the object of this paper is to establish certain generalized fractional integration and differentiation involving generalized mittag-leffler function defined by salim and faraj [25]. the considered generalized fractional calculus operators contain the appell's function $f_3$ [2, p.224] as kernel and are introduced by saigo and maeda [23]. the marichev-saigo-maeda fractional calculus operat...

In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished throu...

We construct a four-parameter family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand-Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(∞). The process has a unique inva...

The aim of this paper is to obtain explicit expressions of the generalized hypergeometric function r+2Fr+1 [ a, b, 1 2 (a+ b+ j + 1), (fr +mr) (fr) ; 1 2 ] for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorial parameters differ by positive integers mr. The results are derived with the help of an expansion in terms of a finite sum of 2F1( 1 2 ) functions and a generalization ...

Finding closed form solutions of differential equations has a long history in computer algebra. For example, the Risch algorithm (1969) decides if the equation y′ = f can be solved in terms of elementary functions. These are functions that can be written in terms of exp and log, where “in terms of” allows for field operations, composition, and algebraic extensions. More generally, functions are...

In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < |q| < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalize...

We present a novel family of paraxial optical beams having a confluent hypergeometric transverse profile, which we name hypergeometric Gauss modes of type-II (HyGG-II). These modes are eigenmodes of the photon orbital angular momentum and they have the lowest beam divergence in the waist of HyGG-II among all known finite power paraxial modes families. We propose to exploit this feature of HyGG-...

We consider a family of character sums as multiplicative analogues Kloosterman sums. Using Gauss sums, Jacobi and Katz’s bound for hypergeometric we establish asymptotic formulae any real (positive) moments the above sum

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in ...

It is well known that X/(X -f Y) has the beta distribution when X and Y follow the Dirichlet distribution. Linear combinations of the form aX + pY have also been studied in Provost and Cheong [24]. In this paper, we derive the exact distribution of the product P = XY (involving the Gauss hypergeometric function) and the corresponding moment properties. We also propose an approximation and show ...