Mathematics Subject Classification: 33D60, 33D90, 26A33Fractional q-integral operators of generalized Weyl type, involving generalized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving various q-functions have been evaluated as applications of the main results.
We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front.
Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20By using integral representations for several Mathieu type series, a number of integral transforms of Hankel type are derived here for general families of Mathieu type series. These results generalize the corresponding ones on the Fourier transforms of Mathieu type series, obtained recently by Elezovic et al. [4], Tomovski [19] and Tomovski and Vu Kim Tuan [20].
In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials are given and differential equations satisfied by them is presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials is proposed.
T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied.
We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on . We also construct a solution of the equation in that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.