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Original Articles

Some families of series identities and associated fractional differintegral formulas

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Pages 737-749 | Published online: 03 Sep 2009

References

  • Buschman , R. G. and Srivastava , H. M. 1982 . Series identities and reducibility of Kampé de Fériet functions . Math. Proc. Cambridge Philos Soc. , 91 : 435 – 440 .
  • Carracedo , C. M. and Alix , M. S. 2001 . The Theory of Fractional Powers of Operators , Vol. 187 , Amsterdam, London and New York : Elsevier (North-Holland) Science Publishers . North-Holland Mathematical Studies
  • Chen , K.-Y. and Srivastava , H. M. 2005 . Series identities and associated families of generating functions . J. Math. Anal. Appl. , 311 : 582 – 599 .
  • Craven , T. and Csordas , G. 2006 . The Fox–Wright functions and Laguerre multiplier sequences . J. Math. Anal. Appl. , 314 : 109 – 125 .
  • Erdélyi , A. , Magnus , W. , Oberhettinger , F. and Tricomi , F. G. 1953 . Higher Transcendental Functions , Vol. I , New York, Toronto and London : McGraw-Hill Book Company .
  • Fox , C. 1928 . The asymptotic expansion of generalized hypergeometric functions . Proc. London Math. Soc. (Ser. 2) , 27 : 389 – 400 .
  • Fox , C. 1961 . The G and H functions as symmetrical Fourier kernels . Trans. Amer. Math. Soc. , 98 : 395 – 429 .
  • Fuenmayor , M. , Matera , J. and Salinas de Romero , S. 2008 . N-Fractional calculus of functions . J. Fract. Calc. , 34 : 87 – 98 .
  • Hilfer , R. 2000 . Applications of Fractional Calculus in Physics , New Jersey, London and Hong Kong : World Scientific Publishing Company, Singapore .
  • Kilbas , A. A. , Srivastava , H. M. and Trujillo , J. J. 2006 . Theory and Applications of Fractional Differential Equations , Vol. 204 , Amsterdam, London and New York : Elsevier (North-Holland) Science Publishers . North-Holland Mathematical Studies
  • Kiryakova , V. 1994 . Generalized Fractional Calculus and Applications , Vol. 301 , Harlow : Longman Scientific and Technical . Pitman Research Notes in Mathematics
  • Lin , S.-D. , Tu , S.-T. , Srivastava , H. M. and Wang , P.-Y. 2006 . Some families of multiple infinite sums and associated fractional differintegral formulas for power and composite functions . J. Fract. Calc. , 30 : 45 – 58 .
  • Lin , S.-D. , Nishimoto , K. , Miyakoda , T. and Srivastava , H. M. 2007 . Some differintegral formulas for power, composite and rational functions . J. Fract. Calc. , 32 : 87 – 98 .
  • Miller , K. S. and Ross , B. 1993 . An Introduction to the Fractional Calculus and Fractional Differential Equations , New York, Chichester, Brisbane, Toronto, , Singapore : A Wiley-Interscience Publication, John Wiley and Sons .
  • Miyakoda , T. 2007 . N-fractional calculus of certain algebraic functions . J. Fract. Calc. , 31 : 63 – 76 .
  • Miyakoda , T. 2007 . N-fractional calculus of functions . J. Fract. Calc. , 32 : 65 – 75 .
  • Miyakoda , T. 2008 . “ N-fractional calculus of some algebraic functions ” . In Study on Geometric Univalent Function Theory , 55 – 65 . Kyoto, , Japan : Research Institute for Mathematical Sciences, Kyoto University . RIMS Kôkyûroku 1579
  • Miyakoda , T. 2008 . N-fractional calculus and -th derivatives of some functions . J. Fract. Calc. , 34 : 23 – 33 .
  • Moriguchi , S. , Udagawa , K. and Hitotsumatsu , S. 1957 . Mathematical Formulae , Vol. 1 and 2 , Iwanami, , Japan : Iwanami Zensho .
  • Nishimoto , K. 1984, 1987, 1989 and 1991 . Fractional Calculus , Vol. I, II, III and IV , Koriyama : Descartes Press .
  • Nishimoto , K. 1991 . An Essence of Nishimoto's Fractional Calculus (Calculus of the 21st Century): Integrations and Differentiations of Arbitrary Order , Koriyama : Descartes Press .
  • Nishimoto , K. 2006 . N-fractional calculus of some composite functions . J. Fract. Calc. , 29 : 35 – 44 .
  • Nishimoto , K. N-fractional calculus of products of some power functions and some doubly infinite sums . Presented at the International Symposium on Analytic Function Theory, Fractional Calculus and Their Applications in Honour of Professor H. M. Srivastava held at the University of Victoria (Victoria, British Columbia, Canada) on 22–27 August 2005 . Appl. Math. Comput. 187 (2007), pp. 340–349
  • Nishimoto , K. 2007 . N-fractional calculus of some elementary functions and their semi-differintegrals . J. Fract. Calc. , 31 : 1 – 10 .
  • Nishimoto , K. 2007 . N-fractional calculus of some composite algebraic functions . J. Fract. Calc. , 31 : 11 – 23 .
