Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.      

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

On some properties of polynomials related to hypergeometric modular forms

We find the discriminants, Galois groups, and prove the irreducibility of certain hypergeometric polynomials, which are closely related to modular forms and supersingular elliptic curves. 2000 Mathematics Subject Classification Primary—33C45; Secondary—11F11     

Explicit representations of biorthogonal polynomials

Given a parametrised weight function (x,) such that the quotients of its consecutive moments are Möbius maps, it is possible to express the underlying biorthogonal polynomials in a closed form [5]. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such obeys (inx) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying biorthogonal polynomials.     

On linearly related sequences of derivatives of orthogonal polynomials

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P_n)_n and (Q_n)_n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as

for all n=0,1,2,…, where M and N are fixed nonnegative integer numbers, and r_i,n and s_i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P_n)_n and (Q_n)_n (resp.). Assuming 0mk, we prove the existence of four polynomials Φ_M+m+i and Ψ_N+k+i, of degrees M+m+i and N+k+i (resp.), such that

the (k−m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k=m, then u and v are connected by a rational modification. If k=m+1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k>m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k−m with polynomial coefficients.     

The relation of the d-orthogonal polynomials to the Appell polynomials

We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.

A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.     

Formulas for recurrence coefficients of orthogonal polynomials related to Lorentzian-like weights

One considers the recurrence relation of orthogonal polynomials related to weights |t|^A(1+t^2r/c^2r)^-B on the whole real line, for various integer exponents 2r, and real A>-1, B>0.     

On the linear functionals associated to linearly related sequences of orthogonal polynomials

An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials n(Pn) and n(Qn), respectively, in the sense

     

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form ²|h(z)|, with h(z) a polynomial without zeros in |z|<1. The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.     

A characterization of some q-orthogonal polynomials

We show that the only orthogonal polynomials satisfying a q-difference equation of the form π(x)D _q P _n (x) = (α _n x + β _n )P _n (x) + γ _n P _n−1(x) where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q-Laguerre, the little and big q-Jacobi, and the q-Bessel polynomials. This is a q-analog of the work carried out in [1]. 2000 Mathematics Subject Classification Primary—33C45, 33D45     

Univalent polynomials and fractional order differences of their coefficients

Using fractional order differences, we find sufficient conditions on the coefficients of certain polynomials defined on the unit disk such that the zeros of these polynomials does not lie in D.     

On a new family related to truncated exponential and Sheffer polynomials

In this article, the truncated exponential and Sheffer polynomials are combined to introduce the 2-variable truncated-exponential based Sheffer polynomials (2VTESP) by using operational methods. Examples of certain special polynomials belonging to this family are considered. Operational correspondence between the 2VTESP and Sheffer polynomials is established, which is applied to derive the results for some members belonging to the 2VTESP family.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

On non-null distributions connected with testing that a real normal distribution is complex

The exact and the asymptotic non-null distribution of the maximal invariant corresponding to testing that the covariance matrix of a 2m-dimensional real normal distribution has complex structure is obtained.