Fractional calculus and special functions

. . . . . . . . . . . . . . . . . . . . . . . . . . . p. 1 A. Historical Notes and Introduction to Fractional Calculus . . . . . . . p. 2 B. The Liouville-Weyl Fractional Calculus . . . . . . . . . . . . . . . p. 8 C. The Riesz-Feller Fractional Calculus . . . . . . . . . . . . . . . . p.12 D. The Riemann-Liouville Fractional Calculus . . . . . . . . . . . . . p.18 E. The Grünwald-Letnikov Fractional Calculus . . . . . . . . . . . . . p.22 F. The Mittag-Leffler Functions . . . . . . . . . . . . . . . . . . . . p.26 G. The Wright Functions . . . . . . . . . . . . . . . . . . . . . . p.42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . p.52 The present Lecture Notes are related to a Mini Course on Introduction to Fractional Calculus delivered by F. Mainardi, at BCAM, Bask Centre for Applied Mathematics, in Bilbao, Spain on March 11-15, 2013, see http://www.bcamath.org/en/activities/courses. The treatment reflects the research activity of the Author carried out from the academic year 1993/94, mainly in collaboration with his students and with Rudolf Gorenflo, Professor Emeritus of Mathematics at the Freie Universtät, Berlin. ii Francesco MAINARDI c © 2013 Francesco Mainardi FRACTIONAL CALCULUS AND SPECIAL FUNCTIONS 1 FRACTIONAL CALCULUS AND SPECIAL FUNCTIONS Francesco MAINARDI Department of Physics, University of Bologna, and INFN Via Irnerio 46, I–40126 Bologna, Italy. francesco.mainardi@unibo.it francesco.mainardi@bo.infn.it