# FURTHER HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY FRACTIONAL CALCULUS

• Gaboury, Sebastien (Department of Mathematics and Computer Science University of Quebec at Chicoutimi) ;
• Rathie, Arjun K. (Department of Mathematics School of mathematical and physical sciences Central University of Kerala Riverside Transit Campus)
• Published : 2014.07.31
• 82 5

#### Abstract

Motivated by the recent investigations of several authors, in this paper we present a generalization of a result obtained recently by Choi et al. ([3]) involving hypergeometric identities. The result is obtained by suitably applying fractional calculus method to a generalization of the hypergeometric transformation formula due to Kummer.

#### Keywords

fractional derivatives;generalized hypergeometric function;Kummer's formula;beta integral method;generalization of Kummer's formula

#### References

1. A. Erdelyi, An integral equation involving Legendre polynomials, SIAM J. Appl. Math. 12 (1964), 15-30. https://doi.org/10.1137/0112002
2. B. C. Berndt, Ramanujan's Notebooks. Parts II, Springer-Verlag, Berlin, Heidelberg and New York, 1989.
3. J. Choi, A. K. Rathie, and H. M. Srivastava, A generalization of a formula due to Kummer, Integral Transforms Spec. Funct. 22 (2011), no. 11, 851-859. https://doi.org/10.1080/10652469.2011.588786
4. J. Choi, A. K. Rathie, and H. M. Srivastava, Certain hypergeometric identities deducible by using the beta integral method, Bull. Korean Math. Soc. 50 (2013), no. 5, 1673-1681. https://doi.org/10.4134/BKMS.2013.50.5.1673
5. S. Gaboury, Some relations involving generalized Hurwitz-Lerch zeta function obtained by means of fractional derivatives with applications to Apostol-type polynomials, Adv. Difference Equ. 2013 (2013), 361.
6. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
7. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Chichester, Brisbane, Toronto and Singapore, John Wiley and Sons, 1993.
8. Y. S. Kim and A. K. Rathie, Some results for terminating $_2F_1$(2) series, J. Inequal. Appl. 365 (2011), 1-12.
9. C. Krattenthaler and K. S. Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math. 160 (2003), no. 1-2, 159-173. https://doi.org/10.1016/S0377-0427(03)00629-0
10. E. E. Kummer, Uber die hypergeometrische Reihe $1+\frac{{\alpha}{\beta}}{1{\cdot}{\gamma}}x+\frac{{\alpha}({\alpha}+1){\cdot}{\beta}({\beta}+1)}{1{\cdot}2{\cdot}{\gamma}({\gamma}+1)}x^2{\cdot}{\cdot}{\cdot}$, J. Reine Angew. Math. 15 (1836), 39-83; 127-172.
11. J.-L. Lavoie, T. J. Osler, and R. Tremblay, Fundamental properties of fractional derivatives via Pochhammer integrals, Lecture Notes in Mathematics, Springer-Verlag, 1975.
12. J. Liouville, Memoire sur le calcul des differentielles a indices quelconques, J. de l'Ecole Polytechnique 13 (1832), 71-162.
13. T. J. Osler, Fractional derivatives of a composite function, SIAM J. Math. Anal. 1 (1970), 288-293. https://doi.org/10.1137/0501026
14. T. J. Osler, Leibniz rule for fractional derivatives and an application to innite series, SIAM J. Appl. Math. 18 (1970), 658-674. https://doi.org/10.1137/0118059
15. T. J. Osler, Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives, PhD thesis, New York University, 1970.
16. T. J. Osler, Fractional derivatives and Leibniz rule, Amer. Math. Monthly 78 (1970), 645-649.
17. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960.
18. M. Riesz, L'integrale de Riemann-Liouville et le probleme de Cauchy, Acta. Math. 81 (1949), 1-223. https://doi.org/10.1007/BF02395016
19. H. M. Srivastava and J. Choi, Series Associated with Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
20. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
21. R. Tremblay, S. Gaboury, and B.-J. Fugere, A new leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), no. 2, 111-128.
22. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwoor Limited), John Wiley and Sons, Chichester and New York, 1985.
23. R. Tremblay, Une contribution a la theorie de la derivee fractionnaire, PhD thesis, Laval University, Canada, 1974.
24. R. Tremblay and B.-J. Fugere, The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions, Appl. Math. Comput. 187 (2007), no. 1, 507-529. https://doi.org/10.1016/j.amc.2006.09.076
25. R. Tremblay, S. Gaboury, and B.-J. Fugere, A new transformation formula for fractional derivatives with applications, Integral Transforms Spec. Funct. 24 (2013), no. 3, 172-186. https://doi.org/10.1080/10652469.2012.672323
26. R. Tremblay, S. Gaboury, and B.-J. Fugere, Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), no. 1, 50-64. https://doi.org/10.1080/10652469.2012.665910
27. C. Wei, X. Wang, and Y. Li, Certain transformations for multiple hypergeometric functions, Adv. Difference Equ. 360 (2013), 1-13.