A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

Matrix calculus and related hypergeometric functions

Using only simple tools of a version of matrix calculus we describe a way how to generate series expansions of hypergeometric functions of one (or two) matrix argument(s) in a straightforward manner without the need of computing the underlying zonal polynomials.     

Accurate and efficient evaluation of Schur and Jack functions

We present new algorithms for computing the values of the Schur and Jack functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data ( 0$">) and run in time that is only linear in .

     

Expressions for some hypergeometric functions of matrix argument with applications

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 × 2 Wishart matrix are also given.     

Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules

We have studied the asymptotics of two special two-matrix hypergeometric functions. The validity of the asymptotic expressions for these functions is seen in several selected numerical comparisons between the exact and asymptotic results. These hypergeometric functions find applications in configuration statistics of macromolecules as well as multivariate statistics.This work was supported by grant DE-FG06-84ER45123 from the Department of Energy, U.S.A.     

Nonnegative rank of a matrix with one negative eigenvalue

We show that a rank-three symmetric matrix with exactly one negative eigenvalue can have arbitrarily large nonnegative rank.     

Some identities on Catalan numbers and hypergeometric functions via Catalan matrix power

In this paper we use the Catalan matrix power as a tool for deriving identities involving Catalan numbers and hypergeometric functions. For that purpose, we extend earlier investigated relations between the Catalan matrix and the Pascal matrix by inserting the Catalan matrix power and particulary the squared Catalan matrix in those relations. We also pay attention to some relations between Catalan matrix powers of different degrees, which allows us to derive the simplification formula for hypergeometric function ₃F₂, as well as the simplification formula for the product of the Catalan number and the hypergeometric function ₃F₂. Some identities involving Catalan numbers, proved by the non-matrix approach, are also given.     

Sharp power mean bounds for the Gaussian hypergeometric function

Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form _q+1F_q(−x) with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of _q+1F_q(−x) at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ₃F₂(1) and leads to an integral representations of ₄F₃(x) in terms of the Appell function F₃. In the last section of the paper we list some open questions and conjectures.     

Asymptotic properties of a new Sturm–Liouville problem with retarded argument

In this work, we study the asymptotic properties of a new Sturm–Liouville problem with retarded argument. Contrary to previous works, differential equation includes eigenparameter as a quadratic function. In the considered problem arise new difficulties. Copyright © 2013 John Wiley & Sons, Ltd.     

The bialternate matrix product revisited

The well known bialternate product of two square matrices is re-examined together with another matrix product defined by means of the permanent function and having similar properties. Old and new results concerning both products are presented in a unified manner. A simple and elegant relation with the Kronecker product of matrices is also given.     

On a Sturm–Liouville type problem with retarded argument

In this work, a Sturm–Liouville‐type problem with retarded argument, which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

Some explicit formulas for the polynomial decomposition of the matrix exponential and applications

This paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix A. More precisely, we suppose that the minimal polynomial M_A(X) of A is known and has degree m. Therefore, e^tA is given in terms of P₀(A),…,P_m−1(A), where the P_j(A) are polynomials in A of degree less than m, and some explicit analytic functions. Examples and applications are given. In particular, the two cases m=5 and m=6 are considered.     

On the hypergeometric matrix functions of two variables

We introduce the generalized hypergeometric function with matrix parameters. We also define two variable Appell matrix functions and find their regions of convergence as well as integral representations.     

A new hypergeometric identity linking coefficients of a certain class of homogeneous polynomials motivated from magnetohydrodynamics

Considerations of a particular limit of the magnetohydrodynamic equations, appropriate for the generation of magnetic field in planetary interiors, lead to a set of constraints involving a certain class of homogeneous polynomials. This set is significantly degenerate owing to an identity satisfied by the polynomial coefficients, which involves linear combinations of simple Gauss hypergeometric functions. A generalised version of this new identity is proved by appealing to Wilf–Zeilberger theory.