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Special functions are ubiquitous in mathematics and applications of mathematics. The term applies to such a wide range of functions that no single source contains all of them. We aim to remedy this problem.

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Special functions

Airy functions
Automorphic forms and functions
Bessel functions
Beta function
Binomial coefficient
Binomial function
Böhmer integral
Carlitz-Goss Gamma function
Clausen functions
Cosine integral
Dawson function
Dedekind eta function
Digamma function
Dirichlet beta function
Dirichlet eta function
Eisenstein series
Error function
Exponential function
Exponential integral
Factorial
Fresnel integral
Gamma function
Generalized hypergeometric function ${}_pF_q$
Hermite function
Hypergeometric $q$-series
Incomplete gamma function
Jacobi elliptic functions
Jacobi theta functions
Klein's j-invariant
Lambert $W$ function
Legendre function
Lerch transcendent
Logarithm
Logarithmic integral
Mathieu function
Maass forms
Minkowski $?$ function
Meijer G-function
Modular forms and functions
Mock modular forms
Mock theta functions
Painlevé transcendents
Polylogarithm
Partition function
Pochhammer symbol
Polygamma functions
Polylogarithms
Riemann theta function
Sinc function
Sine integral
Sum of positive divisors function
Struve functions
Theta functions
Trigonometric integral
Weierstrass elliptic functions

Polynomials

Abel polynomial
Bateman polynomial
Bedient polynomial
Bell polynomial
Bernoulli polynomial
Bessel polynomial
Chebyshev polynomial
Dickson polynomial
Euler polynomial
Fibonacci polynomial
Gegenbauer Polynomials
Hermite polynomial (physicist)
Hermite polynomial (probabilist)
Jacobi polynomial
$q$-Hermite polynomial
Lagrange polynomial
Laguerre polynomial
Legendre polynomial
Lucas polynomial
Mahler polynomial
Mott polynomial
Pidduck polynomial
Spread polynomial
Touchard polynomial
Rook polynomial

Trigonometric functions

Arccos
Arcsin
Arctan
Cosine
Cotangent
Cosecant
Secant
Sine
Tangent

Special functions in number theory

Chebyshev function
Dirichlet $L$-function
Liouville function
Mangoldt function
Mertens function
Möbius function
Prime counting function
Ramanujan's sum
Riemann zeta function
Totient function

Zeta functions

A directory of zeta functions.

Airy zeta function
Arakawa-Kaneko zeta function
Arithmetic zeta function
Artin-Mazur zeta function
Barnes zeta function
Beurling zeta function
Dedekind zeta function
Epstein zeta function
Goss zeta function
Hasse-Weil zeta function
Height zeta function
Hurwitz zeta function
Igusa zeta function
Ihara zeta function
Lefschetz zeta function
Lerch zeta function
Local zeta function
Matsumoto zeta function
Minakshisudaram-Pleijel zeta function
Motivic zeta function
Multiple zeta function
$p$-adic zeta function
Prime zeta function
Riemann zeta function
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function

$q$-Calculus

$q$-Bessel function
$q$-Beta function
$q$-Binomial function
$q$-Factorial
$q$-Gamma function
$q$-Hermite polynomial
$q$-Pochhammer symbol

Special constants

Apéry's constant
Bernoulli numbers
Catalan constant
Chaitin's constant
Champernowne constant
Conway's constant
$e$
Euler-Mascheroni constant
Feigenbaum constants
Fransén–Robinson constant
Gauss' constant
Gelfond constant
Gelfond–Schneider constant
Golden ratio
Kaprekar's constant
Khinchin's constant
Lévy's constant
Mills' constant
$\pi$
Soldner's Constant

Tools from calculus

Integration by parts
Ratio test
Rolle's theorem

Other

Solution manual to Rainville's "Special Functions"
"Special Functions" by Leon Hall
Relations between special functions by John D. Cook
NIST Digital Library of Mathematical Functions
T.H Koornwinder's list of resources
Victor Moll's proofs of Gradshteyn and Ryzhik integrals
Abramowitz and Stegun
LMFDB, the database of L-functions, modular forms, and related objects.
Phillipe Flajolet's /Analytic Combinatorcs/