R. K. Saxena

Integral and computational representations of the extended Hurwitz–Lerch zeta function

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

On the solutions of certain fractional kinetic equations

Fractional kinetic equations have gained importance during the last decade due to their occurrence in certain problems in science and engineering. This article gives a brief survey of various fractional kinetic equations solved by a number of authors. We also present a new method for solving the fractional kinetic equations that is different from that using Laplace transform. The results presented are in compact forms suitable for numerical computation. Some figures are given, to show the behaviour of the derived solutions.

Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics

The object of this article is to present the solution of a fractional generalization of the Schrodinger equation of quantum mechanics in one dimension. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is obtained in a compact and computational form in terms of the H-function. A result given earlier by Debnath for the solution of a generalized Schrodinger equation is obtained in an explicit form in terms of the H-function, as a special case of our findings.

Multivariate analogue of generalized Mittag-Leffler function

Following the results of Saxena and Kalla [Solutions of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels, Int. J. Math. Math. Sci. 8 (2005), pp. 1155–1170], we introduce and develop here a theory of multivariate generalization of the Mittag-Leffler function, which is defined as: where λ, γ j , ρ j ∈C, Re (ρ j )>0, j=1, …, m. Certain properties of this multivariate generalized Mittag-Leffler function associated with fractional calculus are established. Further, an integral operator with this function as a kernel, in the following form: is studied in the space L(a, b). An analogy of the semi-group property for the composition of two such operators with different indices is proved. A composition of the Riemann–Liouville fractional integral operator with two such operators with different indices is established. The results derived in this paper provide generalization of the results given earlier by Kilbas et al. [Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct. 15 (2004), 31–49].

SOLUTION OF VOLTERRA-TYPE INTEGRO-DIFFERENTIAL EQUATIONS WITH A GENERALIZED LAURICELLA CONFLUENT HYPERGEOMETRIC FUNCTION IN THE KERNELS

The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous function f(τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivatives and a generalized Volterra integral equation are also considered. Several special cases are mentioned. A number of results given recently by various authors follow as particular cases of formulas established here.

On a unified mixture distribution

In this article, a new mixture distribution associated with Fox-Wright generalized hypergeometric function has been studied, which generalizes many mixture distributions investigated earlier by many authors. Some basic functions associated with the probability density function of the mixture distribution, such as kth moments, characteristic function and factorial moments are derived. Some special cases are also pointed out.

Asymptotic formulas for unified elliptic-type integrals

In this article, we present a unification and extension of certain families of elliptic-type integrals, which have been discussed in a number of earlier works on the subject due to their importance and applications in problems arising in radiation physics and nuclear technology. The results derived in this article provide extensions of the results established by Srivastava and Siddiqi [Radiat. Phys. Chem. 46(1995), 303–315], Kalla and Tuan [Computers Math. Appl. 32(1996), 49–55], Al-Zamel et al. [Appl. Math. Comput. 114(2000), 13–25] and Saxena et al. [Math. Balkanica 15(2001), 387–396].

A generalization of beta-type distribution involving ω-Lauricella function in several variables

The object of this paper is to define and study a new generalization of beta-type function in the following form: where λ, u, v, a, c, b j (ju2009=u20091, Λ, n)u2009∈u2009C with cu2009≠u20090,−1,−2, Λ; ω, Re(u), Re (au2009+u2009λ), Re (b j u2009+u2009λ)u2009>u20090(ju2009=u20091, Λ, n), | arg v|u2009<u2009π and is the ω-generalization of the Lauricella hypergeometric function of n-variables. The previous defined function provides extensions of the various type of beta-functions studied earlier by many authors, notably by Ben Nakhi and Kalla [Al-Shammery, A.H. and Kalla, S.L., 2000, An extension of some hypergeometric functions of two variables. Revista de la Acade. Canaria de cie., 12, 189–196; Ben Nakhi, Y. and Kalla, S.L., 2002, A generalised beta function and associated probability density. International Journal of Mathematics and Mathematical Science, 8, 467–478.]. A study of this function will provide deeper, general and useful results in the theory of special functions and statistical distributions. A probability density function (p.d.f.) associated with this function is also introduced. The moment generating function, hazard rate function and mean residue life function for this p.d.f. will also be investigated.

Generalized gamma-type functions involving Kummer’s confluent hypergeometric function and associated probability distributions

The object of this paper is to define and study a new generalization of the generalized gamma-type function in the form where Φ (α, β; z) is the well known Kummer’s confluent hypergeometric function and 2 R 1 (·) is a special case of Wright’s generalized hypergeometric function. This generalization provides unification and extension of the various generalizations given earlier by Kobayashi, Al-Musallam and Kalla and Virchenko et al. Incomplete generalized gamma function corresponding to the above defined generalized gamma function are also defined and their basic properties are investigated, which extend the existing results in the field of generalized gamma-type functions. We study a class of probability density functions involving Kummer’s confluent hypergeometric function and 2 R 1 (·).