Integral Transforms and Special Functions Vol. 21, No. 7, July 2010, 515–522
Solution of Abel-type integral equation involving the Appell hypergeometric function R.K. Raina∗,† M. P. University of Agriculture and Technology, Udaipur 313001, Rajasthan, India (Received 13 May 2009; final version received 9 October 2009 ) The present paper gives a formal solution of a certain Abel-type integral equation involving the Appell hypergeometric function in the kernel. The integral equation and its solution give rise to new forms of generalized fractional calculus operators (viz. the generalized fractional integrals and generalized fractional derivatives). These and their various consequences are also mentioned. The concluding remarks briefly point out possibilities of further work concerning the operators studied in this paper. Keywords: Abel-type integral equation; fractional derivatives and fractional integrals; Appell hypergeometric function; analytic functions; Saigo-type fractional integrals and derivatives 2000 Mathematics Subject Classification: 26A33, 30C45, 33A35, 45E10
1.
Introduction and definitions
The familiar Appell hypergeometric function of the third type F3 (or Horn’s F3 function; see, e.g. [14]) is defined by F3 (a, a , b, b ; c; x, y) =
∞ ∞ (a)m (a )n (b)m (b )n x m y n m=0 n=0
=
∞ (a )n (b )n n=0
(c)n n!
(c)m+n
m! n!
(|x| < 1, |y| < 1)
F (a, b; c + n; x)y n ,
where (a)m is the Pochhammer symbol defined by (a)0 = 1,
∗ Email:
(a)m = a(a + 1) · · · (a + m − 1)(a ∈ C; m ∈ N).
[email protected] address: 10/11 Ganpati Vihar, Opposite Sector 5, Udaipur 313002, Rajasthan, India
† Present
ISSN 1065-2469 print/ISSN 1476-8291 online © 2010 Taylor & Francis DOI: 10.1080/10652460903403547 http://www.informaworld.com
(1)
(2)
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The F3 function in (1) reduces to the Gaussian hypergeometric function by means of the following relationships: F3 (a, a , b, b ; c; x, 0) = F3 (a, 0, b, b ; c; x, y) = F3 (a, a , b, 0; c; x, y) = 2 F1 (a, b; c; x) ≡ F (a, b; c; x)
(3)
and
F3 (a, a , b, c − b; c; x, y) = (1 − x)a F (b, a + a ; c; x).
(4)
Consider the integral equation
(α,α ,β,β ,γ ) f (x) = Ia,x
(x − a)−α (γ )
= g(x)
x −t t −x (x − t)γ −1 (t − a)−α F3 α, α , β, β ; γ ; , f (t) dt x −a t −a
x a
(x > a),
(5)
where α, β ∈ R, γ ∈ R+ (0 < γ < 1). The integral equation (5) obviously generalizes the classical Abel equation [12, Section 2] and is deducible from (5) when α = α = 0. In this paper, we obtain a formal solution of the Abel-type integral equation (5) involving the Appell hypergeometric function in the kernel. The method applied in obtaining the solution follows similar works of studying analogous integral equations with the F3 kernel considered earlier by Higgins [2] and Mariˇcev [5]. Both these references of earlier invstigations are cited in the book by Srivastava and Buschman [13], which also describes in a comprehensive manner several other useful applications of the theory of convolution type integral equations. The integral equation (5) and its solution (see below (14)) suggest certain new forms of generalized fractional calculus operators (viz. the generalized fractional integrals and generalized fractional derivatives). Some interesting consequences of these operators are also mentioned. The concluding remarks and suggestions briefly indicate possibilities of further work pertaining to the operators studied in this paper.
2.
