HypergeometricFunctions.jl

This package provides an implementation of the generalized hypergeometric function pFq(α, β, z).

julia> using HypergeometricFunctions

julia> pFq((), (), 0.1) # ≡ exp(0.1)
1.1051709180756477

julia> pFq((0.5, ), (), 1.0+0.001im) # ≡ exp(-0.5*log1p(-1.0-0.001im))
22.360679774997894 + 22.36067977499789im

julia> pFq((), (1.5, ), -π^2/4) # A root of a spherical Bessel function
4.042865030283967e-17

julia> pFq((1/3, ), (2/3, ), -1000) # A confluent hypergeometric with large argument
0.050558053946448855

julia> pFq((1, 2+im), (3.5, ), exp(im*π/3)) # ₂F₁ at that special point in ℂ
0.6786952632946592 + 0.4523504929285015im

References

[1] N. Michel and M. V. Stoitsov, Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions, Comp. Phys. Commun., 178:535–551, 2008.

[2] J. W. Pearson, S. Olver and M. A. Porter, Numerical methods for the computation of the confluent and Gauss hypergeometric functions, Numer. Algor., 74:821–866, 2017.

[3] R. M. Slevinsky, Fast and stable rational approximation of generalized hypergeometric functions, Numer. Algor., 98:587–624, 2025.