On composition of some general fractional integral operators

In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman [1] and Erdélyi [4] follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.     

A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t _n =[(P ^n/p )₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{ ₀(z, ₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz ^p andA _c :=z/z–z+c. This requirement is satisfied, for arbitraryp, if ₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW as a double matrix Laplace transform in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour ^-+ ^- defined by ^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.