• Title/Summary/Keyword: Generalized fractional calculus operators

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ON GENERALIZED WRIGHT'S HYPERGEOMETRIC FUNCTIONS AND FRACTIONAL CALCULUS OPERATORS

  • Raina, R.K.
    • East Asian mathematical journal
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    • v.21 no.2
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    • pp.191-203
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    • 2005
  • In the present paper we first establish some basic results for a substantially more general class of functions defined below. The results include simple differentiation and fractional calculus operators(integration and differentiation of arbitrary orders) for this class of functions. These results are then invoked in determining similar properties for the generalized Wright's hypergeometric functions. Further, norm estimate of a certain class of integral operators whose kernel involves the generalized Wright's hypergeometric function, and its composition(and other related properties) with the fractional calculus operators are also investigated.

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FRACTIONAL CALCULUS OPERATORS AND THEIR IMAGE FORMULAS

  • Agarwal, Praveen;Choi, Junesang
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1183-1210
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    • 2016
  • During the past four decades or so, due mainly to a wide range of applications from natural sciences to social sciences, the so-called fractional calculus has attracted an enormous attention of a large number of researchers. Many fractional calculus operators, especially, involving various special functions, have been extensively investigated and widely applied. Here, in this paper, in a systematic manner, we aim to establish certain image formulas of various fractional integral operators involving diverse types of generalized hypergeometric functions, which are mainly expressed in terms of Hadamard product. Some interesting special cases of our main results are also considered and relevant connections of some results presented here with those earlier ones are also pointed out.

SOME FAMILIES OF INFINITE SERIES SUMMABLE VIA FRACTIONAL CALCULUS OPERATORS

  • Tu, Shih-Tong;Wang, Pin-Yu;Srivastava, H.M.
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.111-125
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    • 2002
  • Many different families of infinite series were recently observed to be summable in closed forms by means of certain operators of fractional calculus(that is, calculus of integrals and derivatives of any arbitrary real or complex order). In this sequel to some of these recent investigations, the authors present yet another instance of applications of certain fractional calculus operators. Alternative derivations without using these fractional calculus operators are shown to lead naturally a family of analogous infinite sums involving hypergeometric functions.

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THE (k, s)-FRACTIONAL CALCULUS OF CLASS OF A FUNCTION

  • Rahman, G.;Ghaffar, A.;Nisar, K.S.;Azeema, Azeema
    • Honam Mathematical Journal
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    • v.40 no.1
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    • pp.125-138
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    • 2018
  • In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

FRACTIONAL CALCULUS OPERATORS OF THE PRODUCT OF GENERALIZED MODIFIED BESSEL FUNCTION OF THE SECOND TYPE

  • Agarwal, Ritu;Kumar, Naveen;Parmar, Rakesh Kumar;Purohit, Sunil Dutt
    • Communications of the Korean Mathematical Society
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    • v.36 no.3
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    • pp.557-573
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    • 2021
  • In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric 2F1(x) function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the n-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential are also deduced.

ERTAIN k-FRACTIONAL CALCULUS OPERATORS AND IMAGE FORMULAS OF GENERALIZED k-BESSEL FUNCTION

  • Agarwal, P.;Suthar, D.L.;Tadesse, Hagos;Habenom, Haile
    • Honam Mathematical Journal
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    • v.43 no.2
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    • pp.167-181
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    • 2021
  • In this paper, the Saigo's k-fractional integral and derivative operators involving k-hypergeometric function in the kernel are applied to the generalized k-Bessel function; results are expressed in term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Bessel functions are considered.

SOME FAMILIES OF INFINITE SUMS DERIVED BY MEANS OF FRACTIONAL CALCULUS

  • Romero, Susana Salinas De;Srivastava, H.M.
    • East Asian mathematical journal
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    • v.17 no.1
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    • pp.135-146
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    • 2001
  • Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

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SOME APPLICATIONS AND PROPERTIES OF GENERALIZED FRACTIONAL CALCULUS OPERATORS TO A SUBCLASS OF ANALYTIC AND MULTIVALENT FUNCTIONS

  • Lee, S.K.;Khairnar, S.M.;More, Meena
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.127-145
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    • 2009
  • In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.

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FRACTIONAL DIFFERENTIATION OF THE PRODUCT OF APPELL FUNCTION F3 AND MULTIVARIABLE H-FUNCTIONS

  • Choi, Junesang;Daiya, Jitendra;Kumar, Dinesh;Saxena, Ram Kishore
    • Communications of the Korean Mathematical Society
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    • v.31 no.1
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    • pp.115-129
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    • 2016
  • Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. [9] evaluated the fractional integrals of the product of Appell function of the third kernel $F_3$ and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function $F_3$ and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, $Erd\acute{e}lyi$-Kober and Saigo operators, associated with multivariable H-function and Appell function $F_3$ are shown to be deduced as special cases of our findings.

SOME PROPERTIES OF GENERALIZED BESSEL FUNCTION ASSOCIATED WITH GENERALIZED FRACTIONAL CALCULUS OPERATORS

  • Jana, Ranjan Kumar;Pal, Ankit;Shukla, Ajay Kumar
    • Communications of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.41-50
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    • 2021
  • This paper devoted to obtain some fractional integral properties of generalized Bessel function using pathway fractional integral operator. We also find the pathway transform of the generalized Bessel function in terms of Fox H-function.