The exponential-type generating function of the Riemann zeta-function revisited

In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.

A recurring [formula omitted]-integral formula

On Ramanujan's inversion formulas

Geophysical Inverse Theory and Regularization Problems

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.

In 1981, Andrews gave a four-variable generalization of Ramanujan's summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two series and Bailey's transformation formula for three series. Then, it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two summation formulas in terms of two limiting relations and Bailey's another transformation formula for three series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo–Schlosser transformation formula and other two transformation formulas.

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular groupSL(2, ℤ). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups ofSL(2, ℤ) and prove similar estimates for the corresponding Dirichlet series.

For a second-order elliptic system with a singular point, we obtain integral representations and inversion formulas for the case in which the singular point is an interior point of the domain. In the integral representations, we clearly extract the singular part of the solutions, which permits one to study the asymptotics of the solutions as r → 0. In addition, we give a well-posed statement of a number of boundary value problems.

Integral transformations of the form from Lp(ℝ+) into Lq(ℝ+); (1 ≤ p ≤ 2, p−1 + q −1 = 1) are studied. Necessary and sufficient conditions on k to ensure that the transformation is unitary on L2(ℝ+) are obtained, and a formula for the inverse transformation is derived in this case.

The problem of diffraction of an E-polarized plane wave by a dielectric wedge has been treated by a number of authors. In this paper, the integral representations are used for inside and outside solutions. The boundary conditions lead to coupled integral equations of the first kind. In any case, the solution of the integral equation of the 1-st kind leads to an ill-conditioned algebraic system. As a rule, the solution of such a system is connected with any ill-posed problem. The integral transformation is found for the system of integral equations of the 1-st kind and this transformation enables one to obtain a stable solution.

Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the origin, we present a transformation formula for triple $q$-series, which includes several interesting special cases, especially a double $q$-series summation formula. Some applications of this transformation formula to Rogers-Hecke type series are discussed. More than 100 Rogers-Hecke type identities including Andrews' identities for the sums of three squares and the sums of three triangular numbers are obtained.

q-Analogues of two cubic summation formulas that have recently caught the attention of Bill Gosper are found by first showing their connection with the q-binomial formula and then using some known transformation formulas. We also find a q-extension of a cubic transformation formula involving Gauss' hypergeometric function, which turns out to be a relation between balanced and very-well-poised 10φ9 series.

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a general polygonal function the requisite integrals may be easily computed by a closed form representation of simple integrals. This paper provides the integration formulas for two-dimensional variogram functions whose domain is represented as a finite collection of rectangles. The integration formulas essential for a fully developed polygonal approach to an extensive statistical evaluation of geostatistical quantities are presented.

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

The main goal of this article is to study the extend Struve and extended modified Struve matrix functions by making use of extended Beta matrix function. In particular, we investigate certain important properties of these extended matrix functions such as integral representation, differentiation formula and hypergeometric representation of these functions. Finally, we obtain some results on the transform and fractional calculus of these extended Struve and extended modified Struve matrix functions.