Solution of integral equation (5)
To obtain the solution of the integral equation (5) formally, we first express in it the F3 function by means of the series representation (2), and employ the well-known Euler’s transformation: x (6) F (a, b; c; x) = (1 − x)−a F a, c − b; c; x−1 to get g(x) =
∞ (x − a)−α (−1)p (α )p (β )p x (x − t)γ +p−1 (γ ) p=0 p!(γ )p a x−t −α −p · (t − a) F α, β; γ + p; f (t) dt. x−a
(7)
Replacing x by t and t by s in (7), and multiplying both sides of the resulting equation by x−t t −x , (m ∈ N) (8) (x − t)m−γ −1 (t − a)α F3 −α , −α, m − β , −β; m − γ ; x−a t −a
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or, equivalently by the expression ∞ (−α )q (m − β )q q=0
q!(m − γ )q
x−t (x − a)α−q (x − t)m−γ +q−1 F −α, m − γ + β + q; m − γ + q; x−a (9)
(which follows easily from (8) upon using the series representation (2) and the transformation (6)), and then integrating over (a, x), we get x−t t −x , dt (x − t)m−γ −1 F3 −α , −α, m − β , −β; m − γ ; x−a t −a a t t −s s−t γ −1 −α · (t − s) (s − a) F3 α, α , β, β ; γ ; , f (s) ds t −a s−a a x ∞ (−α )q (m − β )q = (x − t)m−γ +q−1 (x − a)α−q q!(m − γ )q a q=0 ⎧ ∞ (−α )p (−β )p x−t ⎨ t · F −α, m − γ + β + q; m − γ + q; (t − s)γ −1 x−a ⎩ a (γ )p p! p=0 ⎫ ⎬ t −s · (s − a)−α −p (s − t)p F α, β; γ + p; f (s) ds ⎭ t −a
(x) =
x
= (γ )
x−t t −x (x − t)m−γ −1 (t − a)α F3 −α , −α, m − β , −β; m − γ ; , g(t) dt, x−a t −a (10)
x
a
where, we have denoted the left-hand side of (10) by (x). Interchanging the order of integration, and using the substitution t = s + (1 − y)(x − s) and the known integral formula [1, (9)(3)]: F (a, b; c; x) =
(c) (λ)(c − λ)
1
s λ−1 (1 − s)c−λ−1 (1 − sx)−a
0
x(1 − s) · F (a − a , b; λ; sx)F a , b − λ; c − λ; ds, 1 − sx
then (x) simplifies to the form
(x) =
∞ ∞ (α )p (β )p (−α )q (m − β )q (−1)p (x − a)α−q (γ )p (m − γ )q p!q!
p=0 q=0
x a
t −s (x − t) (t − s) (t − a) F α, β; γ + p; · t −a s x−t · F −α, m − γ + β + q; m − γ + q; dt x−a t
m−γ +q−1
p+γ −1
−α
(s − a)−α −p f (s) ds
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R.K. Raina
=
∞ ∞ (α )p (β )p (−α )p (m − β )p (−1)p p=0 q=0
x
·
(x − a)−q
(γ )p (m − γ )q p!q!
(s − a)−α −p (x − s)m+p+q−1 f (s) ds
1
y m−γ +q−1 (1 − y)γ +p−1
0
a
y(x − s) −α (1 − y)(x − s)/(x − a) · 1− F α, β; γ + p; x−a 1 − y(x − s)/(x − a) y(x − s) · F −α, m − γ + β + q; m − γ + q; f (s) ds x−a
∞
∞
(γ )(m − γ ) (α )p (β )p (−α )p (m − β )p (−1)p (x − a)−q (m − 1)! (m) p!q! p+q p=0 q=0 x x−s · (s − a)−α −p (x − s)m+p+q−1 F 0, m − γ + β + q; m + p + q; f (s) ds x−a a (γ )(m − γ ) x s−x x−s m−1 −α = (x − s) (s − a) F3 α , −α , β , m − β ; m; , (m − 1)! s−a x−a a
=
· f (s) ds.
(11)
Applying (4) to the F3 function in (11) above, we get x (γ )(m − γ ) s−x −α m−1 (x − a) (x) = (x − s) F −α , 0; m; f (s) ds (m − 1)! s−a a (x − a)−α (γ )(m − γ ) x (x − s)m−1 f (s) ds. = (m − 1)! a
(12)
From (10) and (12), we obtain x x−t t −x (x − a)α m−γ −1 α (x − t) (t − a) F3 −α , −α, m − β , −β; m − γ ; , g(t) dt (m − γ ) a x−a t −a x 1 = (x − s)m−1 f (s) ds. (13) (m − 1)! a Differentiating m times both the sides of (13), we arrive at the solution of the integral equation (5) which is given by x (x − a)α dm (x − t)m−γ −1 (t − a)α f (x) = m dx (m − γ ) a x−t t −x · F3 −α , −α, m − β , −β; m − γ ; , g(t) dt. (14) x−a t −a Remark 1 The pair of integral equations (5) and (14) remain valid if we replace x − a by x − h, where h ≤ a, and consequently the integral equation (x − h)−α x x−t t −x (α,α ,β,β ,γ ) f (x) = (x − t)γ −1 (t − h)−α F3 α, α , β, β ; γ ; , f (t) dt Ia,h,x (γ ) x−h t −h a = g(x)
(x > a),
(15)
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where α, β ∈ R, γ ∈ R+ (0 < γ < 1), and h ≤ a yields the solution f (x) given by x (x − h)α dm (x − t)m−γ −1 (t − h)α f (x) = m dx (m − γ ) a x−t t −x · F3 −α , −α, m − β , −β; m − γ ; , g(t) dt. x−h t −h
(16)
For m = 1, α = β = 0 in (5) and (14), and making use of (2) and (6), the above pair of integral equations correspond to the known pair of integral equations due to Raina et al. [9, pp. 294–295].
