§16.2 Definition and Analytic Properties

Referenced by:: §18.20(ii), §18.23, §18.26(i), §18.5(iii), §34.2, §34.4, §5.16
Permalink:: http://dlmf.nist.gov/16.2

§16.2(i) Generalized Hypergeometric Series
§16.2(ii) Case $p\leq q$
§16.2(iii) Case $p=q+1$
§16.2(iv) Case $p>q+1$
§16.2(v) Behavior with Respect to Parameters

§16.2(i) Generalized Hypergeometric Series

Keywords:: generalized hypergeometric function $\mathop{{{}_{{0}}F_{{2}}}\/}\nolimits$ , generalized hypergeometric functions, generalized hypergeometric series
Referenced by:: §13.1, §13.31(iii), §17.4(i), ¶ ‣ §2.10(iii), ¶ ‣ §7.11
Permalink:: http://dlmf.nist.gov/16.2.i

Throughout this chapter it is assumed that none of the bottom parameters $b_{1}$ , $b_{2}$ , $\dots$ , $b_{q}$ is a nonpositive integer, unless stated otherwise. Then formally

16.2.1 $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum _{{k=0}}^{\infty}\frac{\left(a_{1}\right)_{{k}}\cdots\left(a_{p}\right)_{{k}}}{\left(b_{1}\right)_{{k}}\cdots\left(b_{q}\right)_{{k}}}\frac{z^{k}}{k!}.$

Defines:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: ¶ ‣ §10.16, §10.16, ¶ ‣ §10.39, ¶ ‣ §16.2(iv), §16.2(ii), §16.2(iii), §16.2(iii), §16.2(iii), §16.5, ¶ ‣ §18.17(v), ¶ ‣ §18.17(vi), ¶ ‣ §18.17(vii), ¶ ‣ §18.30
Permalink:: http://dlmf.nist.gov/16.2.E1
Encodings:: TeX, pMML, png

Equivalently, the function is denoted by $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ or $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ , and sometimes, for brevity, by $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(z\right)$ .

§16.2(ii) Case $p\leq q$

Keywords:: generalized hypergeometric functions
Referenced by:: §18.34(i), ¶ ‣ §2.10(iii), ¶ ‣ §7.11
Permalink:: http://dlmf.nist.gov/16.2.ii

When $p\leq q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function.

§16.2(iii) Case $p=q+1$

Notes:: See Luke (1975, §5.2.1) and Slater (1966, §2.2).
Keywords:: generalized hypergeometric functions
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.2.iii

Suppose first one or more of the top parameters $a_{j}$ is a nonpositive integer. Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$ .

If none of the $a_{j}$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is 1, and outside the open disk $|z|<1$ the generalized hypergeometric function is defined by analytic continuation with respect to $z$ . The branch obtained by introducing a cut from 1 to $+\infty$ on the real 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 $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ ; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$ , and $\infty$ . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.

On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\realpart{\gamma _{q}}>0$ , convergent except at $z=1$ if $-1<\realpart{\gamma _{q}}\leq 0$ , and divergent if $\realpart{\gamma _{q}}\leq-1$ , where

16.2.2 $\gamma _{q}=(b_{1}+\dots+b_{q})-(a_{1}+\dots+a_{{q+1}}).$

Symbols:: $q$ : nonnegative integer, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.2.E2
Encodings:: TeX, pMML, png

§16.2(iv) Case $p>q+1$

Notes:: See Luke (1975, §5.2.1).
Keywords:: generalized hypergeometric functions
Referenced by:: §13.6(vi), §18.34(i)
Permalink:: http://dlmf.nist.gov/16.2.iv

¶ Polynomials

Keywords:: generalized hypergeometric functions

In general the series (16.2.1) diverges for all nonzero values of $z$ . However, when one or more of the top parameters $a_{j}$ is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in $z$ . Note that if $-m$ is the value of the numerically largest $a_{j}$ that is a nonpositive integer, then the identity

16.2.3 $\mathop{{{}_{{p+1}}F_{{q}}}\/}\nolimits\!\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{\left(\mathbf{a}\right)_{{m}}(-z)^{m}}{\left(\mathbf{b}\right)_{{m}}}\mathop{{{}_{{q+1}}F_{{p}}}\/}\nolimits\!\left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{{p+q}}}{z}\right)$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/16.2.E3
Encodings:: TeX, pMML, png

can be used to interchange $p$ and $q$ .

Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via

16.2.4 $\sum _{{k=0}}^{m}\frac{\left(\mathbf{a}\right)_{{k}}}{\left(\mathbf{b}\right)_{{k}}}\frac{z^{k}}{k!}=\frac{\left(\mathbf{a}\right)_{{m}}z^{m}}{\left(\mathbf{b}\right)_{{m}}m!}\mathop{{{}_{{q+2}}F_{{p}}}\/}\nolimits\!\left({-m,1,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{{p+q+1}}}{z}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/16.2.E4
Encodings:: TeX, pMML, png

¶ Non-Polynomials

See §16.5 for the definition of $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ as a contour integral when $p>q+1$ and none of the $a_{k}$ is a nonpositive integer. (However, except where indicated otherwise in the DLMF we assume that when $p>q+1$ at least one of the $a_{k}$ is a nonpositive integer.)

§16.2(v) Behavior with Respect to Parameters

Notes:: The statement that follows (16.2.5) follows from the uniform convergence of (16.2.5) when $p\leq q$ , and also when $p=q+1$ provided that $|z|<1$ . For other values of $z$ , apply the straightforward generalization (to higher-order differential equations) of Theorem 3.2 in Olver (1997b, Chapter 5).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.2.v

Let

16.2.5 $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\mathop{\Gamma\/}\nolimits\!\left(b_{1}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(b_{q}\right)\right)}=\sum _{{k=0}}^{\infty}\frac{\left(a_{1}\right)_{{k}}\cdots\left(a_{p}\right)_{{k}}}{\mathop{\Gamma\/}\nolimits\!\left(b_{1}+k\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(b_{q}+k\right)}\frac{z^{k}}{k!};$

Defines:: $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.2(v)
Permalink:: http://dlmf.nist.gov/16.2.E5
Encodings:: TeX, pMML, png

compare (15.2.2) in the case $p=2$ , $q=1$ . When $p\leq q+1$ and $z$ is fixed and not a branch point, any branch of $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ is an entire function of each of the parameters $a_{1},\dots,a_{p},b_{1},\dots,b_{q}$ .

§16.2 Definition and Analytic Properties

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