The Moment Problem Associated with the q -Laguerre Polynomials

   Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.     

Plancherel–Rotach Asymptotics for q-Orthogonal Polynomials

We establish the Plancherel–Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q ^?1-Hermite polynomials of Askey, Ismail, and Masson; q-Laguerre polynomials; and the Stieltjes–Wigert polynomials. We also introduce a one-parameter family of solutions to the q-difference equation of the Ramanujan function.     

q-Laguerre polynomials and related q-partial differential equations

In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in [J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.     

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

Expansions in Terms of q-Laguerre Heat Polynomials

In a previous paper M. S. Ben Hammouda and Akram Nemri derived criteria for the expansion of solutions u(x, t) from the generalized q-heat equation, in series of polynomial solutions \({p_{n}^{\alpha}}\) , thus extending an analogous theory of the ordinary heat equation developed by P. C. Rosenbloom and D. V. Widder. It is the goal to carry out a parallel study for the q-Laguerre differential heat equation and establish the region of convergence of the series of q-Laguerre heat polynomials and their temperature transforms.     

A q-Analogue of the Stirling Formula and a Continuous Limiting Behaviour of the q-Binomial Distribution—Numerical Calculations

In this article, we derive an asymptotic formula for the q-factorial number of order n using the saddle point method. This formula is a q-analogue, for 0?<?q?<?1, of the usual Stirling formula for the factorial number of order n. Also, this formula is used to provide a continuous limiting behaviour of the q-Binomial distribution in the sense of pointwise convergence. Specifically, the q-Binomial distribution converges to a continuous Stieltjes–Wigert distribution. Furthermore, we present some numerical calculations, using the computer program MAPLE, indicating a quite strong convergence.     

The moment problem associated with the Stieltjes-Wigert polynomials

We consider the indeterminate Stieltjes moment problem associated with the Stieltjes-Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes-Wigert polynomials, expressions for the entire functions A, B, C, and D from the Nevanlinna parametrization are obtained. We describe T⁽ⁿ⁾(μ) for when μ=μ₀ is a particular N-extremal solution and explain in detail what happens when n→∞.     

Plancherel-Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q∈(0,1), the asymptotics for Euler's q-exponential, q-Gamma function Γq(x), q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function (q-Airy function), Jackson's q-Bessel function of second kind, Ismail-Masson orthogonal polynomials (q⁻¹-Hermite polynomials), Stieltjes-Wigert polynomials, q-Laguerre polynomials could be derived as special cases.     

Integral representations and holonomic q-difference structure of Askey–Wilson type functions

We derive in a unified way the difference equations for Askey–Wilson polynomials and their Stieltjes transforms, by using basic properties of the de Rham cohomology associated with q-integral representations (Jackson integrals of BC ₁ type) of these functions.     

Limit relations between q-Krall type orthogonal polynomials

In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a q-classical linear functional via the addition of one or two mass points. The limit relations between the q-Krall type modification of big q-Jacobi, little q-Jacobi, big q-Laguerre, and other families of the q-Hahn tableau are established.     

On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

Uniform asymptotics of some q-orthogonal polynomials

Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q⁻¹-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials.     

The Schur algorithm for an indefinite Stieltjes moment problem

The nondegenerate truncated indefinite Stieltjes moment problem in the class of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step‐by‐step algorithm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in [11]. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.     

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (s_j)^∞_{j = 0} of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case.     

Plancherel-Rotach asymptotics for some q-orthogonal polynomials with complex scalings

In this work we study the Plancherel-Rotach type asymptotics for Stieltjes-Wigert, q-Laguerre and Ismail-Masson orthogonal polynomials with complex scalings. The main terms of the asymptotics for Stieltjes-Wigert and q-Laguerre polynomials (Ismail-Masson polynomials) contain Ramanujan function Aq(z) for scaling parameters above the vertical line R(s)=2 (); the main terms of the asymptotics involve theta function for scaling parameters in the vertical strip 0<R(s)<2 (). When scaling parameters in the vertical strips, the number theoretical properties of scaling parameters completely determine the orders of the error terms. These asymptotic formulas may provide some insights to new random matrix models and also add a new link between special functions and number theory.     

Stieltjes Moment Problems and the Friedrichs Extension of a Positive Definite Operator

For an indeterminate Stieltjes moment sequence the multiplication operator Mp(x) = xp(x) is positive definite and has self-adjoint extensions. Exactly one of these extensions has the same lower bound as M, the so-called Friedrichs extension. The spectral measure of this extension gives a certain solution to the moment problem and we identify the corresponding parameter value in the Nevanlinna parametrization of all solutions to the moment problem. In the case where σ is indeterminate in the sense of Stieltjes, relations between the (Nevanlinna matrices of) entire functions associated with the measures t^kdσ(t) are derived. The growth of these entire functions is also investigated.     

Stieltjes Sturm-Liouville equations: Eigenvalue dependence on problem parameters

We examine properties of eigenvalues and solutions to a 2n-dimensional Stieltjes Sturm-Liouville eigenvalue problem. Existence and uniqueness of a solution has been established previously. An earlier paper considered the corresponding initial value problem and established conditions which guarantee that solutions depend continuously on the coefficients [L.E. Battle, Solution dependence on problem parameters for initial value problems associated with the Stieltjes Sturm-Liouville equations, Electron. J. Differential Equations 2005 (2) (2005) 1-18]. Here, we find conditions which guarantee that the eigenvalues and solutions depend continuously on the coefficients, endpoints, and boundary data. For a simplified two-dimensional problem, we find conditions which guarantee the eigenvalues to be differentiable functions of the problem data.     

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.