Showing papers on "Basic hypergeometric series published in 2000"
TL;DR: In this article, a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group was studied.
Abstract: We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed in previous works, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel–Darboux kernel for Laguerre polynomials.
215 citations
TL;DR: In this paper, a method for obtaining Ramanujan's mock theta functions from ordinary theta function by performing certain operations on their q-series expansions is developed, and the method is then used to construct several new mock-theta functions, including the first ones of eighth order.
Abstract: A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.
139 citations
TL;DR: The roots of the general equation of degree n satisfy an A -hypergeometric system of differential equations in the sense of Gel'fand, Kapranov and Zelevinsky, and are constructed for each of the 2n−1 triangulations of the Newton segment.
86 citations
1 Oct 2000
TL;DR: In this paper, a short review of the tensor reduction of Feynman integrals based on recurrence relations w.r.t. space-time dimension d is given.
Abstract: A short review of the method for the tensor reduction of Feynman integrals based on recurrence relations w.r.t. space-time dimension d is given. A solution of the difference equation w.r.t. d for the n-point one-loop integrals with arbitrary momenta and masses is presented. The result is written as multiple hypergeometric series depending on ratios of Gram determinants. For the 3-point function a new expression in terms of the Appell hypergeometric function F1 is presented.
51 citations
Posted Content•
TL;DR: In this article, an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases is presented, and the kernel of the transform is given explicitly in terms of basic hypergeometric series.
Abstract: In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provides an analytical continuation of the Askey-Wilson polynomial in its degree. In this paper we establish the $L^2$-theory of the Askey-Wilson function transform, and we explicitely determine its inversion formula.
46 citations
1 Jan 2000
TL;DR: In this article, a good approximation of hypergeometric functions with matricial argument was obtained by calculating zonal polynomials of high degrees and developing the functions in a truncated series.
Abstract: Hypergeometric functions with matricial argument are being used in several fields of mathematics. This article tries to obtain a good approximation of this family of functions, since there are no general expressions for them, calculating zonal polynomials of high degrees and developing the functions in a truncated series.
30 citations
1 Jan 2000
TL;DR: In this paper, the authors considered the problem of finding analytic solutions to the Horn system of equations with polynomial coefficients, and they showed that the Hadamard multiplication of such series corresponds to the Minkowski sum of polynomials whose zero loci contain the singularities of the factors.
Abstract: This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.
22 citations
TL;DR: Some rather elementary techniques are used in order to derive several summation formulas associated with Lauricella's hypergeometric function FA(r) in r variables with known or new consequences.
21 citations
TL;DR: This paper gave elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series, including a simple proof of Bailey's very-well-poised 6-psi-6 summation.
Abstract: We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised 10-psi-10 transformation, by induction, Slater's general transformation for very-well-poised 2r-psi-2r series, and Slater's transformation for general r-psi-r series. Finally, we derive some new transformations for bilateral basic hypergeometric series of Chu-Gasper-Karlsson-Minton-type.
20 citations
TL;DR: A symbolic technique for computing the exact or approximate solutions of linear differential systems with meromorphic coefficients, where the combinatorial decomposition of F allows to get the solution as a polynomial in Dirichlet functions, or hypergeometric functions, that are built from the coefficients of the system.
20 citations
TL;DR: In this article, it was shown that the roots of the trinomial equation x n −x+t = 0 are finite sums of generalized hypergeometric functions for each positive integer n.
TL;DR: In this article, an orthogonal expansion for hypergeometric probabilities in terms of Krawtchouck's polynomials is given, where an adequate choice of the parameters involved in the expansion and truncation yield binomial approximations to hypergeometrical probabilities.
Journal Article•
TL;DR: In this paper, the authors provided twenty five integrals involving hypergeometric functions in the form of a single integral and fifty two interesting integrals follow as special cases of their main findings.
Abstract: The aim of this research is to provide twenty five integrals involving hypergeometric function in the form of a single integral. Fifty two interesting integrals follow as special cases of our main findings. These results are obtained with the help of generalized Watson's theorem on the sum of a F recently obtained by Lavoie, Grondin and Rathie. The integrals given in this paper are simple, interesting and easily established, and they may be useful.
TL;DR: In this article, it was shown that the formulas of operator factorization of hypergeometric functions obtained in the author's previous works can be extended to hypergeometrical series of the most general form, which does not make the technical apparatus of the factorization method more complicated.
