On extended M−series

With the development of factional calculus theory and applications in different fields in recent years, the rational approximation problem of fractional calculus operator has become a hot spot of research. In the early 1950s and 1960s, Carlson and Halijak proposed regular Newton iterating method to implement rational approximation of the one-nth calculus operator. Carlson regular Newton iterating method has a great sense of innovation for the rational approximation of fractional calculus operator, however, it has been used only for certain calculus operators. The aim of this paper is to achieve rational approximation of arbitrary order fractional calculus operator. The realization is achieved via the generalization of Carlson regular Newton iterating method. To construct a rational function sequence which is convergent to irrational fractional calculus operator function, the rational approximation problem of fractional calculus operator is transformed into the algebra iterating solution of arithmetic root of binomial equation. To speed up the convergence, the pre-distortion function is introduced. And the Newton iterating formula is used to solve arithmetic root. Then the approximated rational impedance function of arbitrary order fractional calculus operator is obtained. For nine different operational orders with n changing from 2 to 5, the impedance functions are calculated respectively through choosing eight different initial impedances for a certain operational order. Considering fractional order operation characteristics of the impedance function and the physical realization of network synthesis, the impedance function should satisfy these basic properties simultaneously: computational rationality, positive reality principle and operational validity. In other words, there exists only rational computation of operational variable s in the expression of impedance function. All the zeros and poles of impedance function are located on the negative real axis of s complex plane or the left-half plane of s complex plane in conjugate pairs. The frequency-domain characteristics of impedance function approximate to those of ideal fractional calculus operator over a certain frequency range. Given suitable initial impedance and for an arbitrary operational order, it is proved that the impedance function could meet all properties above by studying the zero-pole distribution and analyzing frequency-domain characteristics of the impedance function. Therefore, the impedance function could take on operational performance of the ideal fractional calculus operator and achieve the physical realization. It is of great effectiveness in the generalization of this kind of method in both theory and experiment. The results educed in this paper are the basis for further theoretic research and engineering application in constructing the arbitrary order fractional circuits and systems.

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The objective of this paper is to explore and establish certain generalized fractional differentiation and integration involving multivariate generalized Mittag-Leffler function introduced by Salim and Faraz. The generalization of Saigo fractional calculus operators that is Marichev-Saigo-Maeda Fractional Calculus Operator is utilized to establish some formulas related to generalized multivariate M-L function. In conclusion, we depict some key characteristics of generalized fractional integrals and differentiation pertaining to Generalized Multivariate Mittag- Leffler Function.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields. The main object of this paper is to consider some miscellaneous properties and applications which are associated with several fractional differintegral operators. We first investigate, in a systematic and unified manner, various families of series identities which emerged in connection with some of these fractional differintegral formulas. By using such operators of fractional calculus, a number of integral formulas as well as fractional differintegral formulas involving inverse hyperbolic functions are also evaluate.

This project aimed to gather together the latest developments in research concerning complex-valued functions from the perspective of geometric function theory. Scholars contributions weresought on topics including, but not limited to: new classes of univalent and bi-univalent functions; studies regarding coefficient estimates including the FeketeSzego functional, Hankel determinants, and Toeplitz matrices; applications of different types of operators in geometric function theory including differential, integral, fractional, or quantum calculus operators; differential subordination and superordination theories in their classical form and also concerning their recent extensionsstrong and fuzzy differential subordination and superordination theories; applications of different hypergeometric functions and orthogonal polynomials in geometric function theory. The presentation of new results obtained by using any other techniques which can be applied in the field of complex analysis were also welcomed. Hopefully, through this project, new lines of research associatedwithgeometric function theory have been highlighted and willserve toboost development inthis field.

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

Log-convexity and log-concavity of hypergeometric-like functions

Because of the many relations connecting the special functions to each other, and to the elementary functions, it is natural to inquire whether more general functions can be developed so that the special functions and elementary functions are merely specializations of these general functions. General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. There are several varieties of these functions, but the most common are the standard hypergeometric function ( which we discuss in this chapter) and the confluent hypergeometric function (Chap. 10).

The object of this paper is to present generalizations of some recent results ot S.S. Miller and P.T. Mocanu on the univalence and starlikeness of Kummer' confluent hypergeometric function. Some applications involving generalized hypergeometric functions associated with the Hardy space of analytic functions are also considered.

Fractional calculus operators of special functions? The result is well predictable!

Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996) introduced and investigated certain generalized fractional integral and derivative operators involving the Appell function . Here we aim at presenting four unified fractional integral and derivative formulas of Saigo and Maeda type, which are involved in a product of ℵ-function and a general class of multivariable polynomials. The main results, being of general nature, are shown to be some unification and extension of many known formulas given, for example, by Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996), Saxena et al. (Kuwait J. Sci. Eng. 35(1A):1-20, 2008), Srivastava and Garg (Rev. Roum. Phys. 32:685-692, 1987), Srivastava et al. (J. Math. Anal. Appl.193:373-389, 1995) and so on. Our main results are also shown to be further specialized to yield a large number of known and (presumably) new formulas involving, for instance, Saigo fractional calculus operators, several special functions such as H-function, I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function. MSC:26A33, 33E20, 33C45, 33C60, 33C70.

Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out [...]

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.