Related papers: When does a hypergeometric function ${}_{p\!}F_q$ …

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved…

Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to…

We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $\ell^1\cap\ell^\Phi_{\phi,w}$ (resp. to $\ell^1\cap\ell^\Psi_{\psi,\varrho}$), where $\ell^\Phi_{\phi,w}$ and…

Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix…

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for $q$, a power of an odd prime, and…

We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1\psi_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection…

Consider a subset $A$ of $\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming…

The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $\alpha_{L/A}$ being…

A class of complex Fourier Transforms of exponential functions which have all their zeros on the real line is explored from a geometric perspective. These transforms belong to the Laguerre - Polya class, and it is proved that all the zeros…

In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local…

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\operatorname{GL}^+(2)$ of invertible $2\times2-\,$matrices is rank-one convex if and only if it is polyconvex. In a 2005…

This work is devoted to study of algebraicty modulo p of Siegel's G-functions. Our goal is to emphasize the relevance of the notion of strong Frobenius structure, clasically studied in the theory of the p-adic diffenrential equations, for…

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. Often we have hypergeometric functions with indices linear dependent on a small parameter with respect to which…

This paper studies the inclusions between different Sobolev-Lorentz spaces $W^{1,(p,q)}(\Omega)$ defined on open sets $\Omega \subset {\mathbf{R}^n},$ where $n \ge 1$ is an integer, $1<p<\infty$ and $1 \le q \le \infty.$ We prove that if $1…

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The…

Let G be a noncompact connected Lie group and $\rho$ be the right Haar measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields which satisfy H\"ormander's condition, and let $\Delta=-\sum_{i=1}^qX_i^2$ be the…

We study the regularity of smooth functions $f$ defined on an open set of $\mathbb{R}^n$ and such that, for certain integers $p\geq 2$, the powers $f^p :x\mapsto (f(x))^p$ belong to a Denjoy-Carleman class $\mathcal{C}_M$ associated with a…

As is known, the class of weights for Morrey type spaces $\mathcal{L}^{p,\lb}(\rn) $ for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class $A_p$ of such weights for the Lebesgue spaces…