Complex inversion and uniqueness theorems for a generalized Laplace transform (Q1190342)

The authors prove a complex inversion theorem for a generalization of the Laplace transform which is given as \[ F(x)={\Gamma(A) \over \Gamma(B)} \int_ 0^ \infty (xt)^ \beta {_ 1F_ 1}(A,B,-xt)f(t)dt. \] A uniqueness theorem is also given. The results of this paper are nor new neither original. These and many more results like analyticity, inversion, multidimensionality, uniqueness, representation etc. for this transformation were discussed very thoroughly long back in 1972 by this reviewer [A study of certain distributional transforms, Ph. D. Thesis, Ranchi Univ. (1972), Bull. Cal. Math. Soc. 68, 267-274 (1976; Zbl 0376.44003), Collect. Math. 29, 119-131 (1978; Zbl 0445.44003), J. Ind. Math. Soc., New Ser. 39, 219-226 (1975; Zbl 0452.46016) and Riv. Math. Univ. Parma, IV. Ser. 4, 63-72 (1978; Zbl 0409.46045)]. A few apparent changes made by the authors in the construction of seminorms etc. are quite insignificant. It is surprising how any one of the above references did not find a place in the bibliography of the paper under review.