19 Elliptic Integrals Symmetric Integrals 19.18 Derivatives and Differential Equations 19.20 Special Cases

§19.19 Taylor and Related Series

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Keywords:: power-series expansions, symmetric elliptic integrals
Notes:: To prove (19.19.2) expand the product in (19.23.10) in powers of . (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).
Permalink:: http://dlmf.nist.gov/19.19
See also:: Annotations for Ch.19

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

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : nonnegative integer, : 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 $m_{1},\dots,m_{n}$ whose sum is . 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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\sum_{N=0}^{\infty}\frac{{\left(a% \right)_{N}}}{{\left(c\right)_{N}}}T_{N}(\mathbf{b},\boldsymbol{{1}}-\mathbf{z% }),$ $c=\sum_{j=1}^{n}b_{j}$ , $|1-z_{j}|<1$ ,

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : nonnegative integer, : 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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=z_{n}^{-a}\sum_{N=0}^{\infty}\frac{{% \left(a\right)_{N}}}{{\left(c\right)_{N}}}\*{T_{N}(b_{1},\dots,b_{n-1};1-(z_{1% }/z_{n}),\dots,1-(z_{n-1}/z_{n}))},$ $c=\sum_{j=1}^{n}b_{j}$ , $|1-(z_{j}/z_{n})|<1$ .

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : nonnegative integer, : 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 , 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 $\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),$

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Defines:: $E_{s}(\mathbf{z})$ : symmetric function (locally)
Symbols:: : 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 -tuple $\mathbf{\tfrac{1}{2}}=(\tfrac{1}{2},\dots,\tfrac{1}{2})$ . Then

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : nonnegative integer, : nonnegative integer, $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial, $E_{s}(\mathbf{z})$ : symmetric function and : 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 $m_{1},\dots,m_{n}$ 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 $R_{J}\left(x,y,z,p\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},% \tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z,p,p\right)$

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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 $\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)$ with chosen to make $E_{1}(\mathbf{Z})=0$ . Then $T_{N}$ has at most one term if $N\leq 5$ in the series for $R_{F}$ . For $R_{J}$ and $R_{D}$ , $T_{N}$ has at most one term if $N\leq 3$ , and two terms if or 5.

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

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