Hypergeometric2F1 with complex arguments in Python by evaluating the power series

hypergeometric_2f1.py

from numba import njit
import numpy as np

@njit('c8(c8,u8)')
def rising_poch(x, n):
    p = 1+0j
    for k in range(n):
        p *= (x+k)
    return p

@njit('u8(u8)')
def factorial(n):
    p = 1
    n = n + 1
    for k in range(1, n):
        p *= k
    return p

@njit('c8(c8,c8,c8,c8,u8)')
def hypergeometric_2F1(a, b, c, z, n):
    if np.abs(z) > 1:
        return np.nan
    s = 0+0j
    for k in range(n):
        s += (rising_poch(a, k)*rising_poch(b, k)/rising_poch(c, k)) * np.power(z,k) / factorial(k)
    return s

hypergeometric_2F1(0 + 2j, 3., 1., 0.25, 20)
# => (-0.13342943787574768+1.6351008415222168j)
# Mathematica result: -0.133429 + 1.6351 I