ON THE MEIJER TRANSFORMATION

Recently [8], an operational calculus for the operator B μ = t − μ D t 1 + μ D with − 1 μ ∞ was developed via the algebraic approach [4], [13], [15]. This paper gives the integral transform version. In particular, a differentiation theorem and a convolution theorem are proved.

His calculus reduces to Ditkin's when 0 and Meller's when e(-l,l).
In this paper, we give an integral transform analogue of [8] via the MeiJer transform of the form k {f}(p) 2p.
The presence of a factor 2p in (3) as opposed to those in (i) and ( 2) is essential r (+) in our convolution theorem.

THE MAIN THEOREMS.
We will define the convolution, *, of two functions, f, g by f(x)g[(1-x)(t-)]dxd, (4) see Koh [8], where D is the Riemann-Liouville derivative of order X, see Ross [17].This convolutlon exists if, for example, f and g are in C [0,oo), the space of infinitely differentlable complex functions on [0,).
The integral then converges absolutely within the parabolic region Re/ > y.This is clear from the asymptotic behaviors (6.1)   and (6.2) The first integral on the right hand side of (8) exists because of (6.1); the second exists because of the local integrabillty of f(t) and the continuity of (pt)K (2/pt); and the last integral exists because of (6.2) provided Re/p > y.We state this result in THEOREM i.
Furthermore, the integral (3) as a function of p is analytic in the region of convergence.
The proof of the analyticity is standard and is omitted.
When a function f(t) satisfies the hypothesis of theorem i, we shall write, for brevity, fEHypl.Clearly, if a function f has continuous derivative on [0, ) and f'Hypl, then fgHypl.
-P-+I 2 f) The limit terms vanish because f'Hypl.We now use (7) and another integration by parts to yield k(Bf) 2E2  THEOREM 3.
If fec2k[0,=) and 2k-l)eHypl, then" where Q(z) is a polynomial, we transform (9) into Q(p)k f P(p) + k g where P(p) is a polynomial of degree less than or equal to that of Q(p).Therefore Q(p'---' (klag)(P) and f(t) is retrieved by means of an inversion formula and possibly a convolution theorem.
The following inversion theorem is obtained from MeiJer's Theorem [18] through a simple change of variables, vlz.x / and y / 2/.

U
q Indeed, letting R:k denote the Riemann- Llouvllle integral of order (15) q f (t)dt f exp(-qy-t/y)y  (19) by the definition of the Riemann-Liouville integral, Rk_ and integrating by parts k-times.The integrated terms vanish at t 0 and t by (6.1), (6.2), and the fact that in the definition of (t), the functions f and g satisfy the hypotheses of theorem 5.