Generalized Extended Mittag-Leffler Transform
by
A.A. Koroleva
Assistent Professor, Belarusian State University, Minsk 220050, Belarus
Coauthors: A.A. Kilbas

The generalized extended Mittag-Leffler function E_{a1,  b1,  a2,  b2}(z) with complex a₁,  a₂ Î C  (a₁+a₂ ¹ 0) and b₁, b₂ Î C is defined by the following Mellin-Barnes integral

Ea1,  b1,   a2,  b2(z)= 1 2pi ó õ L  G(s)G(1-s) G(b1-a1s)G(b2-a2s) (-z)-sds     (z ¹ 0).

(1)

Our report deals with the integral transform

(Ea1,  b1,   a2,  b2f)(x)= ¥ ó õ 0  Ea1,  b1,   a2,  b2(-xt)f(t)dt     (x > 0),

(2)

containing the function (1) in the kernel. We show that this transform is a special case of the so-called H-transform [2]. On the basis of this fact we establish mapping properties such as the boundedness, the representation and the range of the transform (2) and its inversion formulas in the space L_{n,  r} of the Lebesgue measurable functions f on the R₊=(0, ¥) such that

¥ ó õ 0  |tnf(t)|  dt t < ¥      (1 £ r £ ¥, v Î R).

(3)

Similar properties for the transform of the form (2), in which E_{a1,  b1,  a2,  b2}(z) is replaced by the classical Mittag-Leffler functions [3,  Sect. 18.1]

Ea,  b(z)= ¥ å k=0  zk G(ak+b)    (a > 0; b Î R; z Î C)

(4)

with a > 0, were given in [1].

References

[1] Bonilla B., Rivero M., Rodriguez-Germa L., Trujillo J.J., Kilbas A.A., Klimets N.G. Mittag-Leffler integral transform on L_{n,  r} spaces. Rev. Acad. Canar. Cienc. V.14, ¹ 1-2, P. 65 - 77, 2002.

[2] Kilbas A.A., Saigo M. H-Transforms. Theory and Applications Chapman and Hall, Boca Raton, Fl, 2004.

[3] Erdelyi A., Magnus W., Oberhettinger T. and Tricomi T.C Higher Transcendental Function. Vol. 3, McGraw-Hill, New York, 1955.

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Date received: May 20, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqw-58.