# §19.19 Taylor and Related Series

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

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

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

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

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

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

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

where

 19.19.8 $\displaystyle A$ $\displaystyle=\frac{1}{n}\sum_{j=1}^{n}z_{j},$ $\displaystyle Z_{j}$ $\displaystyle=1-(z_{j}/A),$ $\displaystyle E_{1}(\mathbf{Z})$ $\displaystyle=0$, $|Z_{j}|<1$. 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

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