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Special functions

Airy functions
Automorphic forms and functions
Bessel functions
Beta function
Binomial function
Carlitz-Goss Gamma function
Clausen functions
Cosine function
Dawson function
Dedekind eta function
Dedekind zeta function
Digamma function
Dirichlet eta function
Dirichlet $L$-function
Eisenstein series
Epstein zeta function
Error function
Exponential function
Exponential integral
Fresnel integral
Gamma function
Generalized hypergeometric function ${}_pF_q$
Hermite function
Hermite polynomial
Hurwitz zeta function
Hypergeometric $q$-series
Ihara zeta 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
Multiple zeta functions
Painlevé transcendents
Polylogarithm
Prime counting function
$q$-Bessel function
$q$-Beta function
$q$-Binomial function
$q$-Factorial
$q$-Gamma function
$q$-Hermite polynomial
$q$-Pochhammer symbol
Partition function
Pochhammer symbol
Polygamma functions
Polylogarithms
Prime counting function
Riemann theta function
Riemann zeta function
Sum of positive divisors function
Secant zeta function
Sinc function
Sine function
Struve functions
Tangent function
Theta functions
Totient function
Trigonometric integral
Weierstrass elliptic functions

Special constants

Apéry's constant
Chaitin's constant
Conway's constant
$e$
Euler-Mascheroni constant
Feigenbaum constants
Fransén–Robinson constant
Golden ratio
Khinchin's constant
Lévy's constant
Mills' constant
$\pi$

Tools from calculus

Integration by parts
Ratio test

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/