Basic Hypergeometric Polynomials Research Articles

Pastro polynomials and Sobolev-type orthogonal polynomials on the unit circle based on a q-difference operator

ABSTRACT Applications of orthogonal polynomials on the unit circle have attracted the attention of many researchers in recent years. Pastro polynomials, which are basic hypergeometric polynomials, are known to be biorthogonal polynomials on the unit circle. However, with special choice of parameters they provide one of the nicest examples of orthogonal polynomials on the unit circle. Our objective here is to consider some properties of three sequences of polynomials which are related to these Pastro orthogonal polynomials on the unit circle by a q-difference operator. We also provide information regarding connection formulas, bounds for the connection coefficients as well as outer relative asymptotics associated with these sequences of polynomials.

Linearization Coefficients for Some Basic Hypergeometric Polynomials

In this paper, we give a simple and original method based on inverse relation to express explicitly the linearization coefficients for some general classes of basic hypergeometric polynomial set in terms of the basic Kampé de Fériet function. We use symbolic computation algorithms, namely, q -Multisum to find recurrence relations for the resulting linearization coefficients and qsum17 to solve some from the obtained recurrence relations. In some cases, the linearization coefficients are reduced to hypergeometric functions or hypergeometric terms.

Christoffel formula for kernel polynomials on the unit circle

Given a nontrivial positive measure μ on the unit circle T, the associated Christoffel–Darboux kernels are Kn(z,w;μ)=∑k=0nφk(w;μ)¯φk(z;μ), n≥0, where φk(⋅;μ) are the orthonormal polynomials with respect to the measure μ. Let the positive measure ν on the unit circle be given by dν(z)=|G2m(z)|dμ(z), where G2m is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing {Kn(z,w;ν)}n≥0 directly in terms of {Kn(z,w;μ)}n≥0.Furthermore, we consider the special case of w=1; it is known that appropriately normalized polynomials Kn(z,1;μ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters {cn(μ)}n=1∞ and {gn(μ)}n=1∞, with 0<gn<1 for n≥1. The double sequence {(cn(μ),gn(μ))}n=1∞ characterizes the measure μ. A natural question about the relation between the parameters cn(μ), gn(μ), associated with μ, and the sequences cn(ν), gn(ν), corresponding to ν, is also addressed.Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a measure for which the Christoffel–Darboux kernels, with w=1, are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel–Darboux kernels, again with w=1, are given by hypergeometric polynomials.

Connection and inversion coefficients for basic hypergeometric polynomials

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.

Properties of the zeros of generalized basic hypergeometric polynomials

We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property.

Interlacing properties and bounds for zeros of $${}_2\phi _1$$ 2 ϕ 1 hypergeometric and little q-Jacobi polynomials

Let \(\displaystyle \{p_n\}_{n=0}^{\infty }\), where \(p_n\) is a polynomial of degree n, be a sequence of polynomials orthogonal with respect to a positive probability measure. If \(x_{1,n} < \cdots < x_{n,n}\) denotes the zeros of \(p_n\) while \(x_{1,n-1} < \cdots < x_{n-1,n-1}\) are the zeros of \(p_{n-1}\), the inequality $$\begin{aligned} x_{1,n} < x_{1,n-1} < x_{2,n} < \cdots < x_{n-1,n}< x_{n-1,n-1}< x_{n,n}, \end{aligned}$$ known as the interlacing property, is satisfied. We use a consequence of a generalised version of Markov’s monotonicity results to investigate interlacing properties of zeros of contiguous basic hypergeometric polynomials associated with little q-Jacobi polynomials and determine inequalities for extreme zeros of the above two polynomials. It is observed that the new bounds which are obtained in this paper give more precise upper bounds for the smallest zero of little q-Jacobi polynomials, improving previously known results by Driver and Jordaan (Math Model Nat Phenom 8(1):48–59, 2013), and in some cases, those by Gupta and Muldoon (J Inequal Pure Appl Math 8(1):7, 2007). Numerical examples are given in order to illustrate the accuracy of our bounds.

Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras

In this paper we introduce a basic representation for the confluent Cherednik algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials.

Basic hypergeometric polynomials with zeros on the unit circle

The objective in this manuscript is to present two specific sequences of polynomials with elements given in terms of basic hypergeometric polynomials or q-hypergeometric polynomials. It is shown that the zeros of these polynomials are simple and that they lie exactly on the unit circle.

D-Orthogonality of some basic hypergeometric polynomials

d-Orthogonality of some basic hypergeometric polynomials

Formula omitted]-orthogonality of Little [formula omitted]-Laguerre type polynomials

In this paper, we solve a characterization problem in the context of the d -orthogonality. That allows us, on one hand, to provide a q -analog for the d -orthogonal polynomials of Laguerre type introduced by the first author and Douak, and on the other hand, to derive new L q -classical d -orthogonal polynomials generalizing the Little q -Laguerre polynomials. Various properties of the resulting basic hypergeometric polynomials are singled out. For d = 1 , we obtain a characterization theorem involving, as far as we know, new L q -classical orthogonal polynomials, for which we give the recurrence relation and the difference equation.

A non-standard generating function for continuous dual q-Hahn polynomials

We study a non-standard form of generating function for the three-parameter continuous dual q-Hahn polynomials pn(x; a, b, c | q), which has surfaced in a recent work of the present authors on the construction of lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials. We show that the resulting generating function identity for the continuous dual q-Hahn polynomials pn(x; a, b, c | q) can be explicitly stated in terms of Jackson’s q-exponential functions eq(z).

On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials

We construct a q-difference operator that lifts the continuous q-Hermite polynomials Hn(x| q) of Rogers up to the continuous big q-Hermite polynomials Hn(x; a| q) on the next level in the Askey scheme of basic hypergeometric polynomials. This operator is defined as Exton's q-exponential function εq(aqDq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator . We next show that one can move two steps more upwards in order first to reach the Al-Salam–Chihara family of polynomials Qn(x; a, b | q), and then the continuous dual q-Hahn polynomials pn(x; a, b, c| q). In both these cases, lifting operators, respectively, turn out to be convolution-type products of two and three one-parameter q-difference operators of the same type εq(aqDq) at the initial step. At each step, we also determine q-difference operators that lift the weight function for the continuous q-Hermite polynomials Hn(x| q) successively up to the weight functions for Hn(x; a| q), Qn(x; a, b | q) and pn(x; a, b, c| q).

A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or $6 - j$ Symbols

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.