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.     

Hypergeometric functions of two variables

Summary The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric functionF ₁ leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system ofF ₁ or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system ofF ₁.     

Reduction and transformation formulas for the Appell and related functions in two variables

In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions F₁,F₂,F₃, and F₄ in two variables and the corresponding (substantially more general) double‐series identities. In particular, we observe that a certain reduction formula for the Appell function F₃ derived recently by Prajapati et al., together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula 12 . We also present a brief account of several other related results that are closely associated with the Appell and other higher‐order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.     

Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F=R,C,H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.     

Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function ₁F₂(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.     

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.     

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.     

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.     

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.     

Some reformulated properties of Jacobian elliptic functions

Various properties of Jacobian elliptic functions can be put in a form that remains valid under permutation of the first three of the letters c, d, n, and s that are used in pairs to name the functions. In most cases 12 formulas are thereby replaced by three: one for the three names that end in s, one for the three that begin with s, and one for the six that do not involve s. The properties thus unified in the present paper are linear relations between squared functions (16 relations being replaced by five), differential equations, and indefinite integrals of odd powers of a single function. In the last case the unification entails the elementary function RC(x,y)=RF(x,y,y), where RF(x,y,z) is the symmetric elliptic integral of the first kind. Explicit expressions in terms of RC are given for integrals of first and third powers, and alternative expressions are given with RC replaced by inverse circular, inverse hyperbolic, or logarithmic functions. Three recurrence relations for integrals of odd powers hold also for integrals of even powers.     

A normal criterion about two families of meromorphic functions concerning shared values

In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D■C,a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f ∈ F , there exists g ∈ G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D.     

Hypergeometric analogues of the arithmetic-geometric mean iteration

The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iterationa _n +1=(a _n +b _n )/2 and \(b_{n + 1} = \sqrt {a_n b_n } \) witha ₀?1 andb ₀?x. The common limit is₂ F ₁(1/2, 1/2; 1; 1?x ²)^?1 and the convergence is quadratic. This is a rare object with very few close relatives. There are however three other hypergeometric functions for which we expect similar iterations to exist, namely:₂ F ₁(1/2?s ₁, 1/2+s; 1; ·) withs=1/3, 1/4, 1/6. Our intention is to exhibit explicitly these iterations and some of their generalizations. These iterations exist because of underlying quadratic or cubic transformations of certain hypergeometric functions, and thus the problem may be approached via searching for invariances of the corresponding second-order differential equations. It may also be approached by searching for various quadratic and cubic modular equations for the modular forms that arise on inverting the ratios of the solutions of these differential equations. In either case, the problem is intrinsically computational. Indeed, the discovery of the identities and their proofs can be effected almost entirely computationally with the aid of a symbolic manipulation package, and we intend to emphasize this computational approach.     

Products of hypergeometric functions and the zeros of 4F3 polynomials

We recall a known result (cf. [1]) expressing certain ₄ F ₃ hypergeometric functions as products of ₂ F ₁ hypergeometric functions. We consider the polynomial case and show how recent results (cf. [2]) concerning the zero distribution of Gauss hypergeometric polynomials can be used to obtain information about the location of the zeros of three types of ₄ F ₃ hypergeometric polynomials. Numerical and graphical evidence of the zeros is provided with the help of Mathematica.     

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.     

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.     

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.     

Symmetry in c, d, n of Jacobian elliptic functions

The relation connecting the symmetric elliptic integral RF with the Jacobian elliptic functions is symmetric in the first three of the four letters c, d, n, and s that are used in ordered pairs to name the 12 functions. A symbol Δ(p,q)=ps²(u,k)−qs²(u,k), p,q∈{c,d,n}, is independent of u and allows formulas for differentiation, bisection, duplication, and addition to remain valid when c, d, and n are permuted. The five transformations of first order, which change the argument and modulus of the functions, take a unified form in which they correspond to the five nontrivial permutations of c, d, and n. There are 18 transformations of second order (including Landen's and Gauss's transformations) comprising three sets of six. The sets are related by permutations of the original functions cs, ds, and ns, and there are only three sets because each set is symmetric in two of these. The six second-order transformations in each set are related by first-order transformations of the transformed functions, and all 18 take a unified form. All results are derived from properties of RF without invoking Weierstrass functions or theta functions.     

On properties of hypergeometric functions of three variables,F M andF G

The behavior of Lauricella hypergeometric seriesF _M andF _G (see [2]) near the boundary points of their domains of convergence is discussed. Such properties for one variable series, the Gauss₂ F ₁ and the Clausen₃ F ₂, and for two variables, the AppellF ₁,F ₂ andF ₃, are established by the author [5], [7] and the results are effectively applied to solve problems for the Euler-Darboux equation and to calculate multiplications of the fractional calculus operators in the articles by the author [4], [6] and by H. M. Srivastava and the author [9].