# with two variables

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##### 1: Bibliography Y
• Z. M. Yan (1992) Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.
• ##### 2: 19.21 Connection Formulas
19.21.1 $R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_% {F}\left(0,z+1,1\right)=3\pi/(2z),$ $z\in\mathbb{C}\setminus(-\infty,0]$.
19.21.7 $(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3\sqrt{y/(xz)},$
19.21.8 $R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3(xyz)% ^{-1/2},$
19.21.9 $xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_% {F}\left(x,y,z\right).$
##### 3: 16.13 Appell Functions
###### §16.13 Appell Functions
The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
##### 4: 19.27 Asymptotic Approximations and Expansions
19.27.7 $R_{D}\left(x,y,z\right)=\frac{3}{2z^{3/2}}\left(\ln\left(\frac{8z}{a+g}\right)% -2\right)\left(1+O\left(\frac{a}{z}\right)\right),$ $a/z\to 0$.
19.27.8 $R_{D}\left(x,y,z\right)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}R_{G}\left(x,y,0% \right)\left(1+O\left(\frac{z}{g}\right)\right),$ $z/g\to 0$.
19.27.9 $R_{D}\left(x,y,z\right)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(% \frac{b}{x}\ln\frac{x}{b}\right)\right),$ $b/x\to 0$.
19.27.10 $R_{D}\left(x,y,z\right)=R_{D}\left(0,y,z\right)-\frac{3\sqrt{x}}{hz}\left(1+O% \left(\sqrt{\frac{x}{h}}\right)\right),$ $x/h\to 0$.
##### 5: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$, the Appell (two-variable hypergeometric) functions ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)$, ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$, and the Meijer $G$-function ${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$. …
##### 6: 19.20 Special Cases
19.20.19 $R_{D}\left(x,y,z\right)\sim 3(xyz)^{-1/2},$ $z/\sqrt{xy}\to 0$.
19.20.20 $R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{% \sqrt{x}}{y}\right),$ $x\neq y$, $y\neq 0$,
19.20.21 $R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{% \sqrt{z}}\right),$ $x\neq z$, $xz\neq 0$.
19.20.23 $R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\tfrac{1}{2}x+\tfrac{1}{2}y$.
##### 7: 19.28 Integrals of Elliptic Integrals
19.28.3 $\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1\right)\mathrm{d}t=\frac{3}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}.$
19.28.5 $\int_{z}^{\infty}R_{D}\left(x,y,t\right)\mathrm{d}t=6R_{F}\left(x,y,z\right),$
19.28.6 $\int_{0}^{1}R_{D}\left(x,y,v^{2}z+(1-v^{2})p\right)\mathrm{d}v=R_{J}\left(x,y,% z,p\right).$
##### 8: Viewing DLMF Interactive 3D Graphics
In the DLMF, we provide facilities for the interactive display of special functions of two independent variables. …
##### 9: 19.16 Definitions
A fourth integral that is symmetric in only two variables is defined by
19.16.5 $R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\mathrm{d}t}{s(t)(t+z)},$
19.16.15 $R_{D}\left(x,y,z\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac% {3}{2};x,y,z\right),$
19.16.21 $R_{D}\left(0,y,z\right)=\tfrac{3}{4}\pi R_{-\frac{3}{2}}\left(\tfrac{1}{2},% \tfrac{3}{2};y,z\right),$
##### 10: 19.25 Relations to Other Functions
19.25.6 $\frac{\partial F\left(\phi,k\right)}{\partial k}=\tfrac{1}{3}kR_{D}\left(c-1,c% ,c-k^{2}\right).$
19.25.9 $E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}% \left(c-1,c-k^{2},c\right),$
19.25.11 $E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1% \right)+\sqrt{(c-k^{2})/(c(c-1))},$ $\phi\neq\tfrac{1}{2}\pi$.
19.25.12 $\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,% c-k^{2},c\right).$
19.25.13 $D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right).$