Functions Of Hypergeometric Type Research Articles

On some series of special functions

Series of the form ∑ k = 0 ∞ ∏ i = 1 p ( a i ) k ∏ j = 1 q ( b j ) k w k F k ( z ) are considered, where F k ( z ) are special functions of hypergeometric type. Various types of series involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials are represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables.

A functional realization of the Gelfand-Tsetlin base

A realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $\mathrm{GL}_n$ is considered. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which follows from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants, that is, they are values of some hypergeometric functions when instead of all arguments ones are substituted.

A functional realization of the Gelfand-Tsetlin base

A realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $\mathrm{GL}_n$ is considered. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which follows from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants, that is, they are values of some hypergeometric functions when instead of all arguments ones are substituted. Bibliography: 16 titles.

Explicit closed algebraic formulas for Orlov–Scherbinn-point functions

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.

Fractional Hypergeometric Functions

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions.

The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

CDF of non-central [formula omitted] distribution revisited. Incomplete hypergeometric type functions approach

The cumulative distribution function of the non-central chi-square distribution χν′2(λ),ν∈R+ possesses an integral representation in terms of a generalized Marcum Q-function. Regarding some already known-results, here we derive a simpler form of the cumulative distribution function for ν=2n∈N degrees of freedom. Also, we express these representations in terms of an incomplete Fox–Wright function pΨq(γ) and the generalized incomplete hypergeometric functions concerning the important special cases such as 1Γ1,2Γ1 and 2γ1. New identities are established between 1Γ1 and 2Γ1 as well.

Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.

Constellations and $\tau$-functions for rationally weighted Hurwitz numbers

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $\tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $\tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $\tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as "colour" and "flavour" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.

Inclusion properties of hypergeometric type functions and related integral transforms

In this work, conditions on the parameters $a, b$ and $c$ are given so that the normalized Gaussian hypergeometric function $zF(a,b;c;z)$, where \begin{align*} F(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n}z^n, \quad |z|<1, \end{align*} is in certain class of analytic functions. Using Taylor coefficients of functions in certain classes, inclusion properties of the Hohlov integral transform involving $zF(a,b;c;z)$ are obtained. Similar inclusion results of the Komatu integral operator related to the generalized polylogarithm are also obtained. Various results for the particular values of these parameters are deduced and compared with the existing literature.

Weighted Hurwitz Numbers and Topological Recursion

The KP and 2D Toda \(\tau \)-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel–Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.

Sonine Formulas and Intertwining Operators in Dunkl Theory

Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.

From positive geometries to a coaction on hypergeometric functions

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally- regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter ϵ. We show that the coaction defined on this class of integral is consistent, upon expansion in ϵ, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric p+1Fp and Appell functions.

Recurrence Relations of the Hypergeometric-type functions on the quadratic-type lattices

The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratictype lattices. We introduce some recurrence relations for such solutions by also considering their applications to polynomials on the quadratic-type lattices.

Pfaffian Equations and Contiguity Relations of the Hypergeometric Function of Type (&lt;i&gt;k&lt;/i&gt;+1, &lt;i&gt;k&lt;/i&gt;+&lt;i&gt;n&lt;/i&gt;+2) and Their Applications

We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ by using twisted cohomology groups and the intersection form on them. We apply our results to algebraic statistics; numerical evaluation of the normalizing constants of two way contingency tables with fixed marginal sums.

A formula for the nonsymmetric Opdam hypergeometric function of type $A_2$

The aim of this paper is to give an explicit formula for the nonsymmetric Heckman–Opdam hypergeometric function of type $A_2$. This is obtained by differentiating the corresponding symmetric hypergeometric function.

Fermionic Approach to Weighted Hurwitz Numbers and Topological Recursion

A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative and differential recursion relations are deduced for the adapted bases and their duals, and a Christoffel-Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion programme are derived, as well as the generalized cut and join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers.

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and a truncated Haar distributed unitary random matrix. These polynomials form a general class of hypergeometric functions of type 2 F r . We describe the oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel–Rotach type and subsequently use it to obtain the limiting distribution of the suitably rescaled zeros. Moreover, we show that the asymptotic zero distribution lies in the class of Raney distributions and by introducing appropriate coordinates elementary and explicit characterizations are derived for the densities as well as for the distribution functions.

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields R , C , H . \mathbb R, \mathbb C, \mathbb H. These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.