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Register a new userChebyshev type lattice path weight polynomials by a constant term method
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We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated weights as well as an arbitrary `background weight. Our CT theorem, like Viennots lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennots diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennots original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as models of steric stabilization and sensitized flocculation.
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Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three types: C-, S- and E-functions. Orbit functions of the Lie algebras An, or equivalently, of the Lie group SU(n+1), are considered. First, orbit functions in two different bases - one orthonormal, the other given by the simple roots of SU(n) - are written using the isomorphism of the permutation group of n elements and the Weyl group of SU(n). Secondly, it is demonstrated that there is a one-to-one correspondence between classical Chebyshev polynomials of the first and second kind, and C- and $S$-functions of the simple Lie group SU(2). It is then shown that the well-known orbit functions of SU(n) are straightforward generalizations of Chebyshev polynomials to n-1 variables. Properties of the orbit functions provide a wealth of properties of the polynomials. Finally, multivariate exponential functions are considered, and their connection with orbit functions of SU(n) is established.
A word $sigma=sigma_1...sigma_n$ over the alphabet $[k]={1,2,...,k}$ is said to be {em smooth} if there are no two adjacent letters with difference greater than 1. A word $sigma$ is said to be {em smooth cyclic} if it is a smooth word and in addition satisfies $|sigma_n-sigma_1|le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation.
We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.
Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $det((a+j-i)Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $Pf((j-i)Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.
Trace estimators allow to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.
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