19 Elliptic Integrals Symmetric Integrals 19.19 Taylor and Related Series 19.21 Connection Formulas

§19.20 Special Cases

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Keywords:: special cases, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.20
See also:: Annotations for Ch.19

§19.20(i) $R_{F}\left(x,y,z\right)$
§19.20(ii) $R_{G}\left(x,y,z\right)$
§19.20(iii) $R_{J}\left(x,y,z,p\right)$
§19.20(iv) $R_{D}\left(x,y,z\right)$
§19.20(v) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

§19.20(i) $R_{F}\left(x,y,z\right)$

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Keywords:: general lemniscatic case, lemniscate constants, symmetric elliptic integrals
Notes:: In (19.20.2) put $t=1/\sqrt{s+1}$ ; alternatively use (19.29.19). For the second equality replace $t^{4}$ by and apply (5.12.1). For (19.20.3) use Carlson (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
Permalink:: http://dlmf.nist.gov/19.20.i
See also:: Annotations for §19.20 and Ch.19

In this subsection, and also §§19.20(ii)–19.20(v), the variables of all -functions satisfy the constraints specified in §19.16(i) unless other conditions are stated.

19.20.1

$R_{F}\left(x,x,x\right)=x^{-1/2},$

$R_{F}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-1/2}R_{F}\left(x,y,z% \right),$

$R_{F}\left(x,y,y\right)=R_{C}\left(x,y\right),$

$R_{F}\left(0,y,y\right)=\tfrac{1}{2}\pi y^{-1/2},$

$R_{F}\left(0,0,z\right)=\infty.$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/19.20.E1
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png
See also:: Annotations for §19.20(i), §19.20 and Ch.19

The first lemniscate constant is given by

19.20.2 $\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)=\frac% {\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;8777% 1\;46059\;90523\;\dots.$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Notes:: For more digits see OEIS Sequence A085565; see also Sloane (2003).
Referenced by:: §19.20(i), §19.20(iv), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.20.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(i), §19.20 and Ch.19

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. The general lemniscatic case is

19.20.3 $R_{F}\left(x,a,y\right)=R_{-\frac{1}{4}}\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\frac{1}{2}(x+y)$ .

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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 $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(i), §19.20 and Ch.19

§19.20(ii) $R_{G}\left(x,y,z\right)$

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Notes:: For (19.20.4) use (19.20.5) and (19.23.6_5). For (19.20.5) put in (19.21.10).
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.ii
See also:: Annotations for §19.20 and Ch.19

19.20.4

$R_{G}\left(x,x,x\right)=x^{1/2},$

$R_{G}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{1/2}R_{G}\left(x,y,z% \right),$

$R_{G}\left(0,y,y\right)=\tfrac{1}{4}\pi y^{1/2},$

$R_{G}\left(0,0,z\right)=\tfrac{1}{2}z^{1/2},$

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.20(ii)
Permalink:: http://dlmf.nist.gov/19.20.E4
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png
See also:: Annotations for §19.20(ii), §19.20 and Ch.19

19.20.5 $2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}.$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.20(ii)
Permalink:: http://dlmf.nist.gov/19.20.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(ii), §19.20 and Ch.19

§19.20(iii) $R_{J}\left(x,y,z,p\right)$

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Keywords:: circular cases, hyperbolic cases, symmetric elliptic integrals
Notes:: For (19.20.6) substitute in (19.16.2) and (19.16.5). In (19.20.7) see (19.27.12) for $p\rightarrow 0+$ ; for $p\rightarrow 0-$ use (19.20.17) and (19.6.15). In (19.20.8) the third equation is proved by partial fractions, and also implies the first two equations by (19.6.15). For (19.20.9) put in (19.20.13). For (19.20.10) interchange and in (19.27.14) and use (19.6.15). For (19.20.11) use (19.27.13), (19.20.17), and (19.27.2). For (19.20.12) see (19.27.11) and (19.21.12). For (19.20.13) let in (19.21.12). For (19.20.14) exchange and in (19.21.12) and use (19.2.20).
Referenced by:: §19.21(iii)
Permalink:: http://dlmf.nist.gov/19.20.iii
See also:: Annotations for §19.20 and Ch.19

