§19.20 Special Cases

Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.20

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

§19.20(i) $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$

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

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

19.20.1

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

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

$\mathop{R_{F}\/}\nolimits\!\left(x,y,y\right)=\mathop{R_{C}\/}\nolimits\!\left(x,y\right),$

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

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Permalink:: http://dlmf.nist.gov/19.20.E1
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

The first lemniscate constant is given by

19.20.2 $\int _{0}^{1}\frac{dt}{\sqrt{1-t^{4}}}=\mathop{R_{F}\/}\nolimits\!\left(0,1,2\right)=\frac{\left(\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{{1/2}}}=1.31102\; 87771\; 46059\; 90523\dots.$

Notes:: For more digits see OEIS Sequence A085565; see also Sloane (2003).
Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.20(i), §19.20(iv), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.20.E2
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.E3
Encodings:: TeX, pMML, png

§19.20(ii) $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$

Notes:: For (19.20.4) use (19.20.5) and (19.16.3). For (19.20.5) put $z=y$ in (19.21.10).
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.ii

19.20.4

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

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

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

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

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
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

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

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

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

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 $x=0$ in (19.20.13). For (19.20.10) interchange $x$ and $z$ 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 $q=p$ in (19.21.12). For (19.20.14) exchange $x$ and $z$ in (19.21.12) and use (19.2.20).
Keywords:: symmetric elliptic integrals
Referenced by:: §19.21(iii)
Permalink:: http://dlmf.nist.gov/19.20.iii

19.20.6

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

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

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

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

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

19.20.7 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)\to+\infty,$ $p\to 0+$ or $0-$ ; $x,y,z>0$ .

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E7
Encodings:: TeX, pMML, png

19.20.8

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

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

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

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

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : 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

19.20.10

$\lim _{{p\to 0+}}\sqrt{p}\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)=\frac{3\pi}{2\sqrt{yz}},$

$\lim _{{p\to 0-}}\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)={-\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)-\mathop{R_{D}\/}\nolimits\!\left(0,z,y\right)}=\frac{-6}{yz}\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: Figure 19.17.7, Figure 19.17.7, Figure 19.17.8, Figure 19.17.8, §19.20(iii), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.20.E10
Encodings:: TeX, TeX, pMML, pMML, png, png

19.20.11 $\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)\sim\frac{3}{2p\sqrt{z}}\mathop{\ln\/}\nolimits\left(\frac{16z}{y}\right),$ $y\to 0+$ ; $p$ ( $\neq 0$ ) real.

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\sim$ : asymptotic equality
Referenced by:: Figure 19.17.8, Figure 19.17.8, §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E11
Encodings:: TeX, pMML, png

19.20.12 $\lim _{{p\to\pm\infty}}p\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right).$

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

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : 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

where $x,y,z$ may be permuted.

When the variables are real and distinct, the various cases of $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ 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 $x,y,z$ are permuted so that $0\leq x<y<z$ , then the Cauchy principal value of $\mathop{R_{J}\/}\nolimits$ is given by

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

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

valid when

19.20.15

$q>0,$

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

Permalink:: http://dlmf.nist.gov/19.20.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

19.20.16

$p=wy+(1-w)z,$

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

$0<w<1.$

Permalink:: http://dlmf.nist.gov/19.20.E16
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

Since $x<y<p<z$ , $p$ is in a hyperbolic region. In the complete case ( $x=0$ ) (19.20.14) reduces to

19.20.17 $(q+z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,-q\right)=(p-z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right),$ $p=z(y+q)/(z+q)$ , $w=z/(z+q)$ .

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

§19.20(iv) $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$

Notes:: For the third equation in (19.20.18) put $t=y{\mathop{\tan\/}\nolimits^{{2}}}\theta$ 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 $t$ and apply (5.12.1). For (19.20.23) use Carlson (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
Keywords:: lemniscate constants, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.20.iv

19.20.18

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

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

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

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

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
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

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

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\sim$ : asymptotic equality
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.20.E19
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : 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

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

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

The second lemniscate constant is given by

19.20.22 $\int _{0}^{1}\frac{t^{2}dt}{\sqrt{1-t^{4}}}=\tfrac{1}{3}\mathop{R_{D}\/}\nolimits\!\left(0,2,1\right)=\frac{\left(\mathop{\Gamma\/}\nolimits\!\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{{1/2}}}=0.59907\; 0 1173\; 67796\; 10371\dots.$

Notes:: For more digits see OEIS Sequence A085566; see also Sloane (2003).
Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6
Permalink:: http://dlmf.nist.gov/19.20.E22
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.20.E23
Encodings:: TeX, pMML, png

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

Notes:: See Carlson (1977b, (6.2-1),(6.8-15)).
Keywords:: symmetric elliptic integrals
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.v

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

19.20.24

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

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

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial and $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial
Permalink:: http://dlmf.nist.gov/19.20.E24
Encodings:: TeX, TeX, pMML, pMML, png, png

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

19.20.25 $\mathop{R_{{-c}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\prod _{{j=1}}^{n}z_{j}^{{-b_{j}}},$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $n$ : nonnegative integer
Referenced by:: §19.18(ii), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.20.E25
Encodings:: TeX, pMML, png

19.20.26 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\prod _{{j=1}}^{n}z_{j}^{{-b_{j}}}\mathop{R_{{-a^{{\prime}}}}\/}\nolimits\!\left(\mathbf{b};\boldsymbol{{z^{{-1}}}}\right),$ $a+a^{{\prime}}=c$ , $\boldsymbol{{z^{{-1}}}}=(z_{1}^{{-1}},\dots,z_{n}^{{-1}})$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/19.20.E26
Encodings:: TeX, pMML, png

§19.20 Special Cases

Contents

§19.20(i)

§19.20(ii)

§19.20(iii)

§19.20(iv)

§19.20(v)

§19.20(i) $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$

§19.20(ii) $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$

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

§19.20(iv) $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$

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