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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
Automorphic forms and functions
Blancmange function
Carlitz-Goss Gamma function
Clausen functions
Dawson function
Eisenstein series
Heaviside step function
Hermite function
Jacobi theta functions
K-function
Legendre chi
Legendre function
Lerch transcendent
Mathieu function
Maass forms
Meijer G-function
Modular form
Mock modular forms
Mock theta functions
Painlevé transcendents
Riemann theta function
Sievert integral
Signum
Struve functions
Theta functions
Elliptic functions
Polynomials
 Abel $p$ |
 Associated Laguerre L |
 Bateman |
 Bernoulli $B$ |
 Bessel poly |
 Chebyshev poly |
 Dickson |
 Euler $E$ |
 Fibonacci |
 Gegenbauer $C$ |
 Hermite (phys) |
 Hermite (prob) |
 Jacobi $P$ |
 Laguerre $L$ |
 Legendre $P$ |
 Lucas |
Bell polynomial
Charlier polynomial
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange polynomial
Mahler polynomial
Mott polynomial
Pidduck polynomial
Spread polynomial
Touchard polynomial
Rook polynomial
Trigonometric functions
 Cosine integral |
 Cosh integral |
 Fresnel $C$ |
 Fresnel S |
 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
 Euler $\phi$ |
 Mangoldt |
 Mertens |
 Möbius |
 Liouville $\lambda$ |
 Prime counting |
 Riemann zeta |
 Sum of divisors |
Chebyshev function
Dirichlet $L$-function
Greatest prime factor
Ramanujan's sum
Zeta functions
A directory of zeta functions.
 Hurwitz zeta |
 Jacobi zeta |
 Prime zeta $P$ |
Riemann zeta |
 Weierstrass zeta |
Airy zeta function
Arakawa-Kaneko zeta function
Arithmetic zeta function
Artin-Mazur zeta function
Barnes zeta function
Beurling zeta function
Cotangent zeta function
Dedekind zeta function
Epstein zeta function
Goss zeta function
Hasse-Weil zeta function
Height 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
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function
$q$-Calculus
 $q$-Binomial |
 $q$-Gamma |
 $q$-Factorial |
 $q$-Pochhammer |
$q$-Bessel function
$q$-Beta function
q-exponential
$q$-Hermite polynomial
$q$-Hypergeometric function
$q$-Polygamma function
q-shifted factorial
$q$-Theta function
Special sequences
Bernoulli number ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$)
Harmonic number
Mersenne numbers
Special constants
Apéry's constant ($\zeta(3)=1.2020569031595942854 \ldots$)
Catalan constant ($0.915965594177219015054603514932384110774 \ldots$)
Chaitin's constant
Champernowne constant ($0.12345678910111213141516171819202122232425\ldots$)
Conway's constant ($1.303577269034 \ldots$)
Copeland-Erdős constant ($0.23571113171923...$)
Euler's number ($e=2.71828182846 \ldots$)
Erdős-Borwein Constant ($E=1.606695152415291763\ldots$)
Euler-Mascheroni constant ($\gamma=0.5772156649015328606065120900824024310421593359399235\ldots$)
Feigenbaum constants ($4.669201609102990671853203820466201617258185577475768632745651 \ldots$)
Fransén–Robinson constant ($2.807770242028519365221501186557772932308085920930198291220054 \ldots$)
Gauss' constant
Gelfond constant ($e^{\pi}=23.14069263277926900572908636794854738026610624260021199344504 \ldots$)
Gelfond–Schneider constant ($2^{\sqrt{2}}=2.665144142690225188650297249873139848274211313714659492835979 \ldots$)
Glaisher–Kinkelin constant ($A = 1.2824271291 \ldots$)
Golden ratio ($\phi=1.618033988749894848204586834365638117720309179805762862135448 \ldots$)
Kaprekar's constant
Khinchin's constant ($K=2.685452001065306445309714835481795693820382293994462953051152 \ldots$)
Legendre's constant ($B=1$)
Lévy's constant
Mills' constant ($M=1.306377883863080690468614492602605712916784585156713644368053 \ldots$)
Pi ($\pi=3.141592653589793238462643383279502884197169399375105820974944 \ldots$)
Silver ratio ($\delta_s=2.4142135623730950488 \ldots$)
Soldner's Constant ($\mu = 1.45136923488338105028396848589202744949\ldots$)
Tools from calculus
Analytic continuation
Difference equation of hypergeometric type
Integration by parts
Laplace transform
Ratio test
Rolle's theorem
Taylor series
Weierstrass factorization
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
Mathematical Constants by Steven R. Finch
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