19.20.6

$R_{J}\left(x,x,x,x\right)=x^{-3/2},$

$R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p\right)=\lambda^{-3/2}R_{J}% \left(x,y,z,p\right),$

$R_{J}\left(x,y,z,z\right)=R_{D}\left(x,y,z\right),$

$R_{J}\left(0,0,z,p\right)=\infty,$

$R_{J}\left(x,x,x,p\right)=R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}% \left(x,p\right)-\frac{1}{\sqrt{x}}\right),$ $x\neq p$ , $xp\neq 0$ .

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E6
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.7 $R_{J}\left(x,y,z,p\right)\to+\infty,$ $p\to 0+$ or

;

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Symbols:: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.8

$R_{J}\left(0,y,y,p\right)=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})},$

$R_{J}\left(0,y,y,-q\right)=\frac{-3\pi}{2\sqrt{y}(y+q)},$

$R_{J}\left(x,y,y,p\right)=\frac{3}{p-y}(R_{C}\left(x,y\right)-R_{C}\left(x,p% \right)),$ $p\neq y$ ,

$R_{J}\left(x,y,y,y\right)=R_{D}\left(x,y,y\right).$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), §19.26(i), §19.6(iv)
Permalink:: http://dlmf.nist.gov/19.20.E8
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.9 $R_{J}\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z% \right).$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii), §19.22(i)
Permalink:: http://dlmf.nist.gov/19.20.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.10

$\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)=\frac{3\pi}{2\sqrt{y}\sqrt{z}},$

$\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)={-R_{D}\left(0,y,z\right)-R_{D}\left(0% ,z,y\right)}=\frac{-6}{yz}R_{G}\left(0,y,z\right).$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: Figure 19.17.7, Figure 19.17.7, Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), §19.21(i), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.20.E10
Encodings:: TeX, TeX, pMML, pMML, png, png
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.11 $R_{J}\left(0,y,z,p\right)=\frac{3}{2p\sqrt{z}}\ln\left(\frac{16z}{y}\right)-% \frac{3}{p}R_{C}\left(z,p\right)+O\left(y\ln y\right),$ $y\to 0+$ ;

( $\neq 0$ ) real.

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\sim$ : Poincaré asymptotic expansion and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), Erratum (V1.0.26) for Equation (19.20.11)
Permalink:: http://dlmf.nist.gov/19.20.E11
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.26):: The constant term $\frac{-3}{p}R_{C}\left(z,p\right)$ and the order term $O\left(y\ln y\right)$ were added, and hence $\sim$ was replaced by .
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.12 $\lim_{p\to\pm\infty}pR_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right).$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.13 $2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left(% yz,p^{2}\right),$ $p=x\pm\sqrt{(y-x)(z-x)}$ ,

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.20.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

where may be permuted.

When the variables are real and distinct, the various cases of $R_{J}\left(x,y,z,p\right)$ are called circular (hyperbolic) cases if is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If are permuted so that $0\leq x<y<z$ , then the Cauchy principal value of $R_{J}$ is given by

19.20.14 $(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,y,z,p\right)-3R_{F}\left(x,y% ,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{C}\left(xy+pq,pq\right),$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.16(i), §19.16(i), §19.20(iii), §19.20(iii), §19.21(iii), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.20.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

valid when

19.20.15

$p=\frac{z(x+y+q)-xy}{z+q},$

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Permalink:: http://dlmf.nist.gov/19.20.E15
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.16

$w=\frac{z-x}{z+q},$

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Permalink:: http://dlmf.nist.gov/19.20.E16
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

Since , is in a hyperbolic region. In the complete case () (19.20.14) reduces to

19.20.17 $(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3R_{F}\left(0,y% ,z\right),$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

