Composition formulas of Bessel-Struve kernel function

The generalized operators of fractional integration involving Appell's function $F_{3}(.)$ due to Marichev-Saigo-Maeda, is applied to the Bessel Struve kernel function $S_{\alpha }\left( \lambda z\right),\lambda ,z\in \mathbb{C}$ to obtain the results in terms of generalized Wright functions.The pathway integral representations Bessel Struve kernel function and its relation between many other functions also derived in this study.Comment: 13 page

Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups

We compute generalized Bernstein-Reznikov integrals associated with standard complex symplectic forms by studying Knapp-Stein intertwining operators between spherical degenerate principal series of complex symplectic groups.Comment: Final version, to appear in Internat. J. Mat

Linear independent solutions and operational representations for hypergeometric functions of four variables

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. Here we choose the Exton function $K_{2}$ among his 21 functions to show how to find the linearly independent solutions of partial differential equations satisfied by this function $K_{2}$. Based upon the classical derivative and integral operators we introduce a new operational images for hypergeometric function $K_{2}$. By means of these operational images a number of finite series and decomposition formulas are then fund.Comment: 8 page

Paley-Wiener theorems for the Θ\Theta-spherical transform: an overview

The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $\Theta$-spherical functions, the correspondint $\Theta$-spherical Fourier transform, and the Paley-Wiener theorem for this transform. Up to now such a theorem has only been proven in very special cases, and still, its formulation and proof are very technical. In this paper we do not go into details of the proofs, but present an overview which explains the different examples which have inspired and motivated the theory of $\Theta$-spherical functions

Marichev-Saigo-Maeda fractional operator representations of generalized Struve function

The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function due to Marichev-Saigo-Maeda, to the generalized Struve function. The results are expressed in terms of generalized Wright function. The results obtained here are general in nature and can easily obtain various known results

Quadratic Differential Systems and Chazy Equations, I

Generalized Darboux-Halphen (gDH) systems, which form a versatile class of three-dimensional homogeneous quadratic differential systems (HQDS's), are introduced. They generalize the Darboux-Halphen (DH) systems considered by other authors, in that any non-DH gDH system is affinely but not projectively covariant. It is shown that the gDH class supports a rich collection of rational solution-preserving maps: morphisms that transform one gDH system to another. The proof relies on a bijection between (i) the solutions with noncoincident components of any `proper' gDH system, and (ii) the solutions of a generalized Schwarzian equation (gSE) associated to it, which generalizes the Schwarzian equation (SE) familiar from the conformal mapping of hyperbolic triangles. The gSE can be integrated parametrically in terms of the solutions of a Papperitz equation, which is a generalized Gauss hypergeometric equation. Ultimately, the rational gDH morphisms come from hypergeometric transformations. A complete classification of proper non-DH gDH systems with the Painleve property (PP) is also carried out, showing how some are related by rational morphisms. The classification follows from that of non-SE gSE's with the PP, due to Garnier and Carton-LeBrun. As examples, several non-DH gDH systems with the PP are integrated explicitly in terms of elementary and elliptic functions.Comment: 68 pages, 1 figure; slightly expande

A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.Comment: 55p, typos were correcte

Interest rate models and Whittaker functions

I present the technique which can analyse some interest rate models: Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian Motion. All these models have the unified structure of Whittaker function. The main focus of this text is closed-form solutions of the zero-coupon bond value in these models. In text I emphasize the specific details of mathematical methods of their determination such as Laplace transform and hypergeometric functions.Comment: 19 pages, 2 figure

One-dimensional double Hecke algebras and Gaussians

We use one-dimensional double affine Hecke algebras to introduce q-counterparts of the Gauss integrals and new types of Gauss-Selberg sums at roots of unity.Comment: amste

Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles

We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.Comment: AMSTeX, 19 page