CERTAIN INTEGRAL TRANSFORMS AND FRACTIONAL INTEGRAL FORMULAS FOR THE EXTENDED HYPERGEOMETRIC FUNCTIONS

Abstract

In this present paper, we derive various integral transforms, including Euler, Varma, Laplace, and Whittaker integral transforms for the extended hypergeometric functions which has recently been introduced by Choi et al.[3]. Further, we also apply Saigo's fractional integral operators for this extended hypergeometric function. Some interesting special cases of our main results are also considered.

Keywords

• extended beta function,extended Gauss hypergeometric functions,integral transforms,fractional integral operators.

References

1. Chaudhary,M.A., Qadir,A., Srivastava,H.M., and Paris,R.B., (1997), Extended hypergeometric and confluent hypergeometric functions, J. Comput. Appl. Math., 78(1), pp.19-32.
2. Chaudhary,M.A., Qadir,A., Rafique,M., and Zubair,S.M., (2004), Extension of Euler’s beta function, Appl. Math. Comput., 159(2), pp.589-602.
3. Choi,J., Rathie,A.K., and Parmar,R.K., (2014), Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36(2), pp.339-367.
4. Kilbas,A.A., Srivastava,H.M., and Trujillo,J.J., (2006), Theory and Applications of Fractional Dif- ferential Equations, North-Holland Mathematical Studies, Elsevier Science, Amsterdem, The Nether- lands.
5. Kachhia,K.B., Prajapati,J.C., Purohit,S.D., and Parmar,R.K., Integral transforms and fractional in- tegral formulas for the generalized hypergeometric functions, submitted.
6. Lee,D.M., Rathie,A.K., Parmar,R.K., and Kim,Y.S., (2011), Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Math. J., 33(2), pp.187-206.
7. Mathai,A.M. and Saxena,R.K. (1973), Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Lecture Notes Series No. 348, Heidelberg.
8. Mathai,A.M., Saxena,R.K., and Haubold,H.J., (2010), The H-Function Theory and Applications, Springer-Verlag, New York.

9. ¨Ozergin,E., ¨Ozarslan,M.A., and ALtin,A., (2011), Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235(16), pp.4601-4610.
10. Parmar,R.K., (2013), A new generalization of Gamma, Beta, hypergeometric and confluent hyperge- ometric functions, Matematiche (Catania), 69, pp.33-52.
11. Parmar,R.K., (2014), Some generating relations for generalized extended hypergeometric functions involving generalized fractional derivative operator, Journal of Concrete and Applicable Mathematics, 12, pp.217-228.
12. Pohlen,T., (2009), The Hadamard Product and Universal Power Series, Dissertation, Universit¨at Trier. [13] Sneddon,I.N., (1979), The Use of Integral Transform, Tata McGraw-Hill, New Delhi, India.
13. Srivastava,H.M. and Karlsson,P.W., (1985), Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto.