# Hypergeometric Functions¶

Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15.

gsl_sf_hyperg_0F1(c, x)

This routine computes the hypergeometric function $${}_0F_1(c,x)$$.

gsl_sf_hyperg_1F1_int(m, n, x)

This routine computes the confluent hypergeometric function $${}_1F_1(m,n,x) = M(m,n,x)$$ for integer parameters $$m$$, $$n$$.

gsl_sf_hyperg_1F1(a, b, x)

This routine computes the confluent hypergeometric function $${}_1F_1(a,b,x) = M(a,b,x)$$ for general parameters $$a$$, $$b$$.

gsl_sf_hyperg_U_int(m, n, x)

This routine computes the confluent hypergeometric function $$U(m,n,x)$$ for integer parameters $$m$$, $$n$$.

gsl_sf_hyperg_U(a, b, x)

This routine computes the confluent hypergeometric function $$U(a,b,x)$$.

gsl_sf_hyperg_2F1(a, b, c, x)

This routine computes the Gauss hypergeometric function $${}_2F_1(a,b,c,x) = F(a,b,c,x)$$ for $$|x| < 1$$.

If the arguments $$(a,b,c,x)$$ are too close to a singularity then the function can return an error when the series approximation converges too slowly. This occurs in the region of $$x=1, c - a - b = m$$ for integer $$m$$.

gsl_sf_hyperg_2F1_conj(aR, aI, c, x)

This routine computes the Gauss hypergeometric function $${}_2F_1(a_R + i a_I, a_R - i a_I, c, x)$$ with complex parameters for $$|x| < 1$$.

gsl_sf_hyperg_2F1_renorm(a, b, c, x)

This routine computes the renormalized Gauss hypergeometric function $${}_2F_1(a,b,c,x) / \Gamma(c)$$ for $$|x| < 1$$.

gsl_sf_hyperg_2F1_conj_renorm(aR, aI, c, x)

This routine computes the renormalized Gauss hypergeometric function $${}_2F_1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)$$ for $$|x| < 1$$.

gsl_sf_hyperg_2F0(a, b, x)

This routine computes the hypergeometric function $${}_2F_0(a,b,x)$$. The series representation is a divergent hypergeometric series. However, for $$x < 0$$ we have $${}_2F_0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)$$