FURTHER EXPANSION FORMULAS FOR A CLASS OF GENERALIZED HURWITZ-LERCH ZETA FUNCTIONS OBTAINED BY MEANS OF A NEW LEIBNIZ RULE FOR FRACTIONAL DERIVATIVES
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The aim of this paper is to make use of a new generalized Leibniz rule for fractional derivatives obtained recently by Tremblay et al. [Tremblay, Gaboury and Fugere, A new Leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), 111-128] by means of a representation based on the Pochhammer's contour of integration for fractional derivatives in order to derive new expansion formulas for several families of the Hurwitz- Lerch zeta function. Special cases are also given.
Abstract
The aim of this paper is to make use of a new generalized Leibniz rule for fractional derivatives obtained recently by Tremblay et al. [Tremblay, Gaboury and Fugere, A new Leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), 111-128] by means of a representation based on the Pochhammer's contour of integration for fractional derivatives in order to derive new expansion formulas for several families of the Hurwitz- Lerch zeta function. Special cases are also given.
참고문헌 (26)
- A. Bayad, "Reduction and duality of the generalized Hurwitz-Lerch zetas", Fixed Point Theory and Applications, 2013, pp.82~ (2013)
- A. Erdelyi, "An integral equation involving Legendre polynomials", SIAM J. Appl. Math, 12, pp.15~30 (1964)
- A. Erdelyi, "Higher transcendental functions, vols.1-3" (1953), McGraw-Hill Book Company
- A. Erdelyi, "Tables of integral transforms, vols.1-2" (1953), McGraw-Hill Book Company
- M. Garg, "Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions", Integral Transform Spec. Funct, 17(11), pp.803~815 (2006)
- S. P. Goyal, "On the generalized Rieman Zeta function and the generalized Lambert transform", Ganita Sandesh, 11, pp.99~108 (1997)
- P.L. Gupta, "A class of Hurwitz-Lerch zeta distributions and their applications in reliability", Appl. Math. Comput, 196, pp.521~531 (2008)
- J.-L. Lavoie, "Fundamental properties of fractional deriva-tives via Pochhammer integrals" (1976)
- J.-L. Lavoie, "Fundamental properties of fractional derivatives via Pochhammer integrals" (1976), Springer-Verlag
- S.-D. Lin, "Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations", Appl. Math. Comput, 154, pp.725~733 (2004)
- S.-D. Lin, "Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions", Integral Transform Spec. Funct, 17, pp.817~827 (2006)
- J. Liouville, "Memoire sur le calcul des differentielles a indices quelconques", J. de l'Ecole Polytechnique, 13, pp.71~162 (1832)
- Y. Luke, "Mathematics in Science and Engineering" (1969), Academic Press
- K.S. Miller, "An introduction of the fractional calculus and fractional differential equations" (1993), John Wiley and Sons
- 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 innite series", SIAM J. Appl. Math, 18, pp.658~674 (1970)
- T.J. Osler, "Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives", Ph.D., pp.1970
- T.J. Osler, "Fractional derivatives Leibniz rule", Amer.Math.Monthly, 78, pp.645~649 (1971)
- D. Raducanu, "A new class of analytic functions dened by means of a convolution operator involving the Hurwitz-Lerch zeta function", Integral Trans-forms Spec. Funct, 18, pp.933~943 (2007)
- M. Riesz, "L'integrale de Riemann-Liouville et le probleme de Cauchy", Acta. Math, 81, pp.1~233 (1949)
- H.M. Srivastava, "Some formulas for the Bernoulli and Euler polynomials at rational arguments", Math. Proc. Cambridge Philos. Soc, 129, pp.77~84 (2000)
- H.M. Srivastava, "Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions", Russ. J. Math. Phys, 18(1), pp.107~121 (2011)
- H.M. Srivastava, "Series associated with zeta and related functions" (2001), Kluwer Academin Publishers
- R. Tremblay, "Une contribution a la theorie de la derivee fractionnaire", Ph.D., pp.1974
- R. Tremblay, "A new Leibniz rule and its integral ana-logue for fractional derivatives", Integral Transforms Spec. Funct, 24(2), pp.111~128 (2013)
- R. Tremblay, "Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives", Integral Transforms Spec. Funct, 24(1), pp.50~64 (2013)
