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.