UDC 517.946.9 : 517.516

ON A PARTICULAR CASE OF THE DIRICHLET PROBLEM FOR THE DARBOUX EQUATION FOR A SPACE CUT ALONG AN ELLIPSOIDAL DISK

V. I. EVSIN

The following problem is considered: in a space of \(n+1\) dimensions of points \((x_1, x_2,\ldots, x_n, x_{n+1}=z)\), construct a function \(v\), even with respect to \(z\), which satisfies the Darboux equation

\[ \Delta^* v=\Delta v+\frac{k}{z}\cdot \frac{\partial v}{\partial z}=0,\qquad |k|<1, \tag{0.1} \]

vanishes at infinity, and assumes prescribed values on the \(n\)-dimensional ellipsoidal disk:

\[ z=0,\qquad 1-\frac{x_1^2}{a_1^2}-\cdots-\frac{x_n^2}{a_n^2}\geqslant 0. \tag{0.2} \]

For this problem, by the method of separation of variables, eigenfunctions are constructed—polynomials orthogonal on the ellipsoid (0.2) with respect to a certain weight. These functions are related to multidimensional Lamé functions and constitute a generalization to the multidimensional case of the functions constructed in [1] for three-dimensional space. In the case when the boundary values of \(v\) on the disk (0.2) are prescribed in the form of polynomials in the Cartesian coordinates \((x_1,x_2,\ldots,x_n)\), this method makes it possible to solve the stated problem effectively.

1. The Darboux equation (0.1) in ellipsoidal coordinates of an \(n+1\)-dimensional space has the form

\[ \Delta^*v = 4\sum_{\nu=1}^{n+1} \frac{\Psi(\lambda_\nu)} {\displaystyle\prod_{\substack{\mu=1\\ \mu\ne \nu}}^{n+1}(\lambda_\nu-\lambda_\mu)} \times \]

\[ \times \left\{ \frac{\partial^2 v}{\partial \lambda_\nu^2} + \frac{1}{2} \left[ \frac{\Psi'(\lambda_\nu)}{\Psi(\lambda_\nu)} + \frac{k}{\lambda_\nu} \right] \frac{\partial v}{\partial \lambda_\nu} \right\} =0, \tag{1.1} \]

where here and below the prime on the product sign denotes omission of the factor with index \(\mu=\nu\), and all \(\lambda_\nu\) are ellipsoidal coordinates, i.e., the roots of the equation

\[ \Phi(s)=1-\frac{x_1^2}{a_1^2+s}-\cdots-\frac{x_{n+1}^2}{a_{n+1}^2+s}=0, \tag{1.2} \]

where

\[ a_{n+1}=0,\qquad \lambda_1 \geq 0 \geq \lambda_2 \geq -a_n^2 \geq \lambda_3 \geq \cdots \geq \lambda_{n+1} \geq -a_1^2 \tag{1.3} \]

and

\[ \Psi(s)=(a_1^2+s)(a_2^2+s)\ldots(a_n^2+s)s. \]

As a result of separation of variables for functions \(E_\nu(\lambda_\nu)\), such that the product \(v=\prod_{\nu=1}^{n+1}E_\nu(\lambda_\nu)\) is a solution of equation (1.1), we obtain the same differential equation

\[ \frac{d^2E}{ds^2} +\frac{1}{2}\left[ \frac{1+k}{s} +\frac{1}{a_1^2+s} +\cdots +\frac{1}{a_n^2+s} \right]\times \]

\[ \times \frac{dE}{ds} = \frac{P_{n-1}(s)}{4\Psi(s)}\,E, \tag{1.4} \]

where \(P_{n-1}(s)=g_0s^{n-1}+g_1s^{n-2}+\cdots+g_{n-1}\) is a polynomial, all coefficients of which are arbitrary. In what follows it is convenient to pass from symmetric ellipsoidal coordinates to nonsymmetric ones by the formula \(\rho_\nu^2=\lambda_\nu+a_1^2\). Equation (1.2) is then replaced by

\[ \Phi(\sigma^2) = 1-\frac{x_1^2}{\sigma^2-c_1^2} -\cdots -\frac{x_{n+1}^2}{\sigma^2-c_{n+1}^2} =0, \tag{1.5} \]

where \(c_\nu^2=a_1^2-a_\nu^2\). Through the roots \(\rho_\nu\), equation (1.5) is written as

\[ \Phi(\sigma^2) = \prod_{\nu=1}^{n+1} \frac{(\sigma^2-\rho_\nu^2)}{(\sigma^2-c_\nu^2)} =0. \tag{1.6} \]

