Confluent hypergeometric function

Remarks

The ratio of consecutive terms is $\frac{(a + n) z}{(b + n)(n + 1)}$, which tends to $0$, so the series converges for every $z$ and $M(a, b, \cdot)$ is entire. Inserting the series into Kummer's equation shows that $M(a, b, z)$ is its solution that is analytic at $0$ with $M(a, b, 0) = 1$. Example: $M(a, a, z) = e^z$, since $(a)_n / (a)_n = 1$. Example: $M(1, 2, z) = \frac{e^z - 1}{z}$, since $\frac{(1)_n}{(2)_n\, n!} = \frac{1}{(n + 1)!}$. Non-example: for $b = -1$ the series is not defined, since $(b)_2 = 0$ appears in a denominator. If $a = -m$ with $m$ a nonnegative integer, the series stops and $M(-m, b, z)$ is a polynomial of degree $m$; for $b = \alpha + 1$ it is a multiple of the generalized Laguerre polynomial $L_m^{(\alpha)}(z)$. Another notation is $\Phi(a; b; z)$.