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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
Böhmer integral
Carlitz-Goss Gamma function
Clausen functions
Dawson function
Dilogarithm
Dirac delta
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
$p$-adic $L$ function
Painlevé transcendents
Riemann theta function
Rising factorial
Sievert integral
Signum
Struve functions
Theta functions
Bessel functions
Riccati-Bessel
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Kelvin functions
Kelvin ber
Kelvin bei
Kelvin ker
Kelvin kei
Elliptic functions
 Elliptic $E$ |
 Elliptic $K$ |
 Elliptic $\theta$ |
 Modular $\lambda$ |
 Weierstrass $P$ |
Jacobi elliptic functions
Weierstrass $\sigma$
Polynomials
Bell polynomial
Charlier polynomial
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange polynomial
Lommel polynomial
Mahler polynomial
Meixner polynomial
Meixner-Pollaczek polynomial
Mott polynomial
Pidduck polynomial
Spread polynomial
Touchard polynomial
Rook polynomial
Trigonometric functions
Special functions in number theory
 Euler totient |
 Mangoldt |
 Mertens |
 Möbius |
 Liouville $\lambda$ |
 Prime counting |
 Riemann zeta |
 Sum of divisors |
Chebyshev function
Dirichlet $L$-function
Greatest prime factor
Ramanujan's sum
Riemann prime counting
Zeta functions
A directory of zeta functions.
 Hurwitz zeta |
 Jacobi zeta |
 Prime zeta $P$ |
Riemann zeta |
 Weierstrass zeta |
Airy zeta function
Arithmetic 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
Reciprocal zeta function
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function
$q$-Calculus
Basic Fourier series
$q$-analog
$q$-Bernoulli numbers
$q$-derivative
$q$-numbers
$q$-Special functions
Basic hypergeometric series $\phi$
Basic hypergeometric series $\psi$
Continuous $q$-Hermite polynomials
Continuous q-ultraspherical polynomial
Exton $q$-exponential
Generalized $q$-Bessel
Hahn-Exton $q$-Bessel
Gosper $q$-sine
Jackson $q$-Bessel (1)
Jackson $q$-Bessel (2)
Jackson $q$-Bessel (3)
$q$-Bessel function
$q$-Beta function
$q$-$\cos$
$q$-$\mathrm{Cos}$
$q$-Binomial coefficient
$q$-exponential e
$q$-exponential E
$q$-Fibonacci polynomials
$q$-Gaussian distribution
$q$-Hermite polynomial
$q$-Hypergeometric function
$q$-Pochhammer symbol
q-Pollaczek polynomial
$q$-Polygamma function
$q$-shifted factorial
$q$-$\sin$
$q$-$\mathrm{Sin}$
$q$-Theta function
Tsallis $q$-exponential
 $q$-Binomial |
 $q$-Gamma |
 $q$-factorial |
 Euler $\phi$ |
Special sequences
Bell numbers
Bernoulli numbers ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$)
Euler numbers
Genocchi numbers
Harmonic number
Lah numbers
Mersenne numbers
Stirling numbers of the first kind
Stirling numbers of the second kind
Figurate numbers
Square number
Triangular numbers
Tetrahedral 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 ($\phi=1.618033988749894848204586834365638117720309179805762862135448 \ldots$)
Imaginary number ($i=\sqrt{-1}$)
Kaprekar's constant
Khinchin's constant ($K=2.685452001065306445309714835481795693820382293994462953051152 \ldots$)
Legendre's constant ($B=1$)
Lefschetz number
Lévy's constant
Mills' constant ($M=1.306377883863080690468614492602605712916784585156713644368053 \ldots$)
Pi ($\pi=3.141592653589793238462643383279502884197169399375105820974944 \ldots$)
$q$-Bernoulli numbers
$\pi_q$
Silver ratio ($\delta_s=2.4142135623730950488 \ldots$)
Soldner's Constant ($\mu = 1.45136923488338105028396848589202744949\ldots$)
Numbers
Algebraic number $\overline{\mathbb{Q}}$
Complex number $\mathbb{C}$
Integer $\mathbb{Z}$
Irrational number $\mathbb{R} \setminus \mathbb{Q}$
Natural number $\mathbb{N}$
Prime number
Rational number $\mathbb{Q}$
Real number $\mathbb{R}$
Tools from calculus
Absolute maximum
Absolute minimum
Analytic continuation
Continuous
Derivative
Difference equation of hypergeometric type
Integration by parts
Laplace transform
Lebesgue integral
Local maximum
Local minimum
Ratio test
Riemann integral
Rolle's theorem
Taylor series
Weierstrass factorization
Other
Higher transcendental functions (Bateman project)
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
