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Salvatore Pincherle: the pioneer of the Mellin-Barnes integrals. (English) Zbl 1050.33018

This interesting paper on Mellin-Barnes integrals begins with an overview of the work of Pincherle on generalized hypergeometric functions and its relation to work of later authors. The duality principle relating linear differential equations to linear difference equations with rational polynomial coefficients is discussed. It is then shown how these early developments of Pincherle lead to a derivation of the linear differential equation and the Mellin-Barnes integral representation for Meijer’s \(G\)-function.

MSC:

33E30 Other functions coming from differential, difference and integral equations
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
01-XX History and biography
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References:

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