5 research outputs found
Catalan-like numbers and Hausdorff moment sequences
In this paper we show that many well-known counting coefficients, including
the Catalan numbers, the Motzkin numbers, the central binomial coefficients,
the central Delannoy numbers are Hausdorff moment sequences in a unified
approach. In particular we answer a conjecture of Liang at al. which such
numbers have unique representing measures. The smallest interval including the
support of representing measure is explicitly found. Subsequences of
Catalan-like numbers are also considered. We provide a necessary and sufficient
condition for a pattern of subsequences that if sequences are the Stieltjes
Catalan-like numbers, then their subsequences are Stieltjes Catalan-like
numbers. Moreover, a representing measure of a linear combination of
consecutive Catalan-like numbers is studied
Hankel determinants for convolution powers of Catalan numbers
The Hankel determinants
of
the convolution powers of Catalan numbers were considered by Cigler and by
Cigler and Krattenthaler. We evaluate these determinants for by
finding shifted periodic continued fractions, which arose in application of
Sulanke and Xin's continued fraction method. These include some of the
conjectures of Cigler as special cases. We also conjectured a polynomial
characterization of these determinants. The same technique is used to evaluate
the Hankel determinants . Similar results are obtained.Comment: 29 page
Orthogonal polynomials and Hankel Determinants for certain Bernoulli and Euler Polynomials
Using continued fraction expansions of certain polygamma functions as a main
tool, we find orthogonal polynomials with respect to the odd-index Bernoulli
polynomials and the Euler polynomials , for
. In the process we also determine the corresponding Jacobi
continued fractions (or J-fractions) and Hankel determinants. In all these
cases the Hankel determinants are polynomials in which factor completely
over the rationals
Hankel Determinants of sequences related to Bernoulli and Euler Polynomials
We evaluate the Hankel determinants of various sequences related to Bernoulli
and Euler numbers and special values of the corresponding polynomials. Some of
these results arise as special cases of Hankel determinants of certain sums and
differences of Bernoulli and Euler polynomials, while others are consequences
of a method that uses the derivatives of Bernoulli and Euler polynomials. We
also obtain Hankel determinants for sequences of sums and differences of powers
and for generalized Bernoulli polynomials belonging to certain Dirichlet
characters with small conductors. Finally, we collect and organize Hankel
determinant identities for numerous sequences, both new and known, containing
Bernoulli and Euler numbers and polynomials
Hankel determinants of linear combinations of moments of orthogonal polynomials
We prove evaluations of Hankel determinants of linear combinations of moments
of orthogonal polynomials (or, equivalently, of generating functions for
Motzkin paths), thus generalising known results for Catalan numbers.Comment: 28 pages, AmS-TeX; some typos correcte