Hypergeometric Polynomials Research Articles

Hypergeometric polynomials are optimal

With any integer convex polytope $$P\subset {\mathbb {R}}^n$$ we associate a multivariate hypergeometric polynomial whose set of exponents is $${\mathbb {Z}}^{n}\cap P.$$ This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn’s type. We prove that under certain nondegeneracy conditions any such polynomial is optimal in the sense of [7], i.e., that the topology of its amoeba [13] is as complex as it could possibly be. Using this, we derive optimal properties of several classical families of multivariate hypergeometric polynomials.

Open Problem in Orthogonal Polynomials

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line and two of its discrete versions; one with a finite spectrum and another with countably infinite spectrum. A second class of these new orthogonal polynomials appeared recently while solving a Heun-type equation. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the Askey scheme of hyper-geometric orthogonal polynomials. Up to now, these polynomials are defined by their three-term recursion relations and initial values. However, their other properties like the weight functions, generating functions, orthogonality, Rodrigues-type formulas, etc. are yet to be derived analytically. Due to the prime significance of these polynomials in physics and mathematics, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g., in terms of hypergeometric functions).

On some classical type Sobolev orthogonal polynomials

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: 2F2(−n,1;α+1,κ+1;x) and 3F2(−n,n+α+β+1,1;α+1,κ+1;x) (α,β,κ>−1, n=0,1,…), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: Ly+λnDy=0, where L,D are linear differential operators with polynomial coefficients not depending on n. For nonnegative integer values of the parameter κ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained. Some generalizations of these polynomials are discussed as well.

Some expansions of functions based upon two sequences of hypergeometric polynomials

In this paper, we introduce two series expansions whose basis functions are two sequences of hypergeometric polynomials. We then present a systematic and detailed study of the general properties of each of these series expansions. We consider several illustrative examples to show the computational efficiency of our expansions in comparison to the usual Taylor series expansion. We also discuss some expansions of mixed types which are related to the results and assertions presented in this paper.

Characterization of Kummer hypergeometric Bernoulli polynomials and applications

In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials Ba,b,n(x) and Kummer hypergeometric polynomials of the second kind Ka,b,n(x), which are respectively defined by the exponential generating functions:extM(a,a+b;t)=∑n=0∞Ba,b,n(x)tnn! and extU(a,a+b;t)=∑n=0∞Ka,b,n(x)tnn! with M(a,b;t)=∑n=0∞(a)n(b)ntnn!, where U(a,a+b;t) is the Kummer hypergeometric function of the second kind.First we construct Gauss–Weierstrass-type convolution operators Twa,b with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions.

$$\mathbb {C}P^{2S}$$ Sigma Models Described Through Hypergeometric Orthogonal Polynomials

The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb{C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any such projector solutions of the $\mathbb{C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrised in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb{C}P^{2S}$ models defined using the generalised Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the $\mathfrak{su}(2s+1)$ algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the $\mathfrak{su}(2)$ spin-s representation and the $\mathbb{C}P^{2S}$ model is explored in detail.

Product of two hypergeometric functions with power arguments

The hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt, respectively. The result is a power series in t whose coefficients are hypergeometric polynomials. Brychkov has extended the product formula such that in one argument the variable t can be squared. We obtain a further extension such that in both arguments the variable t can be raised to positive integer powers.

Asymptotics of the Wilson polynomials

In this paper, we study the asymptotic behavior of the Wilson polynomials [Formula: see text] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [Formula: see text] is fixed and (ii) when the variable is rescaled as [Formula: see text] with [Formula: see text]. Case (ii) has two subcases, namely, (a) zero-free zone ([Formula: see text]) and (b) oscillatory region [Formula: see text]. Corresponding results are also obtained in these cases (iii) when [Formula: see text] lies in a neighborhood of the transition point [Formula: see text], and (iv) when [Formula: see text] is in the neighborhood of the transition point [Formula: see text]. The expansions in the last two cases hold uniformly in [Formula: see text]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.

Matiyasevich-type identities for hypergeometric Bernoulli polynomials and poly-Bernoulli polynomials

Matiyasevich-type identities for hypergeometric Bernoulli polynomials and poly-Bernoulli polynomials

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

We establish a new connection between moments of {n times n} random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments {mathbb{E}{rm Tr} X_n^{-s}} as a function of the complex variable {s in mathbb{C}} , whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. (J Math Phys 57:111901, 2016) on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order {n rightarrow infty} asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials.

The inverse and derivative connecting problems for some hypergeometric polynomials

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=x^n$ the connection problem is called the inversion problem associated to $\{P_n(x)\}_{n\geq 0}.$ The particular case $Q_n(x)=P'_{n+1}(x)$ is called the derivative connecting problem for polynomial family $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form $${}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, n+a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ \pm n +b \end{array} \right | z \right],$$ where $\displaystyle {}_2 F_1 \left[ \left. \begin{array}{c} a, b \\ c \end{array} \right | z \right] =\sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!},$ is the Gauss hypergeometric function and $(x)_n$ denotes the Pochhammer symbol defined by $$\displaystyle (x)_n=\begin{cases}1, n=0, \\x(x+1)(x+2)\cdots (x+n-1) , n>0.\end{cases}$$ All polynomials are considered over the field of real numbers.

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

The Krein-like r-functionals of the hypergeometric orthogonal polynomials {pn(x)}, with the kernel of the form xs[ω(x)]βpm1(x)…pmr(x) being ω(x) the weight function on the interval Δ∈R, are determined by means of the Srivastava linearization method. The particular 2-functionals, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments and the novel Krein-like moments. Moreover, various related types of exponential and logarithmic functionals are also investigated.

A Note on the Bailey Transform, the Bailey Pair and the WP Bailey Pair and Their Applications

The subject of q-analysis, which is popularly known as quantum analysis, has its roots in such important areas as (for example) Mathematical Physics, Analytic Number Theory, and the Theory of Partitions. Our present investigation is motivated essentially by the potential for applications of q-series and q-polynomials, especially the basic (or q-) and bibasic hypergeometric functions and the corresponding hypergeometric polynomials. The main object of this paper is to investigate the Bailey transform, the Bailey pair and the WP-Bailey pair of sequences and some of their applications in order to establish a number transformation formulas for q-hypergeometric series as well as bi-basic hypergeometric series, together with several other related q-series identities.

Orthogonal Polynomials in Mathematical Physics

This is a review of ([Formula: see text]-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ([Formula: see text]-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra. To the memory of Ludwig Dmitrievich Faddeev

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.

Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of order $k\le n-1$. The polynomials in the sequence $\{Q_{n,k}\}_{n\geq 0}$ are obtained from $P_{n}$, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable and these equations are used to prove quasi-orthogonality of order $k$. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence $\{P_n\}_{n\geq 0}$, where possible.

Truncated euler polynomials

Abstract We define a truncated Euler polynomial E m,n (x) as a generalization of the classical Euler polynomial En (x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-P\'{o}lya class via hyperbolic polynomials play an important role in this paper. Some open problems are also stated, which may be of interest for further research.

Generalized hypergeometric series for Racah matrices in rectangular representations

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group [Formula: see text] through the Askey–Wilson polynomials, associated with the [Formula: see text]-hypergeometric functions [Formula: see text]. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary [Formula: see text] — at least for exclusive Racah matrices [Formula: see text]. The natural question then is what substitutes the conventional [Formula: see text]-hypergeometric polynomials when representations are more general? New advances in the theory of matrices [Formula: see text], provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of [Formula: see text]. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.