Corresponding to inequalities (1.3), we obtain the inequalities

\[ \rho_1^2 \geq c_{n+1}^2 \geq \rho_2^2 \geq \cdots \geq \rho_{n+1}^2 \geq c_1^2=0. \tag{1.7} \]

If \(\rho_1=a_1\), then the \(n+1\)-dimensional ellipsoid \(\Phi(\rho_1^2)=0\) flattens into a certain “disc,” an \(n\)-dimensional ellipsoid

\[ \left.\Phi(\rho_1^2)\right|_{\rho_1=a_1} = 1-\frac{x_1^2}{a_1^2} -\cdots -\frac{x_n^2}{a_n^2} =0. \tag{1.8} \]

On this disc the Cartesian coordinates and Lamé coefficients have the form

\[ x_\nu^2= \frac{ \displaystyle \prod_{\mu=2}^{n+1}\left|\rho_\mu^2-c_\nu^2\right|^{\frac12} }{ \displaystyle c_\nu\prod_{\mu=2}^{n+1}\left|c_\mu^2-c_\nu^2\right| }, \tag{1.9} \]

\[ H_{\rho_\nu}\, \frac{\displaystyle\prod_{\mu=2}^{n+1} |\rho_\mu^2-\rho_\nu^2|^{\frac12}} {\displaystyle\prod_{\mu=2}^{n+1} |\rho_\mu^2-c_\nu^2|^{\frac12}}, \tag{1.10} \]

where \(\nu=2,3,\ldots,n+1\). From (1.6) we find

\[ \Phi(\rho_1^2)\big|_{\rho_1^2=a_1^2} = \prod_{t=1}^{n} \frac{(c_{n+1}^2-\rho_{t+1}^2)} {(c_{n+1}^2-c_t^2)}. \tag{1.11} \]

Consequently,

\[ \prod_{\nu=2}^{n+1} |\rho_\nu^2-c_{n+1}^2| = M\left(1-\frac{x_1^2}{a_1^2}-\cdots-\frac{x_n^2}{a_n^2}\right), \tag{1.12} \]

where \(M\) is a certain constant factor. The substitution \(s+a_1^2=\sigma^2\) in equation (1.4) leads to the equation

\[ \prod_{\nu=2}^{n+1}(\sigma^2-c_\nu^2) \left[ \frac{d^2E}{d\sigma^2} +\sigma\frac{dE}{d\sigma} \left( \frac{1+k}{\sigma^2-c_{n+1}^2} + \sum_{r=2}^{n}\frac{1}{\sigma^2-c_r^2} \right) \right] = P_{n-1}(\sigma^2)E, \tag{1.13} \]

where \(P_{n-1}(\sigma^2)=g_0\sigma^{2n-2}+g_1\sigma^{2n-4}+\cdots+g_{n-1}\). For the point at infinity the determining equation will be:
\(\alpha^2+\alpha(n+k-1)-g_0=0\). Taking \(g_0\) in the form
\(g_0=m(m+n+k-1)\), we obtain, with respect to the point at infinity, exponents equal to \(-m,\ m+n+k-1\). The remaining coefficients of the right-hand side of (1.13) will be regarded as arbitrary accessory parameters, and their set will be denoted by
\(q=\{g_1,g_2,\ldots,g_{n-1}\}\). We shall show that, for characteristic values of \(q\), equation (1.13) admits solutions in the form of quasipolynomials

\[ E=\eta\sqrt{\prod_{t=1}^{p}(\sigma^2-c_{\gamma_t}^2)}, \tag{1.14} \]

where

\[ \eta=\sum_{s=0}^{r} b_s\sigma^{m-p-2s},\qquad r=\left[\frac{m-p}{2}\right], \]

and \(\gamma_t\) assumes the values of arbitrary \(p\) indices among \(2,3,\ldots\), or \(p=0\). Substituting \(\eta\) in the form (1.14) into equation (1.13), we obtain for the polynomial \(\eta\) the following differential equation:

