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

§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 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 -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, : differential of 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 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 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).
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

;

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

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

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

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

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

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

( $\neq 0$ ) 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 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 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 $\mathop{R_{J}\/}\nolimits$ 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

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

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

19.20.16

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

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

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

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

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

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, : differential of 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, : : 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 : 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 : 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) $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$