{title2=$M(a,b,z)={}_1F_1(a;b;z)$}

= Kummer function
{c}
{synonym}

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the <power series>
$$
M(a,b,z)=\sum_{n=0}^{\infty}\frac{(a)_n}{(b)_n}\frac{z^n}{n!}.
$$
Here $(a)_n$ is the <rising factorial>. Substitution into the <Kummer differential equation> proves the coefficient recurrence. This normalization agrees with https://dlmf.nist.gov/13.2[NIST's definitions].


 Confluent hypergeometric function of the first kind

ID: confluent-hypergeometric-function-of-the-first-kind