\[ \prod_{\nu=2}^{n+1}(\sigma^2-c_\nu^2) \left[ \frac{d^2\eta}{d\sigma^2} + \frac{d\eta}{d\sigma}\times \right. \]

\[ \times\left(2\sum_{t=1}^{p}\frac{\sigma}{\sigma^{2}-c_{\nu_t}^{2}} +\sum_{\nu=2}^{n}\frac{\sigma}{\sigma^{2}-c_{\nu}^{2}} +\frac{(1+k)\sigma}{\sigma^{2}-c_{n+1}^{2}}\right)\Bigg]+ \]

\[ +\eta\left\{\prod_{\nu=2}^{n+1}(\sigma^{2}-c_{\nu}^{2}) \left[\left(\sum_{t=1}^{p}\frac{\sigma}{\sigma^{2}-c_{\nu_t}^{2}}\right)^{2} -\sum_{t=1}^{p}\frac{\sigma^{2}+c_{\nu_t}^{2}}{\sigma^{2}-c_{\nu_t}^{2}}+\right.\right. \]

\[ \left.\left. +\left(\frac{\sigma(1+k)}{\sigma^{2}-c_{n+1}^{2}} +\sum_{\nu=2}^{n}\frac{\sigma}{\sigma^{2}-c_{\nu}^{2}}\right) \sum_{t=1}^{p}\frac{\sigma}{\sigma^{2}-c_{\nu_t}^{2}}\right] -P_{n-1}(\sigma^{2})\right\}=0. \tag{1.15} \]

Arranging the coefficients of equation (1.15) in descending powers of \(\sigma\), we obtain the equation

\[ (\sigma^{2n}-d_{1}^{2}\sigma^{2n-2}+d_{2}^{4}\sigma^{2n-4}-\ldots+(-1)^{n}d_{n}^{2n}) \frac{d^{2}\eta}{d\sigma^{2}}+ \]

\[ +\sigma\{(2p+n+k)\sigma^{2n-2}-l_{1}^{2}\sigma^{2n-4} +\ldots+(-1)^{n-1}l_{n-1}^{2n-2}\} \frac{d\eta}{d\sigma}+ \]

\[ +\{[p(p-1)+pn+pk-m(m+n+k-1)]\times \]

\[ \times\sigma^{2n-2}-(g_{1}+f_{1}^{2})\sigma^{2n-4} +\ldots+(-1)^{n}(g_{n-2}+f_{n-2}^{2n-4})\times \]

\[ \times\sigma^{2}+(-1)^{n-1}(g_{n-1}+f_{n-1}^{2n-2})\}\eta=0, \tag{1.16} \]

where \(d_{\nu}^{2\nu}\) is the symmetric function of the roots \(c_{\nu}^{2}\);

\[ l_{\nu}^{2\nu}=2\{\varkappa_{\nu}^{2\nu}(p-\nu) +\varkappa_{\nu-1}^{2\nu-2}\delta_{1}^{2}(p-\nu+1) +\ldots+\delta_{\nu}^{2\nu}p\}+ \]

\[ +(n-\nu)d_{\nu}^{2\nu} +k\left[ \frac{\displaystyle\prod_{\nu=2}^{n+1}(\tau^{2}-c_{\nu}^{2})} {\tau^{2n-2\nu}} \right]_{\tau^{2}=c_{n+1}^{2}}; \]

\[ f_{\nu}^{2\nu}=\varkappa_{\nu}^{2\nu}(p-\nu)(p-\nu-1) +\varkappa_{\nu-1}^{2\nu-2}\delta_{1}^{2}(p-\nu-1)\times \]

\[ \times(p-\nu-2)+\ldots+\delta_{\nu}^{2\nu}p(p-1)+ \]

\[ +(n-\nu)\bigl(p\delta_{\nu}^{2\nu} +(p-1)\varkappa_{1}^{2}\delta_{\nu-1}^{2\nu-2} +\ldots+\varkappa_{\nu}^{2\nu}(p-\nu)\bigr)+ \]

