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Johann Cigler

Publications and source records attributed to Johann Cigler.

At least 55 records · Page 3Linked to original sources
arXiv2016

Some remarks and conjectures related to lattice paths in strips along the x-axis

In the first part of this paper I give an elementary overview about some number sequences which count various sorts of lattice paths in strips along the x-axis and compute their generating functions in terms of Fibonacci and Lucas polynomials. In the second part I generalize these results by introducing suitable weights and study some special cases in more detail. In the course of this work I have been led to curious number triangles and various conjectures

math.CO↗
arXiv2016

Elementary observations on Rogers-Szegö polynomials

The Rogers-Szegö polynomials are natural q-analogues of Newton binomials. In general they have no closed expression. We consider some exceptional cases which are products of a factor with a closed formula and another one with nice values for q=1 and q=-1. These are related to normalized Rogers-Szegö polynomials which have nice expansions and Hankel determinants. The Exposition of the paper is elementary and almost self-contained.

math.CA↗
arXiv2013

Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results

The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.

math.CO↗
arXiv2013

A special class of Hankel determinants

In this expository paper we compute Hankel determinants of some sequences whose generating functions are given by C-fractions and derive orthogonality properties for associated polynomials.

math.CO↗
arXiv2012

Hankel determinants of some polynomial sequences

We give simple new proofs of some Hankel determinant evaluations by Omer Egecioglu and Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic and prove analogous results for sums of moments of symmetric orthogonal polynomials.

math.CO↗
arXiv2012

q-Chebyshev polynomials

In this overview paper a direct approach to q-Chebyshev polynomials and their elementary properties is given. Special emphasis is placed on analogies with the classical case. There are also some connections with q-tangent and q-Genocchi numbers.

math.CO↗
arXiv2011

Some nice Hankel determinants

I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the theory of orthogonal polynomials is not applicable. Even so our theorems and conjectures show some similarity with the relations between Catalan numbers and Fibonacci polynomials or between central binomial coefficients and Lucas polynomials.

math.CO↗