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

Many distributions in multivariate analysis can be expressed in a form involving hypergeometric functions $_pF_q$ of matrix argument e.g. the noncentral Wishart $(_0F_1)$ and the noncentral multivariate $F(_1F_1)$. For an exposition of distributions in this form see James [9]. The hypergeometric function $_pF_q$ has been defined by Constantine [1] as the power series representation \begin{equation*}\tag{1.1} _pF_q(a_1,\cdots, a_p; b_1,\cdots, b_q; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a_1)_\kappa\cdots(a_p)_\kappa}{(b_1)_\kappa\cdots (b_q)_\kappa} \frac{C_\kappa (R)}{k!}\end{equation*} where $a_1,\cdots, a_p, b_1,\cdots, b_q$ are real or complex constants, $(a)_\kappa = \mathbf{prod}^m_{i=1}(a - \frac{1}{2}(i - 1))_{k_i},\quad (a)_n = a(a + 1)\cdots (a + n - 1)$ and $C_\kappa(R)$ is the zonal polynomial of the $m \times m$ symmetric matrix $R$ corresponding to the partition $\kappa = (k_1, k_2,\cdots, k_m), k_1 \geqq k_2 \geqq \cdots \geqq k_m$, of the integer $k$ into not more than $m$ parts. The functions defined by (1.1) are identical with the hypergeometric functions defined by Herz [5] by means of Laplace and inverse Laplace transforms. For a detailed discussion of hypergeometric functions and zonal polynomials, the reader is referred to the papers [1] of Constantine and [7], [8], [9] of James. From a practical point of view, however, the series (1.1) may not be of great value. Although computer programs have been developed for calculating zonal polynomials up to quite high order, the series (1.1) may converge very slowly. It appears that some asymptotic expansions for such functions must be obtained. It is well known that asymptotic expansions for a function can in many cases be derived using a differential equation satisfied by the function (see e.g. Erdelyi [4]), and so, with this in mind, a study of differential equations satisfied by certain hypergeometric functions certainly seems justified. In this paper a conjecture due to A. G. Constantine is verified i.e. it is shown that the function \begin{equation*}\tag{1.2} _2F_1(a, b; c; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a)_\kappa(b)_\kappa}{(c)_\kappa} \frac{C_\kappa(R)}{K!}\end{equation*} satisfies the system of partial differential equations \begin{align*} \tag{1.3} R_i(1 &- R_i)\partial^2F/\partial R_i^2 + \{ c - \frac{1}{2}(m - 1) - (a + b + 1 - \frac{1}{2}(m - 1))R_i \\ &+\frac{1}{2} \sum^m_{j=1,j\neq i}\lbrack R_i(1 - R_i)/(R_i - R_j) \rbrack\}\partial F/\partial R_i \\ &-\frac{1}{2} \sum^m_{j=1,j\neq i} \lbrack R_j(1 - R_j)/(R_i - R_j) \rbrack\partial F/\partial R_j = abF \quad (i = 1,2, \cdots, m)\end{align*} where $R_1, R_2,\cdots, R_m$ are the latent roots of the complex symmetric $m \times m$ matrix $R$. When $m = 1$, the system (1.3) clearly reduces to the classical hypergeometric equation. It appears difficult to establish this conjecture directly, and the method used has necessitated a section devoted to a summary of the argument involved (Section 3). The main result in the paper is summarized in Theorem 3.1 of this section. Section 4 contains proofs referred to in Section 3. Using the fact that $C_\kappa (R)$ satisfies the partial differential equation (James [10])\begin{equation*}\tag{1.4} \sum^m_{i=1} R_i^2\partial^2y/\partial R_i^2 + \sum^m_{j=1,j\neq i}\lbrack R_i^2/(R_i - R_j) \rbrack\partial y/\partial R_i = \sum^m_{i=1}k_i(k_i + m - i - 1)y,\end{equation*} James and Constantine [11] have obtained the effects of certain differential operators on $C_\kappa(R)$. These results are given in Section 2 and are used in many proofs in Section 4. Section 5 is probably of most interest statistically, for here systems of partial differential equations similar to (1.3) are given for $_1F_1(a; c; R)$ and $_0F_1(c; R)$. These two functions occur often in multivariate distributions. The differential equations for $_1F_1(a; c; R)$ have been used by Constantine [3] to obtain an asymptotic expansion for the noncentral likelihood ratio criterion, and by the author [12] to obtain asymptotic distributions of Hotelling's generalized $T_0^2$ statistic, Pillai's $V^{(m)}$ criterion, and for the largest latent root of the covariance matrix. The system for $_0F_1(c; R)$ is a generalization of that given by James [6] for $_0F_1(m/2; R)$.

The hypergeometric functions are one of the most important and special functions in mathematics. They are the generalization of the exponential functions. Particularly the ordinary hypergeometric function 2F1(a, b; c; z) is represented by hypergeometric series and is a solution to a second order differential equation. Similarly, Laplace transform is a form of integral transform that converts linear differential equations to algebraic equations. This paper aims to study the convergence of hypergeometric function and Laplace transform of some hypergeometric functions. Moreover, some relationships between Laplace transformation and hypergeometric functions is established in the concluding section of this paper.

Three methods are discussed for obtaining the asymptotic expansions for the nonnull distributions of multivariate test statistics; (i) the method by the Taylor expansion including the studentization method, (ii) the method by the expansion of exact characteristic functions using the formulas for weighted sums of zonal polynomials, and (iii) the method based on the system of partial differential equations satisfied by the hypergeometric functions. A short note on some other useful methods is also given. The emphasis in the discussion is put on the basic idea in each method.

HypExp 2, expanding hypergeometric functions about half-integer parameters

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q-gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q-analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q-Hadamard expansion. Expansions and asymptotics for modified q-Bessel functions are derived as special cases from the general expansion.

Harmonic Analysis and Special Functions on Symmetric Spaces

Because of the many relations connecting the special functions to each other, and to the elementary functions, it is natural to inquire whether more general functions can be developed so that the special functions and elementary functions are merely specializations of these general functions. General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. There are several varieties of these functions, but the most common are the standard hypergeometric function ( which we discuss in this chapter) and the confluent hypergeometric function (Chap. 10).

On fundamental solutions for 3D singular elliptic equations with a parameter

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and for Gaussian hypergeometric functions F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) , in particular, if the parameters are also viewed as variables.

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

This paper continues the analysis of a class of orthogonal polynomials in two variables on a region bounded by two straight lines and a parabola touching these lines, which was introduced by the first author. An explicit series expansion for these polynomials is obtained, which generalizes Constantine’s expansion of hypergeometric functions of $(2 \times 2)$ matrix argument in terms of James’ zonal polynomials. In two special cases the orthogonal polynomials turn out to be Appell’s hypergeometric $F_4 $-functions and certain hypergeometric functions in two variables of order three, respectively.

: The usual power series representations for hypergeometric functions in two variables have a limited range of validity. In particular, they are of little use when the magnitude of one of the variables becomes very large. Using a Barnes-type integral representation and the concept of analytic continuation, the region of utility is extended to the desired domain. The poles that occur in the Barnes-type integral are assumed to be simple. Thus, explicit asymptotic expansions are obtained for each of the fourteen hypergeometric functions that belong to this class. Hypergeometric functions, Functions with a large argument, Functions of two variables, Contour integration, Power series expansions.

The analytic solution for the power series expansion of Heun function

Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b-gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software.•By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples;•Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics.•Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.