§19.18 Derivatives and Differential Equations

Let ${\partial }_j=\partial /\partial z_j$, and ${𝐞}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. Also define

The next two equations apply to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23).

and two similar equations obtained by permuting $x,y,z$ in (19.18.10).

More concisely, if $v=R_{-a}(𝐛;𝐳)$, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:

and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent):

or equivalently,

Here $j,l=1,2,\mathrm{\dots },n$ and $j\ne l$. For group-theoretical aspects of this system see Carlson (1963, §VI). If $n=2$, then elimination of ${\partial }_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_1,b_2,z_1,z_2)=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1).

The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)).

The function $w=R_{-a}(\frac{1}{2},\frac{1}{2};x+y,x-y)$ satisfies an Euler–Poisson–Darboux equation:

Also $W=R_{-a}(\frac{1}{2},\frac{1}{2};t+r,t-r)$, with $r=\sqrt{x^2+y^2}$, satisfies a wave equation:

Similarly, the function $u=R_{-a}(\frac{1}{2},\frac{1}{2};x+\mathrm{i}y,x-\mathrm{i}y)$ satisfies an equation of axially symmetric potential theory:

and $U=R_{-a}(\frac{1}{2},\frac{1}{2};z+\mathrm{i}\rho ,z-\mathrm{i}\rho )$, with $\rho =\sqrt{x^2+y^2}$, satisfies Laplace’s equation: