Certain hypergeometric series identities deducible by fractional calculus
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Motivated by the recent investigations of several authors, in this paper we present a generalization of a result obtained recently by Ibrahim et al. (A. K. Ibrahim, M. A. Rakha and A. K. Rathie, On certain hypergeometric identities deducible by using the beta integral method, Adv. Di erence Equ. 2013: 341 (2013), 1-8) involving hypergeometric identities. The result is obtained by suitably applying fractional calculus method to a generalization of a quadratic transformation for the Gauss hypergeometric function due to Kummer.
Abstract
Motivated by the recent investigations of several authors, in this paper we present a generalization of a result obtained recently by Ibrahim et al. (A. K. Ibrahim, M. A. Rakha and A. K. Rathie, On certain hypergeometric identities deducible by using the beta integral method, Adv. Di erence Equ. 2013: 341 (2013), 1-8) involving hypergeometric identities. The result is obtained by suitably applying fractional calculus method to a generalization of a quadratic transformation for the Gauss hypergeometric function due to Kummer.
참고문헌 (22)
- B. C. Berndt, "Ramanujan's Notebooks. Parts II" (1989), Springer-Verlag
- 최준상, "Certain hypergeometric identities deducible by using the beta integral method", 대한수학회보, 50(5), pp.1673~1681 (2013), 대한수학회
- A. Erdelyi, "An integral equation involving Legendre polynomials", SIAM J. Appl. Math, 12, pp.15~30 (1964)
- Sebastien Gaboury, "Further hypergeometric identities deducible by fractional calculus", 대한수학회논문집, 29(3), pp.429~437 (2014), 대한수학회
- S. Gaboury, "Remark on certain transformations for multiple hypergeometric functions", Adv. Difference Equ, 2014(252), pp.12~ (2014)
- A. K. Ibrahim, "On certain hypergeometric identities deducible by using the beta integral method", Adv. Difference Equ, 361, pp.1~8 (2013)
- A. A. Kilbas, "Theory and Applications of Frac-tional Differential Equations, North-Holland Mathematical Studies, Vol. 204" (2006), Elsevier (North-Holland) Science Publishers
- Y. S. Kim, "Generalizations of Kummers second theorem with applications", Comput. Math. Math. Phys, 50, pp.387~402 (2010)
- C. Krattenthaler, "Automatic generation of hypergeometric identities by the beta integral method", J. Comput. Appl. Math, 160, pp.159~173 (2003)
- E. E. Kummer, "Uber die hypergeometrische Reihe 1 +αβ", 1·γ x + α(α+1)·β(β+1), J. Reine Angew. Math, pp.1836, 15[DOI]
- J.-L. Lavoie, "Lecture Notes in Mathematics" (1976), Springer-Verlag
- J. Liouville, "Memoire sur le calcul des differentielles a indices quelconques", J. de l' Ecole Polytechnique, 13, pp.71~162 (1832)
- K. S. Miller, "An Introduction of the Fractional Calculus and Fractional Differential Equations" (1993), John Wiley and Sons
- T. J. Osler, "Leibniz rule, the chain rule and Taylor's theorem for fractional deriva-tives", Ph.D., pp.1970
- T. J. Osler, "Fractional derivatives of a composite function", SIAM J. Math. Anal, 1, pp.288~293 (1970)
- T. J. Osler, "Leibniz rule for the fractional derivatives and an application to infinite series", SIAM J. Appl. Math, 18, pp.658~674 (1970)
- T. J. Osler, "Fractional derivatives and Leibniz rule", Amer. Math. Monthly, 78, pp.645~649 (1971)
- E. D. Rainville, "Special functions" (1960), Macmillan Company
- M. Riesz, "L'integrale de Riemann-Liouville et le probleme de Cauchy", Acta. Math, 81, pp.1~222 (1949)
- H. M. Srivastava, "Zeta and q-Zeta Functions and Associated Series and Integrals" (2012), Elsevier Science Publishers
- R. Tremblay, "Une contribution a la theorie de la derivee fractionnaire", Ph.D., pp.1974
- C. Wei, "Certain transformations for multiple hypergeometric functions", Adv. Difference Equ, 360, pp.1~13 (2013)