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(Special functions in number theory)
($q$-Calculus)
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[[Jackson q-Bessel (2) | Jackson $q$-Bessel (2)]]<br />
 
[[Jackson q-Bessel (2) | Jackson $q$-Bessel (2)]]<br />
 
[[Jackson q-Bessel (3) | Jackson $q$-Bessel (3)]]<br />
 
[[Jackson q-Bessel (3) | Jackson $q$-Bessel (3)]]<br />
 +
[[q-analog|$q$-analog]]<br />
 
[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
 
[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
 
[[q-Bessel functions | $q$-Bessel function]]<br />
 
[[q-Bessel functions | $q$-Bessel function]]<br />

Revision as of 00:48, 5 May 2015

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

Airyaithumb.png
Airy
Barnesgthumb.png
Barnes $G$
Betathumb.png
Beta
Binomialthumb.png
Binomial
Dedekindetathumb.png
Dedekind eta
Devilstaircasethumb.png
Devil's staircase
Digammathumb.png
Digamma
Dirichletetathumb.png
Dirichlet eta
Dirichletbetathumb.png
Dirichlet beta
Erfthumb.png
Error function
Expthumb.png
Exponential
Eithumb.png
Exponential integral
Factorialthumb.png
Factorial
Gammathumb.png
Gamma
Hadamardgammathumb.png
Hadamard gamma
Hyperfactorialthumb.png
Hyperfactorial
Hypergeometricthumb.png
Hypergeometric ${}_pF_q$
Incompletegammathumb.png
Incomplete gamma
Kleininvariantjthumb.png
Klein invariant $J$
Lithumb.png
Log integral
Logthumb.png
Logarithm
Thomaethumb.png
Thomae
Digammathumb.png
Polygamma
Polylog1thumb.png
Polylogarithm
Partitionthumb.png
Partition
Reciprocalgammathumb.png
Reciprocal gamma
Lambertwthumb.png
Lambert $W$
Minkowskiqmthumb.png
Minkowski $?$
Planckradiationthumb.png
Planck radiation
Pochhammerthumb.png
Pochhammer
Ramanujantauthumb.png
Ramanujan $\tau$
Riemannsiegelzthumb.png
Riemann-Siegel $Z$
Riemannxithumb.png
Riemann $\xi$

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
$p$-adic $L$ function
Painlevé transcendents
Riemann theta function
Sievert integral
Signum
Struve functions
Theta functions

Bessel functions

Besselj0thumb.png
Bessel $J_{\nu}$

Bessel $Y_{\nu}$
Hankel $H_{\nu}^{(1)}$
Hankel $H_{\nu}^{(2)}$
Modified Bessel $I_{\nu}$
Modified Bessel $K_{\nu}$
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Spherical Bessel $j_{\nu}$
Spherical Bessel $y_{\nu}$
Spherical Hankel $h_{\nu}^{(1)}$
Spherical Hankel $h_{\nu}^{(2)}$

Elliptic functions

Ellipticethumb.png
Elliptic $E$
Elliptickthumb.png
Elliptic $K$
Ellipticthetathumb.png
Elliptic $\theta$
Jacobicnthumb.png
Jacobi $\mathrm{cn}$
Jacobicdthumb.png
Jacobi $\mathrm{cd}$
Jacobicsthumb.png
Jacobi $\mathrm{cs}$
Jacobidcthumb.png
Jacobi $\mathrm{dc}$
Jacobidnthumb.png
Jacobi $\mathrm{dn}$
Jacobidsthumb.png
Jacobi $\mathrm{ds}$
Jacobincthumb.png
Jacobi $\mathrm{nc}$
Jacobindthumb.png
Jacobi $\mathrm{nd}$
Jacobindthumb.png
Jacobi $\mathrm{ns}$
Jacobiscthumb.png
Jacobi $\mathrm{sc}$
Jacobisdthumb.png
Jacobi $\mathrm{sd}$
Jacobisnthumb.png
Jacobi $\mathrm{sn}$
Modularlambdathumb.png
Modular $\lambda$
Weierstrasspthumb.png
Weierstrass $P$

