The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra su(1,1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and the discrete terms are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch-Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.
We introduce some multivariate analogues of Meixner, Charlier and Krawtchouk polynomials, and establish their main properties, that is, duality, degenerate limits, generating functions, orthogonality relations, difference equations, recurrence formulas and determinant expressions. A particularly important and interesting result is that "the generating function of the generating function" for the Meixner polynomials coincides with the generating function of the Laguerre polynomials, which has previously not been known even for the one variable case. Actually, main properties for the multivariat…
Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: generating and determinantal formulae, difference equations. As an application we consider the problem of evaluating moments related to a multivariate Barnes type integral involving the Harish-Chandra $c$-function of a symmetric cone.
We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $ρ_{a,c}$ and Hahn measure $ρ_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre pol…
We derive Mehler--Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $x=0,$ and determine the asymptotic limit of their zeros as the degree $n$ goes to infinity.
In this paper we prove in a constructing way that exceptional Charlier, Meixner, Hermite and Laguerre polynomials satisfy higher order recurrence relations. Our conjecture is that the recurrence relations provided in this paper have minimal order.
In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $Δ$--Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the $Δ$--Meixner--Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for L…
Using Casorati determinants of Meixner polynomials $(m_n^{a,c})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers a sequence of polynomials $m_n^{a,c;\F}$, $n\in σ_\F$, which are eigenfunctions of a second order difference operator, where $σ_\F$ is certain infinite set of nonnegative integers, $σ_\F \varsubsetneq \NN$. When $c$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $m_n^{a,c;\F}$, $n\in σ_\F$, are actually exceptional Meixner polynomials; that is, in addition, they are orthogonal and complete with respect to a positiv…
We study a new family of q-Meixner multiple orthogonal polynomials of the first kind. The discrete orthogonality conditions are considered over a non-uniform lattice with respect to different q-analogues of Pascal distributions. We address some algebraic properties, namely raising and lowering operators as well as Rodrigues-type. Based on the explicit expressions for the raising and lowering operator a high-order linear q-difference equation with polynomial coefficients for the q-Meixner multiple orthogonal polynomials of the first kind is obtained. Finally, we obtain the nearest neighbor recu…
For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous $q^{-1}$-Hahn and $q$-Lague…
Using the steepest descent method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou [Ann. Math. {\bf 137}(1993), 295-368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for $z$ on the positive real line by using the steepest descent method for integrals [Constr. Approx. {\bf 14}(1998), 113-150].
Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $"1"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier r…
For each fixed value of $β$ in the range $-2<β<-1$ and $0<c<1$, we investigate interlacing properties of the zeros of polynomials of consecutive degree for $M_{n}(x;β,c)$ and $M_k(x,β+t,c)$, $k\in\{n-1,n,n+1\}$ and $t\in\{0,1,2\}$. We prove the conjecture in [K. Driver and A. Jooste, Quasi-orthogonal Meixner polynomials, Quaest. Math. 40 (4) (2017), 477-490] on a lower bound for the first positive zero of the quasi-orthogonal order $1$ polynomial $M_n(x;β+1,c)$ and identify upper and lower bounds for the first few zeros of quasi-orthogonal order $2$ Meixner polynomials $M_n(x;β,c)$. We show th…
We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the $(q-)$Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials. These relations are then used to derive generalizations of generating functions for these orthogonal polynomials. The coefficients of these generalized generating functions are in general, given in term of multiple hyperg…
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to $r>1$ different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of th…
The purpose of this note is twofold: firstly to characterize all the sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ such that $$ \frac{\triangle}{{\bf \triangle} x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm{I})P_n(x(s-1/2)), $$ where $\mathrm{I}$ is the identity operator, $x$ defines a class of lattices with, generally, nonuniform step-size, and $\triangle f(s)=f(s+1)-f(s)$; and secondly to present, in a friendly way, a method to deal with these kind of problems.
