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 \({}_2F_0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)\)

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Gegenbauer Functions

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Laguerre Functions