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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	Bessel	Beta	Binomial	Dedekind eta
Digamma	Dirichlet eta	Dirichlet beta	Error function	Exponential	Exponential integral	Factorial
Gamma	Hadamard gamma	Hypergeometric ${}_pF_q$	Log integral	Logarithm	Thomae
Polygamma	Polylogarithm	Reciprocal gamma	Lambert $W$

Automorphic forms and functions
Carlitz-Goss Gamma function
Clausen functions
Dawson function
Dirichlet eta function
Eisenstein series
Heaviside step function
Hermite function
Incomplete gamma function
Jacobi elliptic functions
Jacobi theta functions
Klein's j-invariant
Legendre function
Lerch transcendent
Mathieu function
Maass forms
Minkowski $?$ function
Meijer G-function
Modular forms and functions
Mock modular forms
Mock theta functions
Painlevé transcendents
Partition function
Planck's radiation function
Pochhammer symbol
Riemann theta function
Riemann $\xi$ function
Sievert integral
Struve functions
Theta functions
Weierstrass elliptic functions

Polynomials

Abel polynomial
Associated Laguerre polynomials
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)
Hypergeometric $q$-series
Jacobi polynomial
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange polynomial
Laguerre polynomial
Legendre polynomial
Lucas polynomial
Mahler polynomial
Mott polynomial
Pidduck polynomial
Spread polynomial
Touchard polynomial
Rook polynomial

Trigonometric functions

Böhmer integral

Cosine integral

Cosh integral

Fresnel integral

Sinc

Sine integral

Sinh integral

Standard trigonometric functions

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

Inverse trigonometric functions

Arcsin

Arccos

Arctan

Arccsc

Arcsec

Arccot

Hyperbolic trigonometric functions

Sinh

Cosh

Tanh

Csch

Sech

Coth

Inverse hyperbolic trigonometric functions

Arcsinh

Arccosh

Arctanh

Arccsch

Arcsech

Arccoth

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
Sum of positive divisors 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
$q$-Hypergeometric function
$q$-Pochhammer symbol
$q$-Polygamma function
$q$-Theta function

Special constants

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

Tools from calculus

Analytic continuation
Difference equation of hypergeometric type
Integration by parts
Laplace transform
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/
Gallery of mathematical functions
Online number theory lecture notes and teaching materials
The On-Line Encyclopedia of Integer Sequences