Related papers: A Connection Formula of the Hahn-Exton $q$-Bessel …
We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_\nu(an^px)}{(an^px/2)^\nu}\qquad (x>0),\] in the limit $a\to0$, where…
We verify the continuity of the Riesz transform from the operator related Hardy space to $L^1$ - Lebesgue space of integrable functions. For the standard Euclidean Laplace operator, this is a classical result that plays a significant role…
We prove an analog of the quaternionic Borel-Pompieu formula in the sense of proportional fractional $\psi$-Cauchy-Riemann operators via Riemann-Liouville derivative with respect to another function.
In the present note, we generalize the first part of the Borel-Cantelli lemma. By this generalization, we obtain some strong limit results.
A connection-independent formulation of general relativity is presented, in which the dynamics does not depend on the choice of connection. The gravity action in this formulation includes one additional scalar term in addition to the…
We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for real-valued functions. Analogous results for cosine series…
For an abstract self-adjoint operator $L$ and a local operator $A$ we study the boundedness of the Riesz transform $AL^{-\alpha}$ on $L^p$ for some $\alpha >0$. A very simple proof of the obtained result is based on the finite speed…
The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is a long-standing problem with no known closed-form solution except in specific cases. Taking this as a starting-point, and extending to a general mean-reverting…
In this paper we give a q-analogue of the Hardy's theorem for the $q$-Bessel Fourier transform. The celebrated theorem asserts that if a function $f$ and its Fourier transform $\hat{f}$ satisfying $|f(x)|\leq c.e^{-{1/2} x^2}$ and…
In this paper the discrete Sobolev inner product $$< p,q > =\int p(x) q(x) \,d\mu + \sum_{i=0}^r M_i \, p^{(i)}(c) \, q^{(i)}(c)$$ is considered, where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real…
The aim of this paper is to derive new representations for the Hankel functions, the Bessel functions and their derivatives, exploiting the reformulation of the method of steepest descents by M. V. Berry and C. J. Howls (Berry and Howls,…
We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent…
The study of $\psi-$hyperholomorphic functions defined on domains in $\mathbb R^4$ with values in $\mathbb H$, namely null-solutions of the $\psi-$Fueter operator, is a topic which captured great interest in quaternionic analysis. This…
The second order hypergeometric q-difference operator is studied for the value c=-q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space l^2(Z). The operator has…
We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of…
For processes with gauge bosons in the final state we show how to continuously connect with a single Born-improved amplitude the resonant region, where resummation effects are important, with the asymptotic region far away from the…
Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative…
We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of…
By combining the Gessel--Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the $q$-Dyson constant term identity also known as the Zeilberger--Bressoud $q$-Dyson theorem. This yields a…
We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. Furthermore, the connection…