Fractional calculus operators associated with the F3 function
3.
The pair of integral equations (5) and (14) permits us to define new forms of generalized fractional calculus operators involving the F3 function defined by (1). (α,α ,β,β ,γ ) Let α, α , β, β , γ ∈ R. In view of (5), the generalized fractional integral operator Ia,x of a function f (x) is defined by (α,α ,β,β ,γ ) f (x) Ia,x
(x − a)−α x = (x − t)γ −1 (t − a)−α (γ ) a x−t t −x · F3 α, α , β, β ; γ ; , f (t) dt x−a t −a
(γ > 0, x > a).
(17)
Based upon the solution (14) of the integral equation (5), the generalized fractional derivative (α,α ,β,β ,γ ) of a function f (x) can be defined by operator Da,x
dm −α ,−α,m−β ,−β,m−γ I f (x) (m − 1 ≤ γ < m; m ∈ N) dx m a,x x (x − a)α dm = m (x − t)m−γ −1 (t − a)α dx (m − γ ) a x−t t −x , f (t) dt. · F3 −α , −α, m − β , −β; m − γ ; x−a t −a
(α,α ,β,β ,γ ) f (x) = Da,x
(18)
Indeed, the operators (α,α ,β,β ,γ )
I (α, α , β, β , γ )f (x) = I0,x
f (x)
(19)
and I (α, α , β, β , γ )f (x) =
dm I (α, α , β + m, β , γ + m)f (x) dx m
(γ ≤ 0, m = 1 − [γ ]) (20)
were earlier defined by Saigo and Maeda [11] (see also Kiryakova [4]) as the generalized operators of fractional integral and fractional derivative of a function f (x) involving the F3 function, respectively. The image function under the Saigo–Maeda operators (19) and (20) of a power
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R.K. Raina
function x k are, respectively, given by [11]: (α,α ,β,β ,γ ) k
I (α, α , β, β , γ )x k = I0,x =
x
(1 + k)(1 + k − α + β )(1 + k − α − α − β + γ ) x k−α−α +γ (1 + k + β )(1 + k − α − β + γ )(1 + k − α − α + γ ) · (γ > 0; α, α , β, β ∈ R; k > max(0, α − β , α + α + β − γ ) − 1) (21)
and dm I (α, α , β + m, β , m − γ )x k dx m (1 + k)((1 + k − α + β )(1 + k − α − α − β − γ ) k−α−α −γ = x (1 + k + β )(1 + k − α − β − γ )(1 + k − α − α − γ )
I (α, α , β, β , γ )x k =
· (γ ≥ 0; α, α , β, β ∈ R; k > max(0, α − β , α + α + β + γ ) − 1). (22) On the other hand, it is worth noting here that our generalized fractional derivative operator (18) gives the following image formula for the power function x k : (α,α ,β,β ,γ ) k
D0,x
dm −α ,−α,m−β ,−β,m−γ k I x (m − 1 ≤ γ < m; m ∈ N) dx m 0,x (1 + k)((1 + k + α − β)(1 + k + α + α + β − γ ) k+α+α −γ = x (1 + k − β)(1 + k + α + β − γ )(1 + k + α + α − γ )
x =
· (γ ≥ 0; α, α , β, β ∈ R; k > max(0, β − α, −α − α − β + γ ) − 1). (23) The operators (17) and (18) satisfy the following relationship: −1 (α,α ,β,β ,γ ) (α,α ,β,β ,γ ) (−α ,−α,−β ,−β,−γ ) = Da,x = Ia,x , Ia,x
(24)
which provides improvement to similar types of operational relationships given in [11]. It may be observed that when α = λ + μ, γ = λ, β = −η, α = β = 0
(25)
in (24) above, we get the following Saigo type operational relationship [10] (see also [3, p. 56]): λ,μ,η −1 λ,μ,η −λ,−μ,λ+η } = Da,x = Ia,x . {Ia,x
(26)
The complex variable analogues of the operators (17) and (18) may be formulated in the following forms. (α,α ,β,β ,γ ) of a Let α, α , β, β , γ , a ∈ C and (γ ) > 0, then the fractional integral operator Ia,z function f (z) is defined by (z − a)−α z α,α ,β,β ,γ f (z) = (z − ζ )γ −1 (ζ − a)−α F3 Ia,z (γ ) a z−ζ ζ −z , f (ζ ) dζ, (27) · α, α , β, β ; γ ; z−a ζ −a where the function f (z) is analytic in a simply connected region of the complex z-plane containing the point a, and it is understood that (z − ζ )γ −1 and (ζ − a)−α denote, respectively, the principal values for 0 ≤ arg(z − ζ ) < 2π and 0 ≤ arg(ζ − a) < 2π .
