Special functions and reversible three-term recurrence formula (R3TRF)

Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or q-quadratic lattice

Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's theorem states that the converse is true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or $q$-quadratic lattice is recalled. The Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmers…

Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-β$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-β)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to solutions of the fifth Painlevé equation PV (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.

带对数权函数的正交多项式的递推系数

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

本文利用Riemann-Hilbert方法证明了带对数权函数的正交多项式递推系数的渐近公式，证实了Magnus猜想的特例，并克服了奇点附近缺乏局部参数矩阵的困难。

Fourier transforms of some special functions in terms of orthogonal polynomials on the simplex and continuous Hahn polynomials

In this paper Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained by using the Parseval's identity and several recurrence relations are derived.

Difference equations of q-Appell polynomials

In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results. We also provide the recurrence relations and the q-difference equations for q-Bernoulli polynomials, q-Euler polynomials, q-Genocchi polynomials and for newly defined q-Hermite polynomials, as special cases of q-Appell polynomials

One-Step Recurrences for Stationary Random Fields on the Sphere

Recurrences for positive definite functions in terms of the space dimension have been used in several fields of applications. Such recurrences typically relate to properties of the system of special functions characterizing the geometry of the underlying space. In the case of the sphere ${\mathbb S}^{d-1} \subset {\mathbb R}^d$ the (strict) positive definiteness of the zonal function $f(\cos θ)$ is determined by the signs of the coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials $\{C^λ_n\}$, with $λ=(d-2)/2$. Recent results show that classical differentiation and integ…

Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.

The para-Racah polynomials

New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed persymmetric Jacobi matrix. The orthogonality relation and an explicit expression in terms of hypergeometric functions are also given. Special cases and connections with the para-Krawtchouk polynomials and the dual-Hahn polynomials are also discussed.

变递推关系的初等比率与零点极限

Elementary ratio and zero limits for varying recurrences

本文基于Loureiro和Van Assche的相关定理，研究具有任意系数周期的变高阶三项递推关系的比率与零点极限，运用多种方法完成证明并得出幂律极限等特例结果。

On the Properties of Special Functions on the linear-type lattices

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to $q$-classical polynomials are given.

$q$-Discrete Painlevé equations for recurrence coefficients of modified $q$-Freud orthogonal polynomials

We present an asymmetric $q$-Painlevé equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution which gives the recurrence coefficients. We establish a connection with $α-q-P_V$.

Preud's equations for orthogonal polynomials as discrete Painlevé equations

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2tw_n + w_n(w_n+1 + w_n + w_n-1) for the recurrence coefficients p_n+1 = xp_n - w_np_n-1 of the orthogonal polynomials related to the weight w(x) = exp(-4(tx^3 + x^4)) (notation of [26, pp. 34-36])…

Recurrence Coefficients of a New Generalization of the Meixner Polynomials

We investigate new generalizations of the Meixner polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-β$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-β)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlevé equation P$_{\textup V}$. Initial conditions for different lattices can be transformed to the classical solutions of P$_{\textup V}$ with special values of the parameters. We also study one property of the Bäcklund transformation of P$_{\textup V}$.

有限马尔可夫链与多重正交多项式

Finite Markov chains and multiple orthogonal polynomials

本文利用正交多项式与多重正交多项式的递推矩阵构建有限马尔可夫链，推导其谱理论与多项性质，揭示生灭过程因式分解特性，并结合超几何多项式实例验证，构建对应常返链与反转链。

The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.

Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrenc…