Related papers: A note on the tetrahedral index and the Hahn-Exton…
We generalize some widely used mother wavelets by means of the q-exponential function $e_q^x \equiv [1+(1-q)x]^{1/(1-q)}$ ($q \in {\mathbb R}$, $e_1^x=e^x$) that emerges from nonextensive statistical mechanics. Particularly, we define…
A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller…
The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the…
The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of…
The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p-adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted…
The aim of this paper is to give an explicit formula for the calculation of the Gutzwiller--Maslov index of a Hamiltonian periodic orbit. We identify the index appearing in Gutzwiller's trace formula with a non-trivial extension of the…
This paper is based on the lectures given by the first author at Harvard in February and March, 2009. It begins with an introduction to the classical p-adic theory of the Macdonald, Matsumoto and Whittaker functions. Its major directions…
The paper extends basic Einstein--Hilbert action by adding a newly proposed invariant constructed from a specific contraction between the Einstein tensor and the energy momentum tensor, encoding a non--minimal coupling between the…
The aim of the paper is three-fold. We begin by proving a formula, both global and local versions, relating the number of periodic orbits of an iterated map and the Lefschetz numbers, or indices in the local case, of its iterations. This…
An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $ I_{mn}(\beta,q) = \int_{-\infty}^{\infty} e^{-\beta^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x)…
The aim of this paper is to establish Tur\'an -type inequality for the Hahn-Exton $q$-Bessel functions. The result is obtained by the use of limit transition.
The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…
We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter…
In this paper, the directional derivatives in accordance with the orthonormal frame {T, N, B} are defined in $M_{q}^{3}(c)$, and the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the…
In the case of a system with an unbounded hamiltonian the entropic index q of non-extensive thermodynamics has an upperbound q_c>1 beyond which the formalism becomes meaningless. The expression 1/(q_c-1) is the dimension of the state space…
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order…
A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal…
Some basic facts about Fredholm indices are briefly reviewed, often used in connection with Toeplitz and pseudodifferential operators, and which may be relevant for operators associated to fractals.
We construct the new q-extension of Bernoulli numbers and polynomials in this paper. Finally we consider the q-zeta functions which interpolate the new q-extension of Bernoulli numbers and polynomials.
In this paper, we employ the fractional $q$-calculus in solving a triple system of $q$-Integral equations, where the kernel is the third Jackson $q$-Bessel functions. The solution is reduced to two simultaneous Fredholm $q$-integral…