\[ +k\left\{ \left[ \frac{\displaystyle\prod_{\nu=2}^{n+1}(\tau^{2}-c_{\nu}^{2})} {\displaystyle\prod_{t=1}^{p}(\tau^{2}-c_{\nu_t}^{2})\,\tau^{2n-2p}} \right]_{\tau^{2}=c_{n+1}^{2}} \varkappa_{\nu}(p-\nu)+\right. \]

\[ \left. +\left[ \frac{\displaystyle\prod_{\nu=2}^{n+1}(\tau^{2}-c_{\nu}^{2})} {\displaystyle\prod_{t=1}^{p}(\tau^{2}-c_{\nu_t}^{2})\,\tau^{2n-2p-2}} \right]_{\tau^{2}=c_{n+1}^{2}} \varkappa_{\nu-1}(p-\nu-1)+ \right. \]

\[ +\ldots+ \left| \left. \frac{\displaystyle \prod_{\nu=2}^{n+1}(\tau^2-c_\nu^2)} {\displaystyle \prod_{t=1}^{p}(\tau^2-c_{\nu_t}^2)\tau^{2n-2p-2\nu}} \right|_{\tau^2=c_{n+1}^2} \right\}p, \]

where the square brackets denote the integer-part sign, and \(\delta_r^{2r}\) and \(\chi_r^{2r}\) are the coefficients in the representations:

\[ \prod_{t=1}^{p}(\sigma^2-c_{\nu_t}^2) = \sigma^{2p}-\chi_1^2\sigma^{2p-2}+\chi_2^4\sigma^{2p-4}-\ldots, \]

\[ \frac{\displaystyle \prod_{\nu=2}^{n+1}(\sigma^2-c_\nu^2)} {\displaystyle \prod_{t=1}^{p}(\sigma^2-c_{\nu_t}^2)} = \sigma^{2n-2p}-\delta_1^2\sigma^{2n-2p-2} +\delta_2^4\sigma^{2n-2p-4}-\ldots \]

Substituting \(\eta\) into equation (1.16), we obtain for the coefficients \(b_s\) the following recurrent system:

\[ \begin{aligned} &b_{s+n-1}2(s+n-1)(2m-2s-n+1+k)=\\ &=b_{s+n-2}\{-d_1^2(m-2s-p-2n+4)(m-2s-p-2n+3)-\\ &\qquad\qquad -l_1^2(m-2s-p-2n+4)-g_1-f_1^2\}+\\ &\quad +b_{s+n-3}\{d_2^4(m-2s-p-2n+6)(m-2s-p-2n+5)+\\ &\qquad\qquad +l_2^4(m-2s-p-2n+6)+g_2+f_2^4\}+\\ &\quad +\ldots+b_s(-1)^{n-1}\{(m-2s-p)(m-2s-p-1)d_{n-1}^{2n-2}+\\ &\qquad\qquad +(m-2s-p)l_{n-1}^{2n-2}+g_{n-1}+f_{n-1}^{2n-2}\}+\\ &\quad +b_{s-1}(-1)^n d_n^{2n}(m-2s-p+2)(m-2s-p+1). \end{aligned} \tag{1.17} \]

For \(s=r+1\) we set

\[ b_{r+1}=b_{r+2}=\ldots=b_{r+n-1}=0, \tag{1.18} \]

we obtain \(b_{r+n}=0\), and then all \(b_s=0\) for \(s>r\). Without loss of generality, set \(b_0=1\); after this, from conditions (1.18) we have for the accessory parameters \(q\) an algebraic system of \(n-1\) equations with \(n-1\) unknowns; solving it, we determine the desired polynomial \(\eta\). In the general case, the number of all possible sets \(q\) is obtained by applying the well-known Heine—Stieltjes theorem [5] to the equation that results if, in equation (1.15), the inverse substitution \(\sigma^2=s+d_1^2\) is made. The equation obtained in this way will be of the same form as equation (1.4). By the Heine—Stieltjes theorem, the number of all possible sets \(q\) that generate polynomial solutions of degree \(m'\) of an equation of the form (1.4) is, in the general case, equal to \(C_{m'+n-1}^{\,n-1}\). It is not difficult to establish that the number of polynomial solutions of degree \(m\) is