We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\in\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\in\nn$ and $t=\frac1{p}$, the polynomials $K_n(x;p,N)=(-N)_n\phantom{}_2F_1(-n,-x;-N;\frac1{p})$, $n=0,1,...N$, $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $K_{n}(x;p,a)$, $0<p<1$ and $a>n-1$, the quasi-orthogonal polynomials $K_{n}(x;p,a)$, $k-1<a<k$, $k=1,...,n-1$ and $p>1$ or $p<0$ as well as the non-orthogonal pol…
Given a sequence of polynomials $(p_n)_n$, an algebra of operators $\mathcal{A}$ acting in the linear space of polynomials and an operator $D_p\in \mathcal{A}$ with $D_p(p_n)=np_n$, we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of $m$ consecutive $p_n$: $q_n=p_n+\sum_{j=1}^mβ_{n,j}p_{n-j}$. Using the concept of $\mathcal{D}$-operator, we determine the structure of the sequences $β_{n,j}, j=1,\ldots,m,$ in order that the polynomials $(q_n)_n$ are common eigenfunctions of an operator in the algebra $\mathcal{A}$. As an application, from the classical discret…
We construct a set $M_d$ whose points parametrize families of Meixner polynomials in $d$ variables. There is a natural bispectral involution $b$ on $M_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of $d$ commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the invol…
We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arvesú et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the equilibrium problem with constrain to describe their zero distribution. Then analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.
The aim of this paper is to give identities which are generalizations of the formulas given by Koornwinder [J. Math. Phys. 30, (1989)] and Hamdi-Zeng [J. Math. Phys. 51, (2010)]. Our proofs are much simpler than and different from the previous investigations.
Following the lead of J. Dehesa and his collaborators, we compute the Fisher information of the Meixner-Pollaczek, Meixner, Krawtchouk and Charlier polynomials.
In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+λ\mathscr T^j f (α)\mathscr T^{j}g(α), \] where ${\bf u}^{\tt M}$ is the Meixner linear operator, $λ\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $α\leq 0$, and $\mathscr T$ is the forward difference operator $Δ$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equatio…
An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary $q$-pseudorotation operators. These operators are built from $q$-exponentials of the $\mathcal{U}_q(\mathfrak{su}(1,1))$ generators. The orthogonality, recurrence relation, difference equation, and other properties of the $q$-Mexiner polynomials are systematically obtained in the proposed framework.
In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey-Wilson, Racah and $q$-Racah types. In this paper we explore those of the Meixner-Pollaczek and continuous Hahn types. For the $M$-indexed Meixner-Pollaczek and continuous Hahn polynomials, we present $3+2M$ term recurrence relations with variable dependent coefficients and $1+2L$ term ($L\geq M+1$) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation/annihilation operators of the quantum …
A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional $\mathbb{N}\times \mathbb{N}$ lattice. It is governed by a Hamiltonian expressed as a seven-point difference operator involving three parameters. The exact solutions of the model are given in terms of the two-variable Meixner polynomials orthogonal with respect to the negative trinomial distribution. The constants of motion of the system are constructed using the raising and lowering operators for these polynomials. They are shown to …
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q…
The classical polynomials of Meixner's type--Hermite, Charlier, Laguerre, Meixner, and Meixner--Pollaczek polynomials--are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold $X$ and an intensity measure $σ$ on it. We consider a Jacobi field in the extended Fock space over $L^2(X;σ)$, whose field operator at a point $x\in X$ is of the form $\di_x^†+…
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$_{\textup V}$.
We study multiple orthogonal polynomials of Meixner-Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of block Toeplitz matrices, for which we provide some general results of independent interest. The motivation for this paper is the study…
We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials are $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : a multi-index set), but they form a complete set of orthogonal basis in th…
An explicit bilinear generating function for Meixner-Pollaczek polynomials is proved. This formula involves continuous dual Hahn polynomials, Meixner-Pollaczek functions, and non-polynomial $_3F_2$-hypergeometric functions that we consider as continuous Hahn functions. An integral transform pair with continuous Hahn functions as kernels is also proved. These results have an interpretation for the tensor product decomposition of a positive and a negative discrete series representation of $\su(1,1)$ with respect to hyperbolic bases, where the Clebsch-Gordan coefficients are continuous Hahn funct…
We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek o…
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual stat…
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.
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.
This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials…
In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from t…
Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn).
A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on l^2(N), leading to a proof of Favard's theorem stating that polynomials satisfying a three-term recurrence relation are orthogonal polynomials. We discuss the link to the moment problem. In the second example, the spectral theorem is applied to Jacobi operators on l^2(Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operato…
The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, …
Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kampé de Fériet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kampé de Fériet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Piñeiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.
We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary (vector) orthogonal polynomial systems which are eigenfunctions of a difference operator into other systems of this type. While the extension of Charlier polynomilas is well known it is obtained by different methods. The extension of M…
We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner-Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials.
We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
