Showing papers on "Wilson polynomials published in 2019"

Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations

Abstract: In the paper, the authors discuss the Bell polynomials and a sequence of polynomials applied to the theory of differential equations. Concretely speaking, the authors find four explicit formulas for these polynomials and for derivatives of generating functions of these polynomials, establish four identities between these two kinds of polynomials, and significantly simplify some known results.

Resummation at finite conformal spin

Abstract: We generalize the computation of anomalous dimension and correction to OPE coefficients at finite conformal spin considered recently in [1, 2] to arbitrary space-time dimensions. By using the inversion formula of Caron-Huot and the integral (Mellin) representation of conformal blocks, we show that the contribution from individual exchanges to anomalous dimensions and corrections to the OPE coefficients for “double-twist” operators $$ {\left[{\mathcal{O}}_1{\mathcal{O}}_2\right]}_{\Delta, J} $$ in s-channel can be written at finite conformal spin in terms of generalized Wilson polynomials. This approach is democratic with respect to space-time dimensions, thus generalizing the earlier findings to cases where closed form expressions of the conformal blocks are not available.

Solution of an Open Problem about Two Families of Orthogonal Polynomials

Abstract: An open problem about two new families of orthogonal polynomials was posed by Alhaidari. Here we will identify one of them as Wilson polynomials. The other family seems to be new but we show that they are discrete orthogonal polynomials on a bounded countable set with one accumulation point at 0 and we give some asymptotics as the degree tends to infinity.

Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials

Abstract: We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These "quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

Abstract: In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type $$_4F_{3}(1)$$ . With the aid of some standard reduction formulae such as Pfaff-Saalschutz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

Classical skew orthogonal polynomials in a two-component log-gas with charges $+1$ and $+2$

Abstract: There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$ invariant random matrix ensembles. The solvability of this log-gas system relies on the construction of particular skew orthogonal polynomials, with the skew inner product a linear combination of the $\beta = 1$ and $\beta = 4$ inner products, each involving weight functions. For suitably related classical weight functions, we seek to express the skew orthogonal polynomials as linear combinations of the underlying orthogonal polynomials. It is found that in each case (Gaussian, Laguerre, Jacobi and generalised Cauchy) the coefficients can be expressed in terms of hypergeometric polynomials with argument relating to the fugacity. In the Jacobi case, for example, these are a special case of the Wilson polynomials.

Connections between vector-valued and highest weight Jack and Macdonald polynomials

Abstract: We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of the representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.

Spherical and Cherednik-Opdam transforms of Jacobi-type polynomials

Abstract: The spherical transform maps the orthogonal basis of symmetric Jacobi-type polynomials to an orthogonal basis of (symmetric) Wilson polynomials. The spherical transform is closely related to the Cherednik-Opdam transform, as it is essentially its symmetric version. The symmetric Jacobi-type polynomials can be composed from the non-symmetric Jacobi-type polynomials. These relations, between the symmetric and non-symmetric theory, give an incentive to consider the Cherednik-Opdam transform of non-symmetric Jacobi-type polynomials. This work gives an overview of the symmetric theory about the spherical transform of Jacobi-type polynomials and lays down the groundwork for the Cherednik-Opdam transform of the non-symmetric Jacobi-type polynomials.