of equation (1.15) is equal to \(C_{\left[\frac{m-p}{2}\right]+n-1}^{\,n-1}\). The number of all possible quasipolynomials of the form (1.14) of degree \(m\) is, obviously, equal to
\(C_{n-1}^{p}C_{\left[\frac{m-p}{2}\right]+n-1}^{\,n-1}\). The number of all types of quasipolynomials of degree \(m\) is equal to
\[ \sum_{p=0}^{n-1} C_{n-1}^{p} C_{\left[\frac{m-p}{2}\right]+n-1}^{\,n-1}. \]
From the identity
\[ (1+t)^{n-1}\frac{1+t}{(1-t^{2})^{n}}=\frac{1}{(1-t)^{n}} \]
we find that
\[ \sum_{p=0}^{n-1} C_{n-1}^{p} C_{\left[\frac{m-p}{2}\right]+n-1}^{\,n-1} = C_{m+n-1}^{m}, \]
i.e., this number coincides with the number of homogeneous basis elements of degree \(m\) in the space of polynomials in \(n\) variables \(x_{1}, x_{2}, \ldots, x_{n}\).

1. We now show that the products
   \[ \prod_{\nu=2}^{n+1} E_{\nu}(\rho_{\nu}) \]
   are mutually orthogonal on the “disc” \(\rho_{1}=a_{1}\) with respect to a certain weight and, consequently, are linearly independent.

Let \(L_{m}^{q}(\sigma)\) and \(L_{m'}^{q'}(\sigma)\) be two distinct quasipolynomials corresponding to different characteristic sets \(q\) and \(q'\) and to different degree indices \(m\) and \(m'\). For \(L_{m}^{q}\) we can write the identity
\[ \prod_{\nu=2}^{n+1}(\sigma^{2}-c_{\nu}^{2}) \left\{ \frac{d^{2}L_{m}^{q}}{d\sigma^{2}} +\sigma\left[ \frac{1+k}{\sigma^{2}-c_{n+1}^{2}} +\sum_{\nu=2}^{n}\frac{1}{\sigma^{2}-c_{\nu}^{2}} \right] \times \frac{dL_{m}^{q}}{d\sigma} \right\} = P_{n-1}^{q}(\sigma)L_{m}^{q}. \]
The same identity can also be written for \(L_{m'}^{q'}\). Denote
\[ \Delta(\sigma)= \sqrt{ |\sigma^{2}-c_{n+1}^{2}|^{\,k-1} \prod_{\nu=2}^{n}|\sigma^{2}-c_{\nu}^{2}| }; \]
\[ \Delta_{1}(\sigma)=\Delta(\sigma)\cdot|\sigma^{2}-c_{n+1}^{2}|^{k}. \]
Multiplying both sides of each identity respectively by \(\Delta_{1}(\sigma)L_{m'}^{q'}(\sigma)\) and \(L_{m}^{q}(\sigma)\Delta_{1}(\sigma)\), and then subtracting the second equality from the first, we obtain the identity
\[ \frac{d}{d\sigma}\bigl(H\Delta(\sigma)\bigr) = \varepsilon\, \frac{P_{n-1}^{q}(\sigma^{2})-P_{n-1}^{q'}(\sigma^{2})}{\Delta(\sigma)} \,L_{m}^{q}L_{m'}^{q'}, \tag{2.1} \]
where
\[ H= L_{m'}^{q'}\frac{dL_{m}^{q}}{d\sigma} - L_{m}^{q}\frac{dL_{m'}^{q'}}{d\sigma}, \qquad \varepsilon=\operatorname{sign}\prod_{\nu=2}^{n}(\sigma^{2}-c_{\nu}^{2}). \]

Integrating both sides of (2.1) \(n\) times, respectively over the intervals
\(-c_2 \leqslant \rho_{n+1} \leqslant c_2;\ c_2 \leqslant \rho_n \leqslant c_3;\ \ldots\ c_n \leqslant \rho_2 \leqslant c_{n+1}\), we obtain the system

