Extending the factorization principle to hypergeometric series of general form

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 Erdélyi’s transformation formula relating the seriesG ₃ to the Appell functionF ₂, contains erroneous expressions in the arguments ofG ₃. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG ₃ are presented. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000.     

Decomposition formulas for some triple hypergeometric functions

With the help of some techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Srivastava's hypergeometric functions HA, HB and HC in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 15 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly.     

Valeurs Exceptionnelles de Fonctions Hypergéométriques d’Appell (Exceptional Values of Appell Hypergeometric Functions)

In his article [18], J. Wolfart studied the following exceptionnal set where F is the classical, or Gauss hypergeometric function. The first aim of the present article is to describe the exceptional set in the case of Appell hypergeometric functions, which are a generalization to two variables of the Gauss functions. The link will then be made between, on the one hand, the distribution of complex multiplication points (described by Appell function in the article [5] of P. Cohen and J. Wolfart) on a fixed modular variety, using a André-Oort conjecture, and on the other hand, the arithmeticity of the monodromy group related to this function. Lastly, we will see how the localization of certain complex multiplication points leads to the transcendance of the values of Appell hypergeometric functions, at algebraic points.     

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

Congruences for 2 F 1 Hypergeometric Functions Over Finite Fields

In a recent paper, Ono and Penniston proved a family of congruences for ₃ F ₂ hypergeometric functions over finite fields. They use the relationship between these functions and the arithmetic of a certain family of elliptic curves to obtain their congruences. Here we prove analogous congruences for ₂ F ₁ hypergeometric functions.     

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.     

An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications

A simple proof is given of a new summation formula recently added in the literature for a terminating _{r + 3}F_{r + 2}(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well‐known Saalschütz summation formula for a ₃F₂(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the ₂F₁ series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijovi? & Miller. Copyright © 2015 John Wiley & Sons, Ltd.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Wave Kernels with Bi-Inverse Square Potentials on Euclidean Plane

The Cauchy problemfor the wave equation with bi-inverse square potential on Euclidean plane is solved in terms of the two variables Appell F₂ hypergeometric functions. Our principal tools are the Hankel transforms and the special functions of mathematical physics.     

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.     

On singular univariate specializations of bivariate hypergeometric functions

It is tempting to evaluate F₂(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for ₂F₁(1). But this kind of naive evaluation can lead to errors as the ₂F₁(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications by concrete examples.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

Hypergeometric Functions Related to Schur Q-Polynomials and the BKP Equation

We introduce hypergeometric functions related to projective Schur functions Q and describe their properties. Linear equations, integral representations, and Pfaffian representations are obtained. These hypergeometric functions are vacuum expectations of free fermion fields and are therefore tau functions of the so-called BKP hierarchy of integrable equations.     

Some multiple hypergeometric transformations and associated reduction formulas

The main object of the present paper is to derive various classes of double-series identities and to show how these general results would apply to yield some (known or new) reduction formulas for the Appell, Kampé de Fériet, and Lauricella hypergeometric functions of several variables. A number of closely-related linear generating functions for the classical Jacobi polynomials are also investigated.     

Probability Measures on Dual Objects to Compact Symmetric Spaces and Hypergeometric Identities

We derive, in several different ways, combinatorial identities which are multidimensional analogs of classical Dougall's formula for a bilateral hypergeometric series of the type ₂H₂. These identities have a representation-theoretic meaning. They make it possible to construct concrete examples of spherical functions on inductive limits of symmetric spaces. These spherical functions are of interest to harmonic analysis.     

Two-point Taylor expansions in the asymptotic approximation of double integrals. Application to the second and fourth Appell functions

The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F₂(a,b,b^′,c,c^′;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F₂(a,b,b^′,c,c^′;x,y) for large b, b^′, c and c^′. We also consider a double integral representation of the fourth Appell function F₄(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F₄(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.     

π and the hypergeometric functions of complex argument

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In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function ₂F₁ in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite (1876) [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica^®.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=rQqtVtAf-RQ.     

Realizations of coupled vectors in the tensor product of representations of and

Using the realization of positive discrete series representations of in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of ν+1 representations as polynomials in ν+1 variables z₁,…,z_ν+1. These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch–Gordan coefficients of . In general, these polynomials can be written as (terminating) multiple hypergeometric series. For ν=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The case of is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.     

Analytic continuation of the Appell function <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript> and integration of the associated system of equations in the logarithmic case

The Appell function F ₁ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F ₁ are related in a special way. Formulas for the analytic continuation of F ₁ beyond the unit bicircle are constructed in which F ₁ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F ₁ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.     

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.