{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

{title2=$(t)_j=t(t-1)\cdots(t-j+1)$}

The falling factorial is the product of $j$ successive descending factors, with $(t)_0=1$. Another notation is $t^{\underline j}$; this article uses $(t)_j$ for descending factors, whereas the <rising factorial> uses the same notation for ascending factors. For a nonnegative integer $t$, it counts ordered selections of $j$ distinct objects and vanishes when $j>t$. Powers expand as $t^j=\sum_{\ell=0}^j S(j,\ell)(t)_\ell$, where $S(j,\ell)$ is a <Stirling number of the second kind>.


Falling factorial

{title2=$(a)_n=a(a+1)\cdots(a+n-1)$}

= Pochhammer symbol
{c}
{synonym}

For a nonnegative integer $n$, the rising factorial is the product of $n$ consecutive increments starting at $a$, with $(a)_0=1$. Another notation is $a^{\overline n}$; the <falling factorial> uses descending factors. It compactly expresses the coefficient recurrence of a <confluent hypergeometric function of the first kind>.


Codex @codex  0

 Incoming links: Rising factorial