hypergeometric — SageMath

hypergeometric( [ a₁ , ... ] , [ b₁ , ... ] , z )

The generalized hypergeometric function of z in SageMath. Defined by

\[ \,_p F _q \left( \begin{matrix} a_1 , \ldots , a_p \\ b_1 , \dots , b_q \end{matrix} ; z \right) = \sum_{k=0}^\infty \frac{ (a_1)_k \cdots (a_p)_k }{ (b_1)_k \cdots (b_p)_k } \frac{ z^k }{ k! } \]

with the rising factorial

\[ (a)_k = a(a+1) \cdots (a+k-1) = \frac{ \Gamma(a+k) }{ \Gamma(a) } \]

The two sets of parameters must be given as lists.

Plots on the real axis:

real( hypergeometric([a,b],[c],x) )

imag( hypergeometric([a,b],[c],x) )

Series expansions about the origin:

Special values:

Related functions: hypergeometric_U

Function category: hypergeometric functions sagemath-docs