New identities for the Glasser transform and their applications

Bilinear identities involving the $k$-plane transform and Fourier extension operators

We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$-plane transform. As the estimates are $L^2$-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical $L^2(\mathbb{R}^2)$-bilinear identity for Fourier extension operators associated to curves in $\mathbb{R}^2$.

On Certain Generalizations of Rogers-Ramanujan Type Identities

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by Ramanujan, as well as certain identities with an interesting variable-parameter symmetry based on limiting cases of Heine's transformation of basic hypergeomteric functions.

Operator identities relating sonar and Radon transforms in Euclidean space

We establish new relations which connect Euclidean sonar transforms (integrals taken over spheres with centers in a hyperplane) with classical Radon transforms. The relations, stated as operator identities, allow us to reduce the inversion of sonar transforms to classical Radon inversion.

Several classical identities via Mellin's transform

We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.

Hilbert变换的Rellich恒等式

Rellich identities for the Hilbert transform

本文利用Rellich恒等式和图Lipschitz区域上的Neumann问题证明涉及共形映射的Hilbert变换恒等式，并由此得到新的加权L^2估计及与Helson-Szegö定理相关的精确范数界。

Kernel identities for van Diejen's $q$-difference operators and transformation formulas for multiple basic hypergeometric series

In this paper, we show that the kernel function of Cauchy type for type $BC$ intertwines the commuting family of van Diejen's $q$-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type $BC$. We also construct a new infinite family of commuting $q$-difference operators for which the Koornwinder polynomials are joint eigenfunctions.

Bailey Type Transforms and Applications

The aim of this paper is to establish new series transforms of Bailey type and to show that these Bailey type transforms work as efficiently as the classical one and give not only new $q$-hypergeometric identities, converting double or triple series into a very-well-poised $_{10}Φ_9$ or $_{12}Φ_{11}$ series but can also be utilized to derive new double and triple series Rogers-Ramanujan type identities and corresponding infinite families of Rogers-Ramanujan type identities.

涉及贝塞尔函数平方的积分变换的恒等式与不等式

Identities and inequalities for integral transforms involving squares of the Bessel functions

本文扩展了 Bez 等人关于贝塞尔函数平方积分变换的恒等式到非整数指标，并推导出若干不等式。

A New Form of the Spherical Expansion of Zonal Functions and Fourier Transforms of SO(d)-Finite Functions

This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for problems involving Fourier transforms of functions with rotational symmetry. The method used to derive the expansion formula is based entirely on differential methods and completely avoids the use of various integral identities commonly used in this context. Some new identities for the Fourier transform are derived and as a byproduct seemingly new recurrence rela…

Application of the Efros theorem to the function represented by the inverse Laplace transform of $s^{-μ}\,\exp(-s^ν)$

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag-Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag-Leffler functions and the Volterra functions. Some…