§15.2 Definitions and Analytical Properties

§15.2(i) Gauss Series

The hypergeometric function $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ is defined by the Gauss series

 15.2.1 $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{\left(a% \right)_{s}\left(b\right)_{s}}{\left(c\right)_{s}s!}z^{s}=1+\frac{ab}{c}z+% \frac{a(a+1)b(b+1)}{c(c+1)2!}z^{2}+\cdots=\frac{\mathop{\Gamma\/}\nolimits\!% \left(c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}% \nolimits\!\left(b\right)}\sum_{s=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits% \!\left(a+s\right)\mathop{\Gamma\/}\nolimits\!\left(b+s\right)}{\mathop{\Gamma% \/}\nolimits\!\left(c+s\right)s!}z^{s},$ Defines: $\mathop{F\/}\nolimits\!\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathop{F\/}\nolimits\!\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: hypergeometric function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left(\NVar{a_{1},\dots,a_{p}% };\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left({\NVar{a_{1},\dots,a_{p% }}\atop\NVar{b_{1},\dots,b_{q}}};\NVar{z}\right)$: alternatively $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left(\NVar{\mathbf{a}};\NVar% {\mathbf{b}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left({\NVar{\mathbf{a}}\atop% \NVar{\mathbf{b}}};\NVar{z}\right)$ generalized 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: info for 15.2(i)

on the disk $|z|<1$, and by analytic continuation elsewhere. In general, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ 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 $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$.

For all values of $c$

 15.2.2 $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{% \left(a\right)_{s}\left(b\right)_{s}}{\mathop{\Gamma\/}\nolimits\!\left(c+s% \right)s!}z^{s},$ $|z|<1$, Defines: $\mathop{\mathbf{F}\/}\nolimits\!\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathop{\mathbf{F}\/}\nolimits\!\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z% }\right)$: Olver’s hypergeometric function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\left(\NVar{a}\right)_{\NVar{n}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathop{{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\/}\nolimits\!\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\/}\nolimits\!\left({\NVar{% \mathbf{a}}\atop\NVar{\mathbf{b}}};\NVar{z}\right)$: 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: info for 15.2(i)

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 $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ and $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ 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 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop c};x+i0\right)-\mathop{\mathbf% {F}\/}\nolimits\!\left({a,b\atop c};x-i0\right)=\frac{2\pi i}{\mathop{\Gamma\/% }\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}(x-1)^{c-% a-b}\mathop{\mathbf{F}\/}\nolimits\!\left({c-a,c-b\atop c-a-b+1};1-x\right),$ $x>1$.

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

• (a)

Converges absolutely when $\realpart{(c-a-b)}>0$.

• (b)

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

• (c)

Diverges when $\realpart{(c-a-b)}\leq-1$.

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

§15.2(ii) Analytic Properties

The principal branch of $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ 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 $z$, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\infty$. The same properties hold for $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$, except that as a function of $c$, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ 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.

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

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

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

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

and not

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

which is sometimes used in the literature. (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$, which is analytic at $c=0,-1,-2,\dots$.) For illustration see Figures 15.3.6 and 15.3.7.

In the case $c=-m$ the right-hand side of (15.2.4) becomes the first $m+1$ terms of the Maclaurin series for $(1-z)^{-b}$.