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The generalized fractional integral operator (27) involving the F3 function in the kernel was (in the case a = 0) used in the Geometric Function Theory recently by Kiryakova [4] and Raina [6]. The corresponding complex variable analogue of the generalized fractional derivative operator (α,α ,β,β ,γ ) Da,z of a function f (z) is defined by
(α,α ,β,β ,γ ) f (z) = Da,z
dz −α ,−α,m−β ,−β,m−γ I f (z) dzm a,z
(m − 1 ≤ (γ ) < m; m ∈ N).
(28)
However, the Saigo–Maeda type generalized fractional derivative operator [11] of a function f (z) is given by
α,α ,β,β ,γ f (z) = Da,z
dm α,α ,β+m,β ,m−γ I f (z) dzm a,z
(m − 1 ≤ (γ ) < m; m ∈ N),
(29)
and for a = 0 and f (z) = zk , (29) gives the following corrected formula [6, p. 4, Equation (7)]: α,α ,β,β ,γ k
D0,z
dm α,α ,β+m,β ,m−γ k I z dzm 0,z (1 + k)((1 + k − α + β )(1 + k − α − α − β − γ ) k−α−α −γ = . x (1 + k + β )(1 + k − α − β − γ )(1 + k − α − α − γ )
z =
(γ ≥ 0; α, α , β, β , γ ∈ R; k > max(0, α − β , α + α + β + γ ) − 1). (30) Remark 2 The parameteric substitutions (25) when used in (21) readily gives the Saigo type fractional integral formula [17, p. 415], which has extensively been used in literature (and among many others, see for instance, [4,7,8], and also the various references cited in [11,15,16]). However, the same parameteric substitutions (25) used in (23) (unlike the Saigo type fractional derivative formula given, for example, in [8]) yields the following new formula: (λ+μ,0,−η,0,−η) k
D0,z
λ,μ,η
z = D0,x zk =
(1 + k)((1 + k + λ + μ + η) k+μ z . (1 + k + η)(1 + k + μ))
(λ ≥ 0; k > max(0, −λ − μ − η) − 1)
4.
(31)
Concluding remarks and suggestions
In this paper, we have attempted a formal type of solution (14) for the Abel-type integral equation (5). However, the investigation of the necessary and sufficient conditions concerning the solvability of Equation (5) (though somewhat difficult) can be studied in the space Ł1 (a, b) of summable functions by following the paper [9]. It may also be of some interest to consider the corresponding multidimensional analogue of the Abel-type integral equation (5) over a certain pyramidal domain as considered in [9]. Further, the family of fractional calculus operators (fractional derivaties and fractional integrals) defined by (27) and (28), and particularly, the image formulas corresponding to (21) and (23) (under these operators) can fruitfully be used in the Geometric Function Theory. Several new analytic, multivalent (or meromorphic) function classes can be defined and the various geometric properties like, the coefficient estimates, distortion bounds, radii of starlikeness, convexity and close-to-convexity can be investigated for such contemplated classes of functions. Acknowledgements The author is thankful to the referee for his comments and suggestions.
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