H. Exton, Certain hypergeometric functions of four variables, Bull. Soc. Math. Grece (N.S.) 13, 104–113, 1972.
- H. Exton, Some integral representations and transformations of hypergeometric functions of four variables, Bull. Soc. Math. Grece (N.S.) 14, 132–140, 1973.
- H. Exton, Multiple hypergeometric functions and applications, Ellis Horwood Ltd., John Wiley and Sons, London, New York, Sydney, Toronto, 1976.
- F. I. Frankl, Selected works in gas dynamics, Nauka. Moscow (in Russian), 1973.
- A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Var. Elliptic Equ. 52 (8), 673–-683, 2007.
- A. Hasanov, The solution of the Cauchy problem for generalized Euler–Poisson–Darboux equation, Inter. J. Appl. Math. Statist. 8 (7), 30–-44, 2007.
- A. Hasanov, On a mixed problem for the equation signy|y|muxx + xnuyy = 0, Izv. Akad. Nauk UzSSR, ser. Fiz.-mat. Nauk 2, 28–-32, 1982 (in Russian).
- A. Hasanov, H. M. Srivastava, Decomposition formulas associated with the Lauricella multivariable hypergeometric functions, Comput. Math. Appl. 53 (7), 1119–1128, (2007). https://doi.org/10.1016/j.camwa.2006.07.007
- A. Hasanov, H. M. Srivastava, Some decomposition formulas associated with the Lauricella function FA(r) and other multiple hypergeometric functions, Appl. Math. Lett. 19 (2), 113–121, 2006. https://doi.org/10.1016/j.aml.2005.03.009
- A. Hasanov, H. M. Srivastava, M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl. 324 (2), 955–969, 2006. https://doi.org/10.1016/j.jmaa.2006.01.006
- A. Hasanov, M. Turaev, J. Choi, Decomposition formulas for the generalized hypergeometric 4F3 function, Honam Math. J. 32 (1), 1–16, 2010.
- R. P. Horja, Hypergeometric functions and mirror symmetry in toric varieties, Preprint. math., AG/9912109, 1-103, 1999.
- G. Lohofer, Theory of an electromagnetically levitated metal sphere 1: Absorbed power, SIAM J. Appl. Math. 49 (2), 567–-581, 1989.
- O. I. Marichev, Handbook of integral transforms of higher transcendental functions: Theory and Algorithmic Tables, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Brisbane, Chichester and Toronto, 1982.
- A. M. Mathai, R. K. Saxena, Generalized hypergeometric functions with applications in statistics and physical sciences, Springer-Verlag, Berlin, Heidelberg, New York, Vol. 348, 1973.
- A. W. Niukkanen, Generalised hypergeometric series NF(x1,…,xN) arising in physical and quantum chemical applications, J. Phys. A: Math. Gen. 16, 1813–-1825, 1983. DOI: 10.1088/0305-4470/16/9/007
- M. Passare, A. Tsikh, A. A. Cheshel, Multiple Mellin-Barnes integrals as periods of Calabi-Yau manifolds with several moduli, Theor. Math. Phys. 109 (3), 1544–1555, 1996.
- M. Passare, A. Tsikh, O. Zhdanov, A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals, Aspects Math. E. 26, 233–241, 1994.
- E. G. Poole, Introduction to the theory of linear differential equations, Clarendon (Oxford University) Press, Oxford, 1936.
- S. Saran, Relations between functions contiguous to certain hypergeometric functions of three variables, Ganita 5, 69–76, 1954.
- S. Saran, Transformations of certain hypergeometric functions of three variables, Acta Math. 93 (3-4), 292–312, 1955.
- R. B. Seilkhanova, A. Hasanov, Particular solutions of generalized Euler-Poisson-Darboux equation, Electron. J. Diff. Equ. 9, 1–10, 2015.
- C. Sharma, C. L. Parihar, Hypergeometric functions of four variables (I)., J. Indian Acad. Math. 11 (2), 99–115, 1989.
- I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry, 3rd ed., 182, Longman, London, New York, 1980.
- H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
- H. M. Srivastava, A. Hasanov, J. Choi, Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation, Sohag J. Math. 2 (1), 1–10, 2015.
- H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985.
- H. M. Srivastava, B. R. K. Kashyap, Special functions in queuing theory and related stochastic processes, Academic Prees, 308, New York, London, San Francisco, 1982.
- A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics 21, 371, World Scientific, Singapore, 1995.