= Kummer differential equation
{c}
The <Kummer differential equation> is
$$
xy''+(b-x)y'-ay=0.
$$
When $b\notin\{0,-1,-2,\ldots\}$, its solution analytic at zero and normalized to one is the <confluent hypergeometric function of the first kind>
$$
M(a,b,x)=\sum_{m=0}^{\infty}\frac{(a)_m}{(b)_m\,m!}x^m.
$$
Here $(a)_m$ is the <rising factorial>. Substitution gives $(m+1)(m+b)c_{m+1}=(m+a)c_m$, explaining both the series and the excluded denominator parameters. For $b\notin\mathbb Z$, a second local solution on a chosen logarithm branch is $x^{1-b}M(a-b+1,2-b,x)$. Integer parameter differences require a separate limiting construction and can introduce logarithms. At excluded $b$, the normalized analytic series is not generally defined; special terminating parameter combinations must be treated explicitly rather than by dividing by a zero denominator.https://dlmf.nist.gov/13.2[§13.2(i), equations13.2.1–13.2.2 and integer-parameter discussion]{ref}
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