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$G$-functions as self-reciprocal in an integral transform

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G-functions as self-reciprocal in an integral transform COMPOSITIO MATHEMATICA ROOPNARAINKESARWANI G-functions as self-reciprocal in an integral transform Compositio Mathematica, tome 18, no 1-2 (1967), p. 181-187. <http://www.numdam.org/item?id=CM_1967__18_1-2_181_0> © Foundation Compositio Mathematica, 1967, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 181 G-functions as self-reciprocal in an integral transform by Roop Narain Kesarwani 1 It has been proved [1, p. 298 and 2, p. 396] that the function where G denotes a Meijer’s G-function [3, p. 207], plays the role of a symmetric Fourier kernel. Here 03BC and y are real constants, p and q are integers such that and aj (j = 1, ..., p), bi (j = 1, ..., q) are complex numbers satisfying and obeying conditions mentioned in the following theorem due to Fox [2, p. 399]. See also [4, p. 277]. THEOREM 1. I f then the f ormula defines almost everywhere the function g(x) e L2(0, oo). Also the reciprocal formula 182 holds almost everywhere. The kernel (1.1) is a very general one. It contains as its particular cases Fourier type kernels discovered by various authors from time to time. Some of them have been listed in an earlier paper of the author [5, pp. 957-58]. In case we can differentiate with respect to x under the sign of integral, the formulas (1.2) and (1.3) reduce to The functions f(x) and g(x) may be called G-transforms of each other. If further, f(x) = g(x) so that then f(x) is called self-reciprocal for the kernel (1.

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