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:

• the Euler Gamma Function
• the Beta function
• the incomplete Gamma and Beta functions
• the error function
• Fresnel integrals
• Elliptic integrals
• the sine integral
• the cosine integral
• the exponential integral
• the logarithmic integral

Orthogonal polynomials:

• Legendre polynomials and spherical harmonics
• Laguerre polynomials
• Hermite polynomials
• Chebyshev polynomials
• Gegenbauer polynomials
• Jacobi polynomials

Other functions:

• Bessel functions
• Hankel functions
• Airy functions
• the Riemann zeta function
• elliptic functions, such as the Weierstrass elliptic functions and the Jacobi elliptic functions
• the digamma function and the polygamma functions
• Hypergeometric functions

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