Related papers: Leading coefficient in the Hankel determinants rel…
We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…
We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form $A_{i_1i_2... i_k}=f(i_1+i_2+...+i_k)$). It is found that many classical properties of Hankel determinants can…
We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we…
Using a slightly generalized result of George Andrews and Jet Wimp this note gives a simple computational proof of some Hankel determinants of backwards shifts of convolution powers of Catalan numbers and obtains analogous results for…
A new $q$-analogue of Appell polynomial sequences and their generalizations are introduced and their main characterizations are proved. As consequences new $q$-analogue of Bernoulli and Euler polynomials and numbers is introduced, their…
This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach…
Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix}=\Gamma_1\Gamma_3-\Gamma^2_2, \end{align*}…
In this paper, we give some determinantal and permanental representations of Generalized Lucas Polynomials by using various Hessenberg matrices, which are general form of determinantal and permanental representations of ordinary Lucas and…
The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence…
We study Hankel transforms of sequences, where the transform elements are members of the set {-1,0,1}. We relate these Hankel transforms to special continued fraction expansions. In particular, we posit a conjecture relating the…
In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain an asymptotic…
We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\alpha>0,\;\;t\geq 0. $$ If $t=0$, it is reduced to the classical symmetric…
We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support.…
Applying Johann Cigler's Hankel determinant formula in terms of the binomial coefficient determinants, which is simplified from Christian Krattenthale's, we get an explicit formula of Hankel determinants for general. As far as I know, those…
In this paper we introduce a class of determinants "of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms.
We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2)^{\beta}(t^2-x^2)^\alpha h(x),~x\in(-1,1),$$ where $\beta>-1$, $\alpha+\beta>-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…