\[ \begin{gathered} \bigl[m(m+n+k-1)-m'(m'+n+k-1)\bigr] \int_{-c_2}^{c_2} \frac{\rho_{n+1}^{\,2n+2}}{\Delta(\rho_{n+1})} L_m^q L_{m'}^{q'}\,d\rho_{n+1} + \\[4pt] +\cdots+(g_{n-1}-g'_{n-1}) \int_{-c_2}^{c_2} \frac{1}{\Delta(\rho_{n+1})} L_m^q L_{m'}^{q'}\,d\rho_{n+1} =0, \end{gathered} \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ \begin{gathered} \bigl[m(m+n+k-1)-m'(m'+n+k-1)\bigr] \int_{c_n}^{c_{n+1}} \frac{\rho_2^{\,2n+2}}{\Delta(\rho_2)} L_m^q L_{m'}^{q'}\,d\rho_2 + \\[4pt] +\cdots+(g_{n-1}-g'_{n-1}) \int_{c_n}^{c_{n+1}} \frac{1}{\Delta(\rho_2)} L_m^q L_{m'}^{q'}\,d\rho_2 =0. \end{gathered} \tag{2.2} \]

Since, by assumption,

\[ (m-m')^2+\sum_{i=1}^{n-1}(g_i-g_i')^2\ne 0, \]

the determinant of the homogeneous system (2.2) must be equal to zero. Denoting this determinant, similar to a Vandermonde determinant, by \(J_{mm'}^{qq'}\), we have

\[ J_{mm'}^{qq'} = \int_{c_n}^{c_{n+1}}\cdots \int_{c_2}^{c_3} \int_{-c_2}^{c_2} \prod_{\nu=2}^{n+1} L_m^q(\rho_\nu)L_{m'}^{q'}(\rho_\nu)\,d_1\omega =0, \tag{2.3} \]

where

\[ d_1\omega= \frac{d\omega}{ \displaystyle \prod_{\nu=2}^{n+1}|\sigma^2-c_\nu^2|^{\frac{1-k}{2}} }, \]

and \(d\omega\) is the volume element on the “disc” \(\rho_1=a_1\).

Carrying out arguments analogous to those given in the book [3], we shall show that all roots of the equation \(L_m^q(\sigma)=0\) are distinct, real, and do not exceed \(c_{n+1}\) in absolute value. First we show that the polynomial \(\eta\) cannot have the points \(\pm c_\nu\) as zeros. Indeed, \(\eta\) satisfies equation (1.15) of the form

\[ \prod_{\nu=2}^{n+1}(\sigma^2-c_\nu^2)\frac{d^2\eta}{d\sigma^2} +P\frac{d\eta}{d\sigma}+Q\eta=0, \tag{2.4} \]

where \(P\) and \(Q\) are polynomials not containing factors \(\sigma^2-c_\nu^2\). If \(\eta=0\) at \(\sigma=\pm c_\nu\), then also

\[ \frac{d\eta}{d\sigma}=0 \]

at these points; differentiating (2.4), we conclude that also

\[ \frac{d^2\eta}{d\sigma^2} = \frac{d^3\eta}{d\sigma^3} = \cdots =0, \]

but this cannot be. In the same way …

by the same reasoning, the presence of multiple roots in the polynomial \(\eta\) is ruled out.

In the equality (2.3) that has been proved, \(\prod_{\nu=2}^{n+1} L_m^{q'}(\rho_\nu)\) can be expressed by a constant and by an arbitrary polynomial in the product \(\prod_{\nu=2}^{n+1}\left(\prod_{t=1}^{p} |\sigma^2-c_{\nu_t}^2|\right)^{1/2}\); therefore

\[ \int_{c_n}^{c_{n+1}}\cdots\int_{c_2}^{c_3}\int_{-c_2}^{c_2} \prod_{\nu=2}^{n+1} L_m^q(\rho_\nu)\,d_1\omega=0. \tag{2.5} \]

Since in (2.5) \(d_1\omega>0\), at least once in one of the intervals of integration one of the factors \(L_m^q(\rho_\nu)\) must change sign, i.e., must have some root \(\alpha_1\). Consequently, \(\prod_{\nu=2}^{n+1} L_m^q(\rho_\nu)\), being an even function up to the factor \(\prod_{\nu=2}^{n+1}\rho_\nu\), must contain the factor

\[ \varphi(\rho_2,\rho_3,\ldots,\rho_{n+1}) = \prod_{\nu=2}^{n+1}(\rho_\nu^2-\alpha_1^2). \]

