A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933–2019) and pay tribute Bateman Project. Harry planned his “shoe-boxes” project (accomplished after death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under editorship by A. Erdélyi) a “Guide Functions”. This inspired author use modified title present survey. Most standard (classical) Special Functions are representable in terms Meijer G-function and, specially, generalized hypergeometric functions pFq. These appeared solutions differential equations mathematical physics other applied sciences that integer order, usually second order. However, recently, models fractional order preferred because they reflect more adequately nature various social events, these needs attracted attention “new” classes special their solutions, so-called Fractional Calculus (SF FC). Generally, this notion, we have mind Fox H-functions, most widely used cases Wright p?q particular, Mittag–Leffler type functions, among them “Queen function calculus”, function. indices/parameters extensions classical became an unavoidable tool when fractalized phenomena events treated. Here, try review some basic results on theory SF FC, obtained author’s works for than 30 years, support wide spreading important role several examples.