These are functions that frequently arise in advanced mathematical analysis and in mathematical physics, usually in connection with the solutions of differential equations (in Sturm-Liouville Theory). (this is gradually turning into a metanode...) They fall into several general categories:

Functions defined by integrals. These include:

Orthogonal polynomials:

Other functions:

Most special functions can be thought of as generalizations of the hypergeometric function or confluent hypergeometric function.