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We combine continuous $q^{-1}$-Hermite Askey polynomials with new $2D$ orthogonal polynomials introduced by Ismail and Zhang as $q$-analogs for complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative…
We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of…
The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummer's second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.
Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel--Laplace transformation.…
The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ in the limit when one or more of…
Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded…
After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace…
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple…
We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function which is flat at the origin.…
Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of…
The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$,…
In this paper we study decay of correlations and limit theorems for generalized baker's transformations. Our examples are piecewise non-uniformly hyperbolic maps on the unit square that posses two spatially separated lines of indifferent…
In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial $A_2$-weight. We prove that having a generalized…
We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert…
We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed…
We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to $\psi^\bullet$-quadratic-like Siegel maps. In this form, the bounds are compatible with the $\psi$-quadratic-like renormalization theory and are easily transferable…
We use a degeneration of the 1D double affine Hecke algebra and the Dunkl operator to study systematically nonsymmetric Bessel functions and their truncations.
Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main…
Consider the space B of complex $p\times q$ matrces with norm <1. There exists a standard one-parameter family $S_a$ of unitary representations of the pseudounitary group U(p,q) in the space of holomorphic functions on B (i.e. scalar…
Physically relevant field-theoretic quantities are usually derived from perturbation techniques. These quantities are solved in the form of an asymptotic series in powers of small perturbation parameters related to the physical system, and…