We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2)\in {ℂ}^{n+2}:|z_1{|}^2+\cdots +|z_n{|}^2+|w_1{|}^q<1,|z_1{|}^2+\cdots +|z_n{|}^2+|w_2{|}^r<1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2)\in {ℂ}^3:|z_1{|}^{2/n}+|w_1{|}^q<1,|z_1{|}^{2/n}+|w_2{|}^r<1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.

A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation...

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.

Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline{spt}\left(n\right)$, ${\overline{spt}}_1\left(n\right)$, ${\overline{spt}}_2\left(n\right)$, and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this...

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$