  • Nishimoto , K. 2007 . N-fractional calculus and n-th derivatives of some functions . J. Fract. Calc. , 31 : 77 – 90 .
  • K. Nishimoto, N-fractional calculus of some functions which include a root sign, in Study on Geometric Univalent Function Theory, RIMS Kôkyûroku 1579, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, 2008, pp. 66–78 (see also 29 below)
  • Nishimoto , K. 2008 . “ N-fractional calculus of some multi-powers functions ” . In Study on Geometric Univalent Function Theory , 79 – 87 . Kyoto, , Japan : Research Institute for Mathematical Sciences, Kyoto University . RIMS Kôkyûroku 1579
  • Nishimoto , K. 2008 . N-Fractional calculus of some functions, which have a root sign . J. Fract. Calc. , 33 : 1 – 12 . (see also [27] above)
  • Nishimoto , K. 2008 . N-Fractional calculus of some functions, which have multiple root signs . J. Fract. Calc. , 33 : 35 – 46 .
  • Nishimoto , K. and Miyakoda , T. 2006 . N-fractional calculus of some products . J. Fract. Calc. , 30 : 13 – 22 .
  • Nishimoto , K. and Miyakoda , T. 2007 . N-fractional calculus and n-th derivatives of some algebraic functions . J. Fract. Calc. , 31 : 53 – 62 .
  • Nishimoto , K. and Miyakoda , T. 2008 . N-fractional calculus of some multiple power functions and some identities . J. Fract. Calc. , 34 : 11 – 22 .
  • Nishimoto , K. , Matera , J. , Fuenmayor , M. and Salinas de Romero , S. 2008 . N-fractional calculus of some functions which have some root signs . J. Fract. Calc. , 33 : 59 – 70 .
  • Ozaktas , H. M. , Zalevsky , Z. and Kutay , M. A. 2000 . The Fractional Fourier Transform with Applications in Optics and Signal Processing , New York, Chichester, Brisbane, Toronto, , Singapore : Wiley Series in Pure and Applied Optics, John Wiley and Sons .
  • Podlubny , I. 1999 . Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications , Vol. 198 , New York, London, Tokyo and Toronto : Academic Press . Mathematics in Science and Engineering
  • A.P. Prudnikov and Yu.A. Bryčkov and O.I. Maričev, Integrals and Series, Vol. 2: Special Functions, “Nauka”, Moscow, 1986 (in Russian); Translated from the Russian by N.M. Queen, 2nd ed., Gordon and Breach Science Publishers. New York, Philadelphia, London, Paris, Montreux, Tokyo and Melbourne, 1988
  • Rainville , E D. 1960 . Special Functions , New York : Macmillan Company . Reprinted by Chelsea Publishing Company, Bronx, New York, 1971
  • S.G. Samko and A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications (“Nauka i Tekhnika,” Minsk, 1987). Gordon and Breach Science Publishers, Reading, Tokyo, Paris, Berlin and Langhorne (Pennsylvania), 1993
  • Saxena , R. K. and Nishimoto , K. 2006 . N-fractional calculus of power functions . J. Fract. Calc. , 29 : 57 – 64 .
  • Saxena , R. K. and Nishimoto , K. 2007 . N-fractional calculus of the multivariable H-functions . J. Fract. Calc. , 31 : 43 – 52 .
  • Srivastava , H. M. and Buschman , R. G. 1992 . Theory and Applications of Convolution Integral Equations , Kluwer Series on Mathematics and Its Applications Vol. 79 , Dordrecht, Boston and London : Kluwer Academic Publishers .
  • Srivastava , H. M. and Karlsson , P. W. 1985 . Multiple Gaussian Hypergeometric Series , New York, Chichester, Brisbane and Toronto : John Wiley and Sons . Halsted Press (Ellis Horwood Limited, Chichester)
  • Srivastava , H. M. and Manocha , H. L. 1984 . A Treatise on Generating Functions , New York, Chichester, Brisbane and Toronto : John Wiley and Sons . Halsted Press (Ellis Horwood Limited, Chichester)
  • Srivastava , H. M. , Gupta , K. C. and Goyal , S. P. 1982 . The H-Functions of One and Two Variables with Applications , New Delhi and Madras : South Asian Publishers .
  • Srivastava , H. M. , Owa , S. and Nishimoto , K. 1985 . Some fractional differintegral equations . J. Math. Anal. Appl. , 106 : 360 – 366 .
  • Srivastava , H. M. , Lin , S.-D. and Wang , P.-Y. 2006 . Some fractional-calculus results for the -function associated with a class of Feynman integrals . Russian J. Math. Phys. , 13 : 94 – 100 .
  • E.T. Whittaker and G.N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, 4th ed. (reprinted). Cambridge University Press, Cambridge, London and New York, 1973
  • Wright , E. M. 1935 . The asymptotic expansion of the generalized hypergeometric function . J. London Math. Soc. , 10 : 286 – 293 .
  • Wright , E. M. 1940 . The asymptotic expansion of the generalized hypergeometric function . Proc. London Math. Soc. (Ser. 2) , 46 : 389 – 408 .

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