Related papers: Unique positive solutions to $q$-discrete equation…
The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of q-difference lattice equations satisfied by the universal…
New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal…
Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of…
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product \begin{equation*} \left\langle p,q\right\rangle…
We derive the discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$ that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity…
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this…
In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the…
Let $U_q(\mathfrak{b})$ be the Borel subalgebra of a quantum affine algebra of type $X^{(1)}_n$ ($X=A,B,C,D$). Guided by the ODE/IM correspondence in quantum integrable models, we propose conjectural polynomial relations among the…
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate…
We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$\gamma$-positivity using group actions. Motivated by the sign-balance identity of…
In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The…
We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling…
In this paper, we derive a pair of finite univariate biorthogonal polynomials suggested by the finite univariate orthogonal polynomials $M_{n}^{(p,q)}(x)$. The corresponding biorthogonality relation is given. Some useful relations and…
A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal…
The paper is devoted to the further study of the remarkable classes of orthogonal polynomials recently discovered by Bender and Dunne. We show that these polynomials can be generated by solutions of arbitrary quasi - exactly solvable…
Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving…
We show that the alternative discrete Painlev\'e I equation (alt-dP$_{\rm I}$) has a unique solution which remains positive for all $n \geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second…
We propose a method to characterize discrete time evolution equations, which generalize discrete time soliton equations, including the $q$-difference Painlev\'e IV equations discussed recently by Kajiwara, Noumi and Yamada.