§15.2 Definitions and Analytical Properties

Referenced by:: §18.12
Permalink:: http://dlmf.nist.gov/15.2

§15.2(i) Gauss Series
§15.2(ii) Analytic Properties

§15.2(i) Gauss Series

Notes:: See Andrews et al. (1999, §2.1). (15.2.3) is a consequence of (15.8.4).
Keywords:: Gauss series, convergence, hypergeometric function
Referenced by:: ¶ ‣ §10.16, §10.22(iv), ¶ ‣ §10.39, §10.43(iv), §13.10(ii), §13.16(i), §13.23(i), §13.4(i), §13.4(ii), §14.13, ¶ ‣ §18.12, ¶ ‣ §18.17(vii), §18.26(iv), ¶ ‣ §18.33(iv), §18.35(i), §19.5, ¶ ‣ §2.6(iii), §29.5, §32.10(vi), §33.6, §5.16, §8.14, §8.17(ii), §8.19(x), Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/15.2.i

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:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $s$ : 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

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:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $s$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.2(i)
Permalink:: http://dlmf.nist.gov/15.2.E3
Encodings:: TeX, pMML, png

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

Notes:: See Olver (1997b, Chapter 5, Theorem 9.1) and Temme (1996b, §5.1).
A&S Ref:: 15.4
Keywords:: hypergeometric function, singularities
Referenced by:: Figure 15.3.6, Figure 15.3.6, §15.4(i), ¶ ‣ §15.9(i)
Permalink:: http://dlmf.nist.gov/15.2.ii

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $n$ : integer, $z$ : complex variable, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : integer
A&S Ref:: 15.4.1
Referenced by:: §15.2(ii)
Permalink:: http://dlmf.nist.gov/15.2.E4
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\ell$ : integer and $m$ : integer
Referenced by:: ¶ ‣ §15.10(i)
Permalink:: http://dlmf.nist.gov/15.2.E5
Encodings:: TeX, pMML, png

and not

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $\ell$ : integer and $m$ : integer
Permalink:: http://dlmf.nist.gov/15.2.E6
Encodings:: TeX, pMML, png

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 $(1-z)^{{-b}}$ .

§15.2 Definitions and Analytical Properties

Contents

§15.2(i) Gauss Series

§15.2(ii) Analytic Properties