Research Article
Fractional Integration of the Product of Two H-functions and a General Class of Polynomials
Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India
The study is developed generalized fractional integral operators of a function f(x) introduced by Saigo (1978) (for details see also Kilbas and Saigo (2004)).
Let α, β, η be complex numbers. The fractional integral (Re (α)>0) of a function f(x) defined on (0, ∞) is given by:
(1) |
and
(2) |
where, F is the Gauss hypergeometric function.
When, β = -α, the above Eq. 1 and 2 reduce to the following classical Riemann-Liouville fractional integral operator (Samko et al., 1993):
(3) |
and
(4) |
Again, if β = 0, the Eq. 1 and 2 reduce to the following Erdelyi-Kober fractional integral operator (Samko et al., 1993):
(5) |
and:
(6) |
The H-function occurring in the paper is defined and represented in the following manner (Srivastava et al., 1982):
(7) |
(8) |
The nature of the contour L of the integral 7, the conditions of existence of the H-function defined by 7 and other details can be referred in the book mentioned above.
The H-function of several variables is defined and represented as follows (Srivastava et al., 1982):
(9) |
where:
(10) |
(11) |
It is assumed that the various H-functions of several variables occurring in the paper always satisfy the appropriate existence and convergence conditions corresponding appropriately to those recorded in the book by Srivastava et al. (1982). In case r = 2, it reduce to the H-function of two variables (Srivastava et al., 1982).
Also, Smn [x] occurring in the sequel denotes the general class of polynomials introduced by Srivastava (1972):
(12) |
where, M is an arbitrary positive integer and the coefficients AN, K(N, K≥0) are arbitrary constants, real or complex. On suitably specializing the coefficients AN, K, SMN [x] yields a number of known polynomials as its special cases. These include, among others, the Hermite polynomials, the Jacobi polynomials, the Laguerre polynomials, the Bessel polynomials, the Gould-Hopper polynomials, the Brafman polynomials and several others (Srivastava and Singh, 1983).
Preliminary lemmas: The following lemmas will be required to establish our main results.
Lemma 1: (Kilbas and Sebastain, 2008): Let α, β, η be complex numbers. The fractional integral (Re(α)>0) and (Re(μ)>max{0, Re(β-η)}) then there holds the following relation of a function f(x):
(13) |
In particular, if β = -α and β = α in Eq. 13, We have:
(14) |
(15) |
Lemma 2: (Kilbas and Sebastain, 2008): Let α, β, η be complex numbers. The fractional integral (Re(α)>0) and (Re(μ)<1+min {Re(β) Re(η)}) then there holds the following relation of a function f(x):
(16) |
In particular, if β = -α and β = α in Eq. 16, Author has:
(17) |
(18) |
MAIN RESULTS
Image (1):
(19) |
Where:
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
Proof: To establish the Image 1, first express the general class of polynomials occurring on its left-hand side in the series form given by Eq. 12, replace the product of two H-functions occurring therein by its well-known Mellin-Barnes contour integral given by Eq. 7, interchange the order of summations, (ξ1, ξ2)-integrals and taking the generalized fractional integral operator inside (which is permissible under the conditions stated with Eq. 19) and make a little simplification. Next, we express the following binomial expansion for (b -αx)-v:
So, obtained in terms of MellinBarnes contour integral (Srivastava et al., 1982). Now, interchanging the orders of integrals (which are also permissible under the conditions stated with Eq. 19), it takes the following form after a little simplification:
(26) |
Finally, applying the lemma 1 and re-interpreting the Mellin-Barnes type contour integral thus obtain in terms of the H-function of three variables defined by Eq. 9, we arrive at the right hand side of Eq. 19 after little simplifications.
If we put β = -α in Image 1, Author arrives at the following new and interesting corollary concerning Riemann-Liouville fractional integral operator defined by Eq. 3.
Corollary 1:
(27) |
Where:
(28) |
where, the conditions of existence of the above corollary follow easily with the help of Image 1.
Again, if we put β = 0 in Image 1, we get the following result which is also beloved to be new and pertains to Erdelyi-Kober fractional integral operators defined by Eq. 5.
Corollary 2:
(29) |
Where:
(30) |
and provided that:
(31) |
and the conditions (ii) to (iv) and (vi) in Image 1 are also satisfied.
Image (2):
(32) |
Where:
(33) |
provided that:
(34) |
and the conditions (ii) to (iv) and (vi) in Image 1 are also satisfied.
Proof: We proceed on similar lines as adopted in Image 1 and using lemma 2.
If we put β = -α and β = 0 in Image 2, in succession we shall easily arrive at the corresponding corollaries concerning Riemann-Liouville and Erdelyi-Kober fractional integral operators, respectively.
Corollary 3:
(35) |
Corollary 4:
(36) |
The conditions of validity of the above results follow easily from the conditions given with Image 2, Corollary 1 and 2, respectively.
SPECIAL CASES AND APPLICATIONS
The generalized fractional integral operator result 1 and 2 established here are unified in nature and act as key formulae. Thus the general class of polynomials involved in image 1 and 2 reduce to a large spectrum of polynomials listed by Srivastava and Singh (1983) and so from result 1 and 2 we can further obtain various fractional integral results involving a number of simpler polynomials. Again, the H-function of one variable occurring in these results can be suitably specialized to a remarkably wide variety of useful functions (or product of several such functions) which are expressible in terms of generalized Wright hypergeometric function, generalized Mittag-Laffler function and Bessel functions of one variable. For example:
• | If we reduce the first H-function in Image 1 to the exponential function by taking σ1, λ1, ω1, δ→0 and SMN to the Hermite polynomial (Srivastava and Singh, 1983; Szego, 1975) and by setting: |
in which case M = 2, AN, K = (-1)K, we have the following interesting consequences of the main results after a little simplification.
(37) |
Where:
(38) |
The conditions of validity of the above result can be easily derived from Image 1.
Further on letting w1 and ω2→0in the above result, it takes the following form:
(39) |
If, we put β = -α and v, N→0 and make suitable adjustment in the parameters in the Eq. 37, we arrive at the known result (Kilbas and Saigo, 2004).
If, we reduce the H-function of one variable to generalized Wright hypergeometric function (Srivastava et al., 1982) in the result given by Eq. 39, we get the following new and interesting result after little simplification.
(40) |
The conditions of validity of the above result can be easily derived from Image 1.
If, we put β = -α and v, N→0 and make suitable adjustment in the parameters in the above result, we arrive at the known result (Kilbas, 2005).
If, we take we = -1 and σ2 =-1 in the Eq. 39 and reducing the H-function of one variable occurring therein to, generalized Mittag-Laffler function (Prabhakar, 1971), we easily get after little simplification the following new and interesting result .
(41) |
The conditions of validity of the above result can be easily followed directly from those given with Eq. 19.
If, we put β = -α and v, N→0 and make suitable adjustment in the parameters in the above result, we arrive at the known result (Saxena et al., 2009).
If, we take β = -α and v, N→0, we = 1/4, σ2 = and reduce the H-function to the Bessel function of first kind in theEq. 39, we also get known result (Kilbas and Sebastain, 2008).
A number of special case of Image 1 and 2 can also be obtained but we do not mention them here on account of lack of space.