Special Functions and Middle Convolution
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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 13, Issue 3 Ver. V (May - June 2017), PP Special Functions and Middle Convolution M. Nobert Hounkonnou 1, Kassim, A.M 2 1 University of Abomey-Calavi Republic of Benin 2 Lagos State University, Ojo, Lagos State, Nigeria. Abstract: This article is a survey on recent studies on middle convolution for linear and nonlinear special functions [J. Phys. A: Math. Theor. 43 (2010), ; J. Phys. A: Math. Theor. 42 (2009), ; Kumamoto J. Math. 19 (2006), 15 23]. Part of this survey is also based on the joint work with Y. Haraoka [J. Lond. Math. Soc. 76 (2) (2007), ]. After recalling some basic facts concerning linear and nonlinear special functions, we give an overview of the recent results and current studies on middle convolution for special functions. Keywords: Linear and nonlinear special functions; middle convolution MSC (2010): 34M55. The main objective of this paper is to survey recent results on middle con- volution based on the papers [9, 10, 11, 15] and discuss computational aspects. The paper is organized as follows. In the first section we overview linear special functions such as the Gauss hypergeometric functions and the Heun functions. Next we survey nonlinear special functions, such as the Painlevé transcendents. Next we explain the notion of middle convolution. Finally, we show how to use middle convolution to derive transformations for linear and nonlinear special functions. We also explain the notion of Schlesinger systems and study how they are changed under middle convolution. 1 Special functions There is no strict mathematical definition of the notion special functions. Wikipedia 1 defines special functions as particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications. One can also find a lot of information about special functions at Digital Library of Mathematical Functions at which is a recent update of Abramovich-Stegun s book Handbook of Mathematical Functions. DOI: / Page
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16 M. Nobert Hounkonnou. "Special Functions and Middle Convolution." IOSR Journal of Mathematics (IOSR-JM) 13.3 (2017): DOI: / Page
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