§19.19 Taylor and Related Series

Notes:: To prove (19.19.2) expand the product in (19.23.10) in powers of $u$ . (19.19.3) is derived from (19.16.11) and (19.19.2). For (19.19.5) see Carlson (1979, (A.12)). For (19.19.6) compare (19.16.2) and (19.16.9).
Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.19

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

19.19.1 $T_{N}(\mathbf{b},\mathbf{z})=\sum\frac{\left(b_{1}\right)_{{m_{1}}}\cdots\left(b_{n}\right)_{{m_{n}}}}{m_{1}!\cdots m_{n}!}z_{1}^{{m_{1}}}\cdots z_{n}^{{m_{n}}},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $m$ : 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

where the summation extends over all nonnegative integers $m_{1},\dots,m_{n}$ 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 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\sum _{{N=0}}^{{\infty}}\frac{\left(a\right)_{{N}}}{\left(c\right)_{{N}}}T_{N}(\mathbf{b},\boldsymbol{{1}}-\mathbf{z}),$ $c=\sum _{{j=1}}^{n}b_{j}$ , $|1-z_{j}|<1$ ,

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $n$ : 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

19.19.3 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=z_{n}^{{-a}}\sum _{{N=0}}^{{\infty}}\frac{\left(a\right)_{{N}}}{\left(c\right)_{{N}}}\*{T_{N}(b_{1},\dots,b_{{n-1}};1-(z_{1}/z_{n}),\dots,1-(z_{{n-1}}/z_{n}))},$ $c=\sum _{{j=1}}^{n}b_{j}$ , $|1-(z_{j}/z_{n})|<1$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $n$ : 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

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 $\prod _{{j=1}}^{n}(1+tz_{j})=\sum _{{s=0}}^{n}t^{s}E_{s}(\mathbf{z}),$

Defines:: $E_{s}(\mathbf{z})$ : elementary symmetric function
Symbols:: $n$ : nonnegative integer
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E4
Encodings:: TeX, pMML, png

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

19.19.5 $T_{N}(\mathbf{\tfrac{1}{2}},\mathbf{z})=\sum(-1)^{{M+N}}\left(\tfrac{1}{2}\right)_{{M}}\frac{E_{1}^{{m_{1}}}(\mathbf{z})\cdots E_{n}^{{m_{n}}}(\mathbf{z})}{m_{1}!\cdots m_{n}!},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $E_{s}(\mathbf{z})$ : elementary symmetric function, $!$ : $n!$ : factorial, $m$ : nonnegative integer, $n$ : nonnegative integer, $N$ : nonnegative integer, $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial and $M$ : sum
Referenced by:: §19.19, §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E5
Encodings:: TeX, pMML, png

where $M=\sum _{{j=1}}^{n}m_{j}$ and the summation extends over all nonnegative integers $m_{1},\dots,m_{n}$ such that $\sum _{{j=1}}^{n}jm_{j}=N$ .

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 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=\mathop{R_{{-\frac{3}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z,p,p\right)$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.19.E6
Encodings:: TeX, pMML, png

as well as (19.16.5) and (19.16.6). The number of terms in $T_{N}$ can be greatly reduced by using variables $\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)$ with $A$ chosen to make $E_{1}(\mathbf{Z})=0$ . Then $T_{N}$ has at most one term if $N\leq 5$ in the series for $\mathop{R_{F}\/}\nolimits$ . For $\mathop{R_{J}\/}\nolimits$ and $\mathop{R_{D}\/}\nolimits$ , $T_{N}$ has at most one term if $N\leq 3$ , and two terms if $N=4$ or 5.

19.19.7 $\mathop{R_{{-a}}\/}\nolimits\!\left(\boldsymbol{{\tfrac{1}{2}}};\mathbf{z}\right)=A^{{-a}}\sum _{{N=0}}^{{\infty}}\frac{\left(a\right)_{{N}}}{\left(\tfrac{1}{2}n\right)_{{N}}}T_{N}(\boldsymbol{{\tfrac{1}{2}}},\mathbf{Z}),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $n$ : 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

where

19.19.8

$A=\frac{1}{n}\sum _{{j=1}}^{n}z_{j},$

$Z_{j}=1-(z_{j}/A),$

$E_{1}(\mathbf{Z})=0$ , $|Z_{j}|<1$ .

Symbols:: $E_{s}(\mathbf{z})$ : elementary symmetric function, $n$ : nonnegative integer, $\mathbf{Z}$ : variable, $Z_{j}$ : components and $A$
Permalink:: http://dlmf.nist.gov/19.19.E8
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

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