Special functions summary

This page is meant to track support for special functions in Arb and outstanding issues.

Long-term, it would be nice to support most of the functions known to mpmath, Pari/GP, Mathematica, Maple, DLMF, Abramowitz & Stegun, etc.

Table of Contents Legend Elementary functions Combinatorial and integer sequence functions Hypergeometric functions Elliptic functions and modular forms Zeta, Dirichlet L-functions, and polylogarithms

Legend

A available
(A) implicitly available (trivial conversion from another function)
P partial implementation, missing major optimizations or not supporting the whole domain
X not available
- not applicable
RR - real variable(s)
CC - complex variable(s)
RR - real power series / derivatives
CC - complex power series / derivatives

Function	RR	CC	RRx	CCx	Notes
exp	A	A	A	A

Function	RR	CC	RRx	CCx	Notes
Fibonacci numbers	P(A)	(A)	(A)	(A)	No real/complex extension (trivial via elementary functions)
Factorials	P(A)	(A)	(A)	(A)	No real/complex extension (trivial via gamma function)
Double factorials	P(A)	(A)	(A)	(A)	No real/complex extension (trivial via gamma function)
Multiple factorials	X	X	X	X
Rising factorials	A	A	A	A	Need to use gamma function in some cases for real/complex extension
Falling factorials	(A)	(A)	(A)	(A)	Equivalent to rising factorials
Binomial coefficients	(A)	(A)	(A)	(A)	No real/complex extension (trivial via gamma function)
Multinomial coefficients	X	-	-	-
Gamma function	A	A	A	A
Log-gamma function	A	A	A	A
Digamma function	A	A	A	A
Harmonic numbers	(A)	(A)	(A)	A	Trivial via gamma function
Bernoulli numbers	A	-	-	-
Euler numbers	A	-	-	-
Bernoulli polynomials	A	A	(A)	(A)	Exact computation possible with FLINT
Euler polynomials	(A)	(A)	(A)	(A)	Exact computation possible with FLINT
Eulerian numbers	X	-	-	-
Bell numbers	A	-	-	-
Partition function	A	-	-	-
Stirling numbers	(A)	-	-	-	Exact computation possible with FLINT