§19.19 Taylor and Related Series

For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial

19.19.1		$$T_N(\mathbf{b},\mathbf{z})=\sum \frac{{\left(b_1\right)}_{m_1}\mathrm{\cdots }{\left(b_n\right)}_{m_n}}{m_1!\mathrm{\cdots }{m}_n!}{z}_1^{m_1}\mathrm{\cdots }{z}_n^{m_n},$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer and $T_N(\mathbf{b},\mathbf{z})$: polynomial Referenced by: §19.20(v) Permalink: http://dlmf.nist.gov/19.19.E1 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

where the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ whose sum is $N$. The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23):

19.19.2		$$R_{-a}(\mathbf{b};\mathbf{z})=\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(\mathbf{b},\mathbf{1}-\mathbf{z}),$$
$c={\sum }_{j=1}^n{b}_j$, ,
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(\mathbf{b},\mathbf{z})$: polynomial Referenced by: §19.19 Permalink: http://dlmf.nist.gov/19.19.E2 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

19.19.3		$$R_{-a}(\mathbf{b};\mathbf{z})=z_n^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(b_1,\mathrm{\dots },b_{n-1};1-(z_1/z_n),\mathrm{\dots },1-(z_{n-1}/z_n)),$$
$c={\sum }_{j=1}^n{b}_j$, .
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(\mathbf{b},\mathbf{z})$: polynomial Referenced by: §19.19, §19.19, §19.28 Permalink: http://dlmf.nist.gov/19.19.E3 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

If $n=2$, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)).

Define the elementary symmetric function $E_s(\mathbf{z})$ by

19.19.4		$$\prod _{j=1}^n(1+t{z}_j)=\sum _{s=0}^n{t}^s{E}_s(\mathbf{z}),$$
ⓘ Defines: $E_s(\mathbf{z})$: symmetric function (locally) Symbols: $n$: nonnegative integer Referenced by: §19.36(i) Permalink: http://dlmf.nist.gov/19.19.E4 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

and define the $n$-tuple $\frac{\mathbf{1}}{\mathbf{2}}=(\frac{1}{2},\mathrm{\dots },\frac{1}{2})$. Then

19.19.5		$$T_N(\frac{\mathbf{1}}{\mathbf{2}},\mathbf{z})=\sum {(-1)}^{M+N}{\left(\frac{1}{2}\right)}_M\frac{E_1^{m_1}(\mathbf{z})\mathrm{\cdots }{E}_n^{m_n}(\mathbf{z})}{m_1!\mathrm{\cdots }{m}_n!},$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer, $T_N(\mathbf{b},\mathbf{z})$: polynomial, $E_s(\mathbf{z})$: symmetric function and $M$: sum Referenced by: §19.19, §19.36(i) Permalink: http://dlmf.nist.gov/19.19.E5 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

where $M={\sum }_{j=1}^n{m}_j$ and the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ such that ${\sum }_{j=1}^{n}j{m}_j=N$.

This form of $T_N$ can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use

19.19.6		$$R_J(x,y,z,p)=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};x,y,z,p,p)$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.19 Permalink: http://dlmf.nist.gov/19.19.E6 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

as well as (19.16.5) and (19.16.6). The number of terms in $T_N$ can be greatly reduced by using variables $\mathbf{Z}=\mathbf{1}-(\mathbf{z}/A)$ with $A$ chosen to make $E_1(\mathbf{Z})=0$. Then $T_N$ has at most one term if $N\le 5$ in the series for $R_F$. For $R_J$ and $R_D$, $T_N$ has at most one term if $N\le 3$, and two terms if $N=4$ or 5.

19.19.7		$$R_{-a}(\frac{\mathbf{1}}{\mathbf{2}};\mathbf{z})=A^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(\frac{1}{2}n\right)}_N}{T}_N(\frac{\mathbf{1}}{\mathbf{2}},\mathbf{Z}),$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer, $T_N(\mathbf{b},\mathbf{z})$: polynomial, $\mathbf{Z}$: variable and $A$ Referenced by: §19.15, §19.36(i), §19.36(i), §19.36(i) Permalink: http://dlmf.nist.gov/19.19.E7 Encodings: TeX, pMML, png See also: Annotations for §19.19 and Ch.19

where

19.19.8		$A$	$={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n}{z}_j,$
		$Z_j$	$=1-(z_j/A),$
		$E_1(\mathbf{Z})$	$=0$,
	.
ⓘ Symbols: $n$: nonnegative integer, $E_s(\mathbf{z})$: symmetric function, $\mathbf{Z}$: variable, $Z_j$: components and $A$ Permalink: http://dlmf.nist.gov/19.19.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.19 and Ch.19

Special cases are given in (19.36.1) and (19.36.2).