Kummer differential equation

Kummer's equation isIts solution analytic at zero is the confluent hypergeometric function of the first kindWhen $b$ is not an integer, a second local solution is

Confluent hypergeometric functions solve the Kummer differential equation $z{y}^{\prime \prime }+(b-z)y^{\prime }-ay=0$. The regular solution is the confluent hypergeometric function of the first kind; a second standard solution is conventionally denoted $U(a,b,z)$. The family arises by coalescing singular points of the Gauss hypergeometric equation.

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the power seriesHere $(a{)}_n$ is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.