Related papers: On a relation between certain $q$-hypergeometric s…
In our previous paper we gave proofs of Ibukiyama's correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic…
We describe the construction and properties of a singular theta lift for the orthogonal group $\SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann-Kane-Kohnen with singular sets along geodesics in the upper half…
It is known that among Siegel modular forms of degree $2$ and level $1$ the only functions that violate the Ramanujan conjecture are Saito-Kurokawa lifts of modular forms of level $1$. These are precisely the functions whose Fourier…
Given an arrangement of hyperplanes in $\P^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$…
Interpolated multiple $q$-zeta values are deformation of multiple $q$-zeta values with one parameter, $t$, and restore classical multiple zeta values as $t = 0$ and $q \to 1$. In this paper, we discuss generating functions for sum of…
By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth…
Karabulut and Sibert (\textit{J. Math. Phys}. \textbf{38} (9), 4815 (1997)) have constructed an orthogonal set of functions from linear combinations of equally spaced Gaussians. In this paper we show that they are actually eigenfunctions of…
A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a…
Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the xi-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the…
We discuss the problem of the vanishing of Poincar\'e series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the…
Random matrix model for the Nakagami-q (Hoyt) fading in multiple-input multiple-output (MIMO) communication channels with arbitrary number of transmitting and receiving antennas is considered. The joint probability density for the…
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the…
We demonstrate some 10-periodic properties of the coefficients of two $q$-series related to partition numbers signed by the Legendre symbol $(\frac{a}{5})$ and to the Rogers-Ramanujan continued fraction $R(q)$.
In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families…
This paper aims at developing new shape functions adapted to smooth vanishing coefficients for scalar wave equation. It proposes the numerical analysis of their interpolation properties. The interpolation is local but high order convergence…
The iso-singlet scalar resonance \sigma(600), found in the recent \pi\pi phase shift analysis, is pointed out to have the properties of the \sigma meson in the \sigma model. The role of the \lambda\phi^4 interaction as a source of the…
Within the context of nonrelativistic potential models, we obtain several formulas (with varying degrees of rigor) relating the wave functions at the origin of the $c{\bar c}$, $b{\bar c}$ and $b{\bar b}$ S-wave quarkonium systems. One of…
We study q-integral representations of the q-gamma and the q-beta functions. This study leads to a very interesting q-constant. As an application of these integral representations, we obtain a simple conceptual proof of a family of…
In the present paper we review the $q$-analogue of the Quantum Theory of Angular Momentum based on the $q$-algebra $su_q(2)$, with a special emphasis on the representation of the Clebsch-Gordan coefficients in terms of $q$-hypergeometric…
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…