Inayat-Hussain (1987a) introduced generalization form
of Fox's H-function which is popularly known as -function.
stands on fairly firm footing through the research contributions of various
authors like Inayat-Hussain (1987b), Rathie
(1997), Gupta and Soni (2006), Gupta
et al. (2007), Agarwal and Jain (2009), Agarwal
(2011) and Buschman and Srivastava (1990).
is defined and represented in the following manner by Gupta
et al. (2007):
It may be noted that the
contains fractional powers of some of the gamma function and m, n, p, q are
integers such that 1≤m≤q, 1≤n≤p, (αj)1,p,
(βj)1,q are positive real numbers and (Aj)1,n,
(Bj)m+1,q may take non-integer values which we assume
to be positive for standardization purpose. (αj)1,p
and (βj)1,q are complex numbers.
The nature of contour L, sufficient conditions of convergence of defining integral
(1) and other details about the -function
can be seen in the papers of Gupta and Soni (2006) and
Gupta et al. (2007).
The behavior of the -function
for small values of |z| follows easily from a result given by Rathie
The following series representation for the -function
given by Saxena et al. (2002) will be required
The following function which follows as special cases of the -function
will be required in the sequel (Gupta et al., 2007):
The Srivastava polynomials Snm[x] will be defined and
represented as follows (Srivastava, 1972, Eq.
where, n = 0, 1, 2,
, m is an arbitrary positive integer, the coefficients
An,l (n, l≥0) are arbitrary constants, real or complex. Snm[x]
yields number of known polynomials as its special cases. These include, among
other, the Jacobi polynomials, the Bessel Polynomials, the Lagurre Polynomials,
the Brafman Polynomials and several others (Srivastava and
The following well known Euler Integral Formula is required to establish the
main integral (Srivastava and Karlsson, 1985, Eq.
Let ψ (z) denote the logarithmic derivative of gamma function Γ (z) i.e.:
The following interesting integral will be required to establish the results
from Eq. 11-16:
The above result Eq. 17 will be valid under the following conditions:
where, Ω is given by Eq. 7.
To evaluate the above integral we express
in its series form with the help of Eq. 9 and -function
in terms of Mellin-Barnes type of contour integral by Eq.1
and then interchanging the order of integration and summation, we get:
Further using the result Eq. 10 the above integral becomes:
Then interpret with the help of Eq. 1 and 21,
we have the required result (Eq. 17) and if we express -function
in series form with the help of Eq. 4 we easily arrive at
DERIVATION OF THE MAIN INTEGRALS
The result in Eq. 11 is established by taking the partial
derivative on both sides of Eq. 18 with respect to a. Equation
12 and 13 are similarly established by taking the partial
derivative of Eq. 18 with respect to b and c, respectively.
Equation 14 is established by adding Eq. 11,
12 and 13; Eq. 15 is
established by subtracting Eq. 11 and 12
from 13; Eq. 16 is established by subtracting Eq.
12 and Eq. 13 from Eq. 11.
If we put Aj = Bj = 1, -function
reduces to Foxs H-function (Srivastava et al.,
1982), then the Eq. 17 takes the following form:
If we put Aj = Bj = 1; αj = βj
= 1, then the -function
reduces to general type of G-function (Meijer, 1946)
which is also believe to be new.
The conditions of convergence of Eq. 22 can be easily obtained from those of Eq. 17.
By applying the our result given in Eq. 17 to the case of
Hermite polynomials (Srivastava and Singh, 1983) by
in which mi = 2; ni = n1, r = 1; ,
we have the following interesting results:
The conditions of convergence of Eq. 23 can be easily obtained from those of Eq. 17.
By applying the our results given in Eq. 17 to the case
of Lagurre polynomials (Srivastava and Singh, 1983)
We have the following interesting results:
The conditions of convergence of Eq. 24 can be easily obtained from those of Eq. 17.
If we put n = p, m = 1, q = q+1, b1 = 0, β1 = 1,
aj = 1-aj, bj = 1-bj, in Eq.
17 then the -function
reduces to generalized wright hypergeometric function (Wright,
the Eq. 17 takes the following form:
The conditions of convergence of Eq. 25 can be easily obtained from those of Eq. 17.
The authors express their sincerest thanks to the referees for some valuable