§19.20(iv) $R_{D}\left(x,y,z\right)$

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Keywords:: general lemniscatic case, lemniscate constants, symmetric elliptic integrals
Notes:: For the third equation in (19.20.18) put $t=y{\tan}^{2}\theta$ in (19.16.5); for the fourth equation see (19.27.7). For (19.20.19) see (19.27.8). For (19.20.20) and (19.20.21) use (19.16.15), (19.16.9), and Carlson (1977b, Table 8.5-1). In (19.20.22) put $t=1/\sqrt{s+1}$ ; alternatively use (19.29.20). For the second equality replace $t^{4}$ by and apply (5.12.1). For (19.20.23) use Carlson (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
Permalink:: http://dlmf.nist.gov/19.20.iv
See also:: Annotations for §19.20 and Ch.19

19.20.18

$R_{D}\left(x,x,x\right)=x^{-3/2},$

$R_{D}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-3/2}R_{D}\left(x,y,z% \right),$

$R_{D}\left(0,y,y\right)=\tfrac{3}{4}\pi\,y^{-3/2},$

$R_{D}\left(0,0,z\right)=\infty.$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.20.E18
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.19 $R_{D}\left(x,y,z\right)\sim 3x^{-1/2}y^{-1/2}z^{-1/2},$ $z/\sqrt{xy}\to 0$ .

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $\sim$ : asymptotic equality
Referenced by:: §19.20(iv), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.20.E19
Encodings:: TeX, pMML, png
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.20 $R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{% \sqrt{x}}{y}\right),$ $x\neq y$ , $y\neq 0$ ,

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.20.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.21 $R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{% \sqrt{z}}\right),$ $x\neq z$ , $xz\neq 0$ .

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.20.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

The second lemniscate constant is given by

19.20.22 $\int_{0}^{1}\frac{t^{2}\,\mathrm{d}t}{\sqrt{1-t^{4}}}=\tfrac{1}{3}R_{D}\left(0% ,2,1\right)=\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2% }}=0.59907\;01173\;67796\;10371\dots.$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Notes:: For more digits see OEIS Sequence A085566; see also Sloane (2003).
Referenced by:: §19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6
Permalink:: http://dlmf.nist.gov/19.20.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. Compare (19.20.2). The general lemniscatic case is

19.20.23 $R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\tfrac{1}{2}x+\tfrac{1}{2}y$ .

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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 $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.20.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

§19.20(v) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

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Keywords:: special cases, symmetric elliptic integrals
Notes:: See Carlson (1977b, (6.2-1), (6.8-15)).
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.v
See also:: Annotations for §19.20 and Ch.19

Define $c=\sum_{j=1}^{n}b_{j}$ . Then

19.20.24

$R_{0}\left(\mathbf{b};\mathbf{z}\right)=1,$

$R_{N}\left(\mathbf{b};\mathbf{z}\right)=\frac{N!}{{\left(c\right)_{N}}}T_{N}(% \mathbf{b},\mathbf{z}),$ $N=0,1,2,\dots$ ,

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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), : factorial (as in ) and $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial
Permalink:: http://dlmf.nist.gov/19.20.E24
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.20(v), §19.20 and Ch.19

where $T_{N}$ is defined by (19.19.1). Also,

19.20.25 $R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}},$

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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 : nonnegative integer
Referenced by:: §19.18(ii), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.20.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(v), §19.20 and Ch.19

19.20.26 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}R_{-a^{% \prime}}\left(\mathbf{b};\boldsymbol{{z^{-1}}}\right),$ $a+a^{\prime}=c$ , $\boldsymbol{{z^{-1}}}=(z_{1}^{-1},\dots,z_{n}^{-1})$ .

ⓘ

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/19.20.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §19.20(v), §19.20 and Ch.19

§19.20 Special Cases

Contents

§19.20(i)

§19.20(ii)

§19.20(iii)

§19.20(iv)

§19.20(v)

§19.20(i) $R_{F}\left(x,y,z\right)$

§19.20(ii) $R_{G}\left(x,y,z\right)$

§19.20(iii) $R_{J}\left(x,y,z,p\right)$

§19.20(iv) $R_{D}\left(x,y,z\right)$

§19.20(v) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$