# SOME DECOMPOSITION FORMULAS ASSOCIATED WITH THE SARAN FUNCTION FE

Kim, Yong-Sup;Hasanov, Anvar;Lee, Chang-Hyun

• Accepted : 2010.11.03
• Published : 2010.12.25
• 25 2

#### Abstract

With the help of some techniques based upon certain inverse pairs of symbolic operators initiated by Burchnall-Chaundy, the authors investigate decomposition formulas associated with Saran's function $F_E$ in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By employing their decomposition formulas, we also present a new group of integral representations for the Saran function $F_E$.

#### Keywords

Generalized hypergeometric series;Inverse pairs of symbolic operators;Decomposition formulas;Srivastava's triple hypergeometric functions;Gauss function;Appell functions;Integral representations

#### References

1. P. Appell and J. Kampe de Feriet, Finctions Hypergeometriques et Hyper-spheriques; Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
2. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. Oxford Ser. 11 (1940), 249-270. https://doi.org/10.1093/qmath/os-11.1.249
3. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions. II, Quart. J. Math. Oxford Ser. 12(1941), 112-128. https://doi.org/10.1093/qmath/os-12.1.112
4. T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942). 159-171. https://doi.org/10.1093/qmath/os-13.1.159
5. 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.
6. A. Erdelyi, W. Megnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2. McGraw-Hill Book Company, New York, Toronto and London, 1953.
7. A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations 52(8) (2007), 673-683. https://doi.org/10.1080/17476930701300375
8. A. Hasanov and E. T. Karimov, Fundamental solutions for a class of three-dimemsional elliptic equations with singular coefficients, Appl. Math. Lett. 22 (2009), 1828-1832. https://doi.org/10.1016/j.aml.2009.07.006
9. A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricella function $F_{A}^{(r)}$ and other multiple hypergeometric functions, Appl. Math. Lett. 19 (2006), 113-121. https://doi.org/10.1016/j.aml.2005.03.009
10. A. Hasanov and H. M. Srivastava, Decomposition formulas associated with the Lauricella multivariable hypergeometric functions, Comput. Math. Appl. 53(7) (2007), 1119-1128. https://doi.org/10.1016/j.camwa.2006.07.007
11. A. Hasanov, H. M. Srivastava, and M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl. 324 (2006), 955-969. https://doi.org/10.1016/j.jmaa.2006.01.006
12. G. Lauricella, Sulle funzioni ipergeomtriche a piu variablli, Rend. Cire. Mat. Palermo 7 (1893), 111-158. https://doi.org/10.1007/BF03012437
13. 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.
14. R. C. Pandey, On the expansions of hypergecmetric functions, Agra Univ. J. Res. Sci. 12 (1963), 159-169.
15. E. G. Poole, Introduction to the Theory of Linear Differential Equations, Clarendon (Oxford University Press), Oxford, 1936.
16. S. Saran, Hypergeometric functions of three variables, Ganita 5 (1956), 77-91.
17. J . P. Singhal and S. S. Bhaiti, Certain expansions associated with hypergeometric functions of n variables, Glasnik Math. Ser. III 11(31) (1976), 239-245.
18. H. M. Srivastava, Some integrals representing triple hypergeometric functions, Rend. Circ. Mat. Palermo(Ser. 2) 7 (1967), 99-115. https://doi.org/10.1007/BF02844089
19. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, 1985.
20. H. M. Srivastava and H. L. Monocha, A treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, 1984.

#### Cited by

1. APPLICATION OF THE OPERATOR H (α, β) TO THE SARAN FUNCTION FEAND SOME OTHER RESULTS vol.33, pp.4, 2011, https://doi.org/10.5831/HMJ.2011.33.4.441