Double Hypergeometric Functions Research Articles

Q-Series Identities and Reducibility of Basic Double Hypergeometric Functions

For real or complex q, |q| < 1, let(1.1)for arbitrary λ and μ, so that(1.2)and(1.3)Define, as usual, a generalized q-hypergeometric function by (cf. [26, Chapter 3]; see also [18])(1.4)where, for convergence, |q| < 1 and |z| < ∞ when r is a positive integer, or |z| < 1 when r = 0, provided that no zeros appear in the denominator.

A certain class of [formula omitted]-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of q -extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.

Compound gamma bivariate distributions

Bivariate distributions, which may be of special relevance to the lifetimes of two components of a system, are derived using the following approach. As the two components are part of one system and therefore exposed to similar conditions of service, there will be similarity between their lifetimes that is not shared by components belonging to different systems. The lifetime distribution for a given system is assumed to be Gamma in form (this includes the exponential as a special case; extension to the Stacey distribution, which includes the Weibull distribution, is straightforward). The scale parameter of this distribution is itself a random variable, with a Gamma distribution. We thus obtain what might be termed a compound Gamma-Gamma bivariate distribution. The cumulative distribution function of this may be expressed in terms of one of the double hypergeometric functions of Appell.

A note on the generalization of the summation formulas for Appell's and Kampé de Fériet's hypergeometric functions

In the present paper, a summation formula of a general triple hypergeometric series F (3)( x, y, z) introduced by Srivastava [10] is obtained. A particular case of this formula corresponds to a result of Shah [7] involving Kampé de Fériet's double hypergeometric function which can further be specialized to yield summation formulas of Srivastava [11] and Bhatt [2] for Appell's function F 2.

Analytic Continuation of the Euler Transform

The Euler transform is defined by an integral over the unit interval $[0,1]$ if the transform variables have positive real parts. If the function to be transformed is holomorphic on a neighborhood of $[0,1]$, its transform can be represented for all complex values of the transform variables by an integral around a contour which encircles $[0,1]$. The integrand contains a ${}_2 F_1 $-function. This type of contour integral represents the principal branch of Appell’s double hypergeometric function $F_1$ for all values of the parameters and variables.

Generating functions for Jacobi and related polynomials

In the present paper, we have established the following relation involving Kampe de Feriet's double hypergeometric function of superior order _ al,***, a. F I .b cx I (b)f F_Oci,, cq -y= nE J(b)=1 (X)(b)n F ?bX-; yn rr (bj). LI + b n; j=l which yields a number of interesting generating formulae for Jacobi and related polynomials. A large number of special cases have been also discussed.

Hidden Symmetries of Special Functions

Several familiar special functions possess a hidden permutation symmetry which accounts for some of their transformation properties. For example the transformation of the elliptic integral $K(k)$ into $K({{ik} / {k'}})$ expresses the symmetry of an integral representing Gauss’ arithmetic-geometric mean of two variables. Likewise, a transformation of the hypergeometric function ${}_2 F_1 $ with argument z into the same function with argument ${z / {(z - 1)}}$ can be viewed as a symmetry under interchange of two variables, but in this case two parameters must be interchanged as well. A similar remark applies to Kummer’s transformation of the confluent hypergeometric function. The five transformations which change the modulus k of an incomplete elliptic integral into $k'$, ${1 / k}$, ${1 / {k'}}$, ${{ik} / {k'}}$, and ${{k'} / {ik}}$ are equivalent to the five nontrivial permutations of three variables, and the theory of elliptic integrals can be simplified by choosing standard integrals which are explicitly symmetric. Appell’s double hypergeometric function $F_1 $ has a partly concealed symmetry under simultaneous permutations of three parameters and three variables. The feature common to all these examples is a function $F(b_1 ,b_2 , \cdots ,b_k ;z_1 ,z_2 , \cdots ,z_k )$ which is symmetric in the indices $1,2, \cdots ,k$ of the parameters b and variables z and represents a weighted average of a function of one variable over the convex hull of $\{ {z_1 ,z_2 , \cdots ,z_k } \}$. The advantage of using explicitly symmetric functions is illustrated by generalizing Borchardt’s iterative algorithm for computing an inverse cosine to an algorithm for computing an inverse elliptic cosine.

Some identities involving Kampe de Feriet’s double hypergeometric functions—I

Some identities involving Kampe de Feriet’s double hypergeometric functions—I

Infinite series of Kampé De Feriet’s Double hypergeometric function of higher order

Infinite series for Kampe de Feriet’s hypergeometric functions of two variables are established. Particular cases involving expansions of Appell’s functionsF[1],F[2],F[3] andF[4] are deduced.

Expansions involving hypergeometric functions of two variables

In ?2 of this paper a systematic attempt has been made to extend the expansions concerning Appell functions to the Kampe de F6riet's double hypergeometric functions by using the two symbolic operators of Burchnall and Chaundy. In ?4 using the method of iteration of series some of the expansions due to Chaundy [5], [6], Niblett [10], Wimp and Luke [8] have been extended to the Kamp6 de Feriet's function. The paper is concluded by showing how the induction by using the Laplace transform and its inverse can be employed to extend these results to G-functions of two variables defined recently by Agarwal [1]*. It may be pointed out that these expansions are very general in nature and incorporate a very large number of expansions for the functions of two variables.

BASIC DOUBLE HYPERGEOMETRIC FUNCTIONS (II)

BASIC DOUBLE HYPERGEOMETRIC FUNCTIONS (II) Get access F. H. JACKSON F. H. JACKSON Eastbourne Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume os-15, Issue 1, 1944, Pages 49–61, https://doi.org/10.1093/qmath/os-15.1.49 Published: 01 January 1944 Article history Received: 06 October 1943 Published: 01 January 1944

ON BASIC DOUBLE HYPERGEOMETRIC FUNCTIONS

ON BASIC DOUBLE HYPERGEOMETRIC FUNCTIONS Get access F. H. JACKSON F. H. JACKSON Eastbourne Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume os-13, Issue 1, 1942, Pages 69–82, https://doi.org/10.1093/qmath/os-13.1.69 Published: 01 January 1942 Article history Published: 01 January 1942 Received: 14 January 1942 Revision received: 29 April 1942