Jacobi elliptic functions

Polynomials

Abelpolynomialthumb.png
Abel $p$
Associatedlaguerrelthumb.png
Associated Laguerre L
Batemanpolynomialthumb.png
Bateman $Z$
Bernoullibthumb.png
Bernoulli $B$
Besselpolynomialsthumb.png
Bessel poly
Chebyshevuthumb.png
Chebyshev U
Chebyshevtthumb.png
Chebyshev T
Dicksonthumb.png
Dickson
Eulerethumb.png
Euler $E$
Fibonaccithumb.png
Fibonacci
Gegenbauercthumb.png
Gegenbauer $C$
Hermitephysthumb.png
Hermite (phys)
Hermiteprobthumb.png
Hermite (prob)
Jacobipthumb.png
Jacobi $P$
Laguerrelthumb.png
Laguerre $L$
Legendrepthumb.png
Legendre $P$
Lucasthumb.png
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

Böhmer integral
Sincthumb.png
Sinc

Standard trigonometric functions

Sinethumb.png
Sine
Cosinethumb.png
Cosine
Tangentthumb.png
Tangent
Cosecantthumb.png
Cosecant
Secantthumb.png
Secant
Cotangentthumb.png
Cotangent

Inverse trigonometric functions

Arcsinthumb.png
Arcsin
Arccosthumb.png
Arccos
Arctanthumb.png
Arctan
Arccscthumb.png
Arccsc
Arcsecthumb.png
Arcsec
Arccotthumb.png
Arccot

Hyperbolic trigonometric functions

Sinhthumb.png
Sinh
Coshthumb.png
Cosh
Tanhthumb.png
Tanh
Cschthumb.png
Csch
Sechthumb.png
Sech
Coththumb.png
Coth

Inverse hyperbolic trigonometric functions

Arcsinhthumb.png
Arcsinh
Arccoshthumb.png
Arccosh
Arctanhthumb.png
Arctanh
Arccschthumb.png
Arccsch
Arcsechthumb.png
Arcsech
Arccoththumb.png
Arccoth

$*$-integral functions

Cosintegralthumb.png
Cosine integral
Coshintegralthumb.png
Cosh integral
Fresnelcthumb.png
Fresnel $C$
Eithumb.png
Exponential integral
Fresnelsthumb.png
Fresnel S
Sinintegralthumb.png
Sine integral
Sinhintegralthumb.png
Sinh integral

Special functions in number theory

Eulerphithumb.png
Euler totient
Mangoldtthumb.png
Mangoldt
Mertensthumb.png
Mertens
Möbiusthumb.png
Möbius
Liouvillelambdathumb.png
Liouville $\lambda$
Primepithumb.png
Prime counting
Riemannzetathumb.png
Riemann zeta
Divisorsumthumb.png
Sum of divisors

Chebyshev function
Dirichlet $L$-function
Greatest prime factor
Ramanujan's sum
Riemann prime counting

Zeta functions

A directory of zeta functions.

Hurwitzzetathumb.png
Hurwitz zeta
Jacobizetathumb.png
Jacobi zeta
Primezetapthumb.png
Prime zeta $P$
Riemannzetathumb.png
Riemann zeta
Weierstrasszetathumb.png
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

Qbinomialthumb.png
$q$-Binomial
Qgammathumb.png
$q$-Gamma
Qfactorialthumb.png
$q$-factorial
Eulerqphi.png
Euler $\phi$

Continuous $q$-Hermite polynomials
Hahn-Exton $q$-Bessel
Gosper $q$-sine
Jackson $q$-Bessel (1)
Jackson $q$-Bessel (2)
Jackson $q$-Bessel (3)
$q$-analog
$q$-Bernoulli numbers
$q$-Bessel function
$q$-Beta function
$q$-$\cos$
$q$-$\mathrm{Cos}$
$q$-derivative
$q$-exponential e
$q$-exponential E
$q$-Fibonacci polynomials
$q$-Hermite polynomial
$q$-Hypergeometric function
$q$-numbers
$q$-Pochhammer symbol
$q$-Polygamma function
$q$-shifted factorial
$q$-$\sin$
$q$-$\mathrm{Sin}$
$q$-Theta function

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

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