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Special functions

Airy functions
Bessel functions
Beta function
Binomial function
Cosine function
Dawson function
Dedekind eta function
Digamma function
Dirichlet eta function
Dirichlet $L$-function
Error function
Exponential function
Exponential integral
Fresnel integral
Gamma function
Generalized hypergeometric function ${}_pF_q$
Hermite function
Hurwitz zeta function
Hypergeometric $q$-series
Ihara zeta function
Jacobi elliptic functions
Jacobi theta functions
Lambert $W$ function
Legendre function
Logarithm
Logarithmic integral
Minkowski $?$ function
Polylogarithm
Prime counting function
$q$-Bessel function
$q$-Beta function
$q$-Binomial function
$q$-Factorial
$q$-Gamma function
$q$-Pochhammer symbol
Partition function
Pochhammer symbol
Polylogarithm
Prime counting function
Riemann zeta function
Sum of positive divisors function
Sinc function
Sine function
Tangent function
Theta functions
Totient function
Trigonometric integral
Weierstrass elliptic functions

Special constants

Apéry's constant
Chaitan's constant
Conway's constant
$e$
Euler-Mascheroni constant
Feigenbaum constants
Golden ratio
Khinchin's constant
Lévy's constant
Mills' constant
$\pi$

Tools from calculus

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/