Abstract: It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained in the paper, we analyze transformation properties of the Horn seriesG
3, whose structure is typical for general hypergeometric functions. It is shown that Erdelyi’s transformation formula relating the seriesG
3 to the Appell functionF
2, contains erroneous expressions in the arguments ofG
3. The correct analog of Erdelyi’s formula is found, and some new transformations of the seriesG
3 are presented.
1 Oct 2000
TL;DR: The generalized hypergeometric equation (GHE) as discussed by the authors is a generalized version of the GHE with two regular and one irregular singularity, and it can be shown that it includes the Mathieu equation.
Abstract: It is well-known that a second-order differential equation with three regular singularities can be reduced to the Gaussian hypergeometric equation. A documented extension is the generalized hypergeometric equation, which also has three regular singularities, but is a differential equation of order higher than the second. In this paper a distinct generalization of the Gaussian hypergeometric equation is introduced (section 1), namely the extended hypergeometric equation, which is of second-order, and has two regular and one irregular singularity. It can be shown (Section 2) that it includes the Mathieu equation. The solution in power series, and with logrithmic singularities, are obtained in the neighborhood of the two regular singularities, as for the Gaussian type, with the diffrence (section 3) that the recurence formulas for the coefficients are not two-term but rather multiple-term. The solutions in the neighborhood of the irregular singularity at infinity are obtained (setion 4) by three methods, viz...
TL;DR: In this paper, the generating relations for a set of hypergeometric functions ψα,β,γ,m(x) are obtained by using the representation of the Lie group SL(2,C) giving a suitable interpretation to the index m in order to derive the elements of Lie algebra.
Abstract: In this paper, the generating relations for a set of hypergeometric functions ψα,β,γ,m(x) are obtained by using the representation of the Lie group SL(2,C) giving a suitable interpretation to the index m in order to derive the elements of Lie algebra. The principle interest in our results lies in the fact that a number of special cases would inevitably yield too many new and known results of the theory of special functions, namely the Laguerre, even and odd generalized Hermite, Meixner, Gottlieb, and Krawtchouk polynomials.
TL;DR: In this article, an algorithmic criterion for finding reducible cases was proposed, which holds that any non-trivial reduction formula for a multiple series is a corollary of a trivial (self-obvious) reduction for an appropriately transformed series.
TL;DR: In this paper, various expressions for the 9-j coefficient of su (1,1), with an emphasis on multiple hypergeometric series, were studied, dealing with formulas in which the 9j coefficient appears as a connection or expansion coefficient.
TL;DR: In this article, the arithmetic property which allows to sharpen number-theoretic estimates was studied and applied to generalized hypergeometric functions. But the application of their general qualitive theorems to generalized Hypergeometric Functions is restricted to the set of irrational numbers which are the values of these functions.
Abstract: We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to generalized hypergeometric functions extends the set of irrational numbers which are the values of these functions.
TL;DR: In this paper, a stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered, and criterion for selecting regularizing parameter and error estimates are obtained.
TL;DR: Using Chebyshev polynomials, the authors generate an algorithm for the efficient calculation of hypergeometric probabilities for a fixed population size N and fixed sample size n, such calculations simultaneously produce distributions for all possible values of the population number of successes M.
Abstract: Using Chebyshev polynomials we generate an algorithm for the efficient calculation of hypergeometric probabilities. For a fixed population size N and fixed sample size n, such calculations simultaneously produce distributions for all possible values of the population number of “successes” M.
1 Jan 2000
TL;DR: In this paper, the authors introduced a generalization of the secant integral in the following form I a ψ, b, λ =b a ∫ ψ 0 e − b sec ϕsec ϕ a tan ϕ 2λ−1 d ϕ with a≥0, b>0, 0 π 2, and λ>0.
TL;DR: In this article, a first-order differential system Y′(x) = A(x, Y(x)) on [a, ∞] particular cases of which are equivalent to standard forms of the generalized hypergeometric equation was introduced.
Abstract: We introduce a first-order differential system Y′(x) =A(x)Y(x) on [a, ∞) particular cases of which are equivalent to standard forms of the generalized hypergeometric equation. Our purpose is to obtain the asymptotic solution of the system as x ∞ by defining suitable transformations of the solution vector Y and using ideas from a unified asymptotic theory of differential systems. Thus our methods place the system within the scope of this unified theory, and they are independent of specialized properties of the Meijer G-function solutions of generalized hypergeometric equations. As such, our methods are also capable of extension to other situations not covered by these special functions.
TL;DR: In this paper, the authors considered vector potential and inductance for simple current distributions with rotational symmetry, in terms of those hypergeometric functions that yield the simplest expressions, rather than complete elliptic integrals.