Differentiation identities for hypergeometric functions

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.     

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.     

Refinements of transformation inequalities for zero-balanced hypergeometric functions

In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.     

Summation formulae for elliptic hypergeometric series

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

     

Partial fraction expansion of the hypergeometric functions

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature.     

Extension of gamma, beta and hypergeometric functions

The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations.     

Elliott's identity and hypergeometric functions

Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively.     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

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.     

On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r ²) < (2−r ²)F(a,b;a +b;r ²) holds for all r∈(0,1).     

Duality for hypergeometric functions and invariant Gauss-Manin systems

We present some basic identities for hypergeometric functions associatedwith the integrals of Euler type. We give a geometrical proof for formulaesuch as the identity between the single and double integrals expressingAppell's hypergeometric series F1 (a, b, b' c; x, y).     

Mapping properties of the zero-balanced hypergeometric functions

In this paper, the order of convexity of $z_2{F}_1(a,b;c;z)$ is given under certain conditions on the positive real parameters a, b and c. We show also that the image domains of the unit disc $\mathbb{D}$ under some shifted zero-balanced hypergeometric functions $z_2{F}_1(a,b;a+b;z)$ are convex and bounded by two horizontal lines. This solves a problem raised by Ponnusamy and Vuorinen in [10].     

Jacobi matrices generated by ratios of hypergeometric functions

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.     

Unit Circle Elliptic Beta Integrals

We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations. 2000 Mathematics Subject Classification: Primary—11F50, 11L05, 33D05, 33D67     

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.