Related papers: A simple approach to some Hankel determinants

By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan…

We present a formula that expresses the Hankel determinants of a linear combination of length $d+1$ of moments of orthogonal polynomials in terms of a $d\times d$ determinant of the orthogonal polynomials. This formula exists somehow hidden…

The aim of this paper is to obtain an upper bound to the second Hankel the determinant for starlike and convex functions of order.

This short note contains elementary evaluations of some Euler sums.

In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.

Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.

In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.

We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…

In this paper, we consider a q-analogue of Laplace transform and we investigate some properties of q-Laplace transform. From our investigation, we derive some interesting formulae related to q-Laplace transform.

We prove some symmetric $q$-congruences.

We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group.

We give an overview about some elementary properties of Hoggatt matrices, which are generalizations of Pascal triangle, and study q-analogs and Fibonacci analogs and derive a common generalization.

The estimates for the second Hankel determinant a_2a_4-a_3^2 of analytic function f(z)=z+a_2 z^2+a_3 z^3+...b for which either zf'(z)/f(z) or 1+zf"(z)/f'(z) is subordinate to certain analytic function are investigated. The estimates for the…

We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and derive the asymptotic behavior for the special case that F(s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue…

An elementary proof of an identity by Lyons, Paule and Riese is given. It is simpler than all the 3 published proofs.

We consider the Hankel determinant formula of the $\tau$ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the $\tau$…

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a…

We give a q-analogue of Gauss' divisibility theorem

The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities:…

In this note, we prove some combinatorial identities and obtain a simple form of the eigenvalues of $q$-Kneser graphs.