# §15.2 Definitions and Analytical Properties

## §15.2(i) Gauss Series

The hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss series

 15.2.1 $F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b% \right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}% {c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{\Gamma\left(a\right)\Gamma% \left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s% \right)}{\Gamma\left(c+s\right)s!}z^{s},$ ⓘ Defines: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter A&S Ref: 15.1.1 Referenced by: §15.19(i), §15.4(iii), §19.19 Permalink: http://dlmf.nist.gov/15.2.E1 Encodings: TeX, pMML, png See also: Annotations for 15.2(i), 15.2 and 15

on the disk $|z|<1$, and by analytic continuation elsewhere. In general, $F\left(a,b;c;z\right)$ does not exist when $c=0,-1,-2,\dots$. The branch obtained by introducing a cut from $1$ to $+\infty$ on the real $z$-axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $F\left(a,b;c;z\right)$.

For all values of $c$

 15.2.2 $\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s},$ $|z|<1$, ⓘ Defines: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: Olver’s hypergeometric function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter Referenced by: §15.12(i), §16.2(v) Permalink: http://dlmf.nist.gov/15.2.E2 Encodings: TeX, pMML, png See also: Annotations for 15.2(i), 15.2 and 15

again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way.

Except where indicated otherwise principal branches of $F\left(a,b;c;z\right)$ and $\mathbf{F}\left(a,b;c;z\right)$ are assumed throughout the DLMF.

The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by

 15.2.3 $\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-\mathbf{F}\left({a,b\atop c}% ;x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma\left(a\right)\Gamma\left(b% \right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-x\right),$ $x>1$.

On the circle of convergence, $|z|=1$, the Gauss series:

• (a)

Converges absolutely when $\Re(c-a-b)>0$.

• (b)

Converges conditionally when $-1<\Re(c-a-b)\leq 0$ and $z=1$ is excluded.

• (c)

Diverges when $\Re(c-a-b)\leq-1$.

For the case $z=1$ see also §15.4(ii).

## §15.2(ii) Analytic Properties

The principal branch of $\mathbf{F}\left(a,b;c;z\right)$ is an entire function of $a$, $b$, and $c$. The same is true of other branches, provided that $z=0$, $1$, and $\infty$ are excluded. As a multivalued function of $z$, $\mathbf{F}\left(a,b;c;z\right)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\infty$. The same properties hold for $F\left(a,b;c;z\right)$, except that as a function of $c$, $F\left(a,b;c;z\right)$ in general has poles at $c=0,-1,-2,\dots$.

Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. In particular

 $\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),$ $n=0,1,2,\dots$.

For example, when $a=-m$, $m=0,1,2,\dots$, and $c\neq 0,-1,-2,\dots$, $F\left(a,b;c;z\right)$ is a polynomial:

 15.2.4 $F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right% )_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}^{m}(-1)^{n}\genfrac{(}{)}{0.% 0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}z^{n}.$

This formula is also valid when $c=-m-\ell$, $\ell=0,1,2,\dots$, provided that we use the interpretation

 15.2.5 $F\left({-m,b\atop-m-\ell};z\right)=\lim_{c\to-m-\ell}\left(\lim_{a\to-m}F\left% ({a,b\atop c};z\right)\right),$

and not

 15.2.6 $F\left({-m,b\atop-m-\ell};z\right)=\lim_{a\to-m}F\left({a,b\atop a-\ell};z% \right),$

which sometimes needs to be used in §15.4. (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $\mathbf{F}\left(a,b;c;z\right)$, which is analytic at $c=0,-1,-2,\dots$.)

For comparison of $F\left(a,b;c;z\right)$ and $\mathbf{F}\left(a,b;c;z\right)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7.

Let $m$ be a nonnegative integer. Formula (15.4.6) reads $F\left(a,b;a;z\right)=(1-z)^{-b}$. The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F\left(-m,b;-m;z\right)$ is equal to the first $m+1$ terms of the Maclaurin series for $(1-z)^{-b}$.