{wiki}

Confluent hypergeometric functions solve the <Kummer differential equation> $zy''+(b-z)y'-ay=0$. 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>.


Confluent hypergeometric function

{c}

Kummer's equation is
$$
xy''+(b-x)y'-ay=0.
$$
Its solution analytic at zero is the <confluent hypergeometric function of the first kind>
$$
M(a,b,x)=\sum_{m=0}^{\infty}
\frac{(a)_m}{(b)_m\,m!}x^m.
$$
When $b$ is not an integer, a second local solution is
$$
x^{1-b}M(a-b+1,2-b,x).
$$


Kummer differential equation

{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>.


Rising factorial

{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].


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 Incoming links: Confluent hypergeometric function of the first kind