This factor can be represented in the form of a polynomial in \(\prod_{\nu=2}^{n+1}\sqrt{|\rho_\nu^2-c_2^2|}\) of degree less than \(m\); therefore

\[ \int_{c_n}^{c_{n+1}}\cdots\int_{c_2}^{c_3}\int_{-c_2}^{c_2} \prod_{\nu=2}^{n+1} L_m^q(\rho_\nu)(\rho_\nu^2-\alpha_1^2)\,d_1\omega=0. \tag{2.6} \]

If we denote

\[ \prod_{\nu=2}^{n+1}\widetilde L_m^q(\rho_\nu) = \prod_{\nu=2}^{n+1} L_m^q(\rho_\nu)(\rho_\nu^2-\alpha_1^2)^{-1}, \]

then (2.6) can be written in the form

\[ \int_{c_n}^{c_{n+1}}\cdots\int_{c_2}^{c_3}\int_{-c_2}^{c_2} \prod_{\nu=2}^{n+1}\widetilde L_m^q(\rho_\nu) \prod_{\nu=2}^{n+1}(\rho_\nu^2-\alpha_1^2)\,d_1\omega=0. \]

Noting that \(\prod_{\nu=2}^{n+1}(\rho^2-\alpha_1^2)\,d_1\omega>0\), and repeating the reasoning carried out above a finite number of times, we ultimately arrive at the representation

\[ L_m^q(\sigma)= \sqrt{ \prod_{t=1}^{p}|\sigma^2-c_{\nu_t}^2| \prod_{\mu=1}^{s}(\sigma^2-\alpha_\mu^2) }, \tag{2.7} \]

where all roots \(\pm \alpha_\mu\) are real, distinct, and lie in the interval \((-c_{n+1}, c_{n+1})\), with \(1+2s+p=m\) or \(2s+p=m\). Formula (1.9) shows that
\(\prod_{\nu=2}^{n+1} L_m^q(\sigma)\), up to a constant factor, is a certain polynomial \(Q_m^q(x_1,x_2,\ldots,x_n)\), whose form is

\[ Q_m^q(x_1,x_2,\ldots,x_n) = \prod_{t=1}^{p} x_{\nu_t} \prod_{\mu=1}^{s} \left( 1-\frac{x_1^2}{\alpha_\mu^2}-\ldots-\frac{x_n^2}{\alpha_\mu^2-c_n^2} \right), \tag{2.8} \]

where \(1+2s+p=m\) or \(2s+p=m\). Further, to shorten the notation we shall denote:

\[ x=(x_1,x_2,\ldots,x_n); \qquad 1-\frac{x^2}{a^2} = 1-\frac{x_1^2}{a_1^2}-\ldots-\frac{x_n^2}{a_n^2}; \]

\[ dx=dx_1dx_2\ldots dx_n . \]

Passing to Cartesian coordinates (formulas (1.10) and (1.12)), from (2.3) we obtain

\[ \underbrace{\int \cdots \int}_{1-\frac{x^2}{a^2}\ge 0}^{n\ \text{times}} \frac{Q_m^q(\xi)\,Q_{m'}^{q'}(\xi)\,d\xi} {\left(1-\dfrac{\xi}{a^2}\right)^{\frac{1-k}{2}}} =0. \tag{2.9} \]

Let, for \(m=m'\) and \(q=q'\), in the integrand of (2.9),

\[ \bigl(Q_m^q(x)\bigr)^2 = \sum \Lambda_{p_1p_2\ldots p_n} x_1^{p_1-1}x_2^{p_2-1}\ldots x_n^{p_n-1}, \]

where the summation is carried out over all possible combinations \(p_1,p_2,\ldots,p_n\). Then, using formula \([n^0\,676,\,(8)]\) from [4], for the normalization we obtain

\[ J_{mm}^{qq} = \sum \Lambda_{p_1p_2\ldots p_n} a_1^{p_1}a_2^{p_2}\ldots a_n^{p_n} \times \]

\[ \times \frac{ \Gamma\!\left(\dfrac{p_1}{2}\right) \Gamma\!\left(\dfrac{p_2}{2}\right) \ldots \Gamma\!\left(\dfrac{p_n}{2}\right) \Gamma\!\left(\dfrac{k+1}{2}\right) }{ \Gamma\!\left(\dfrac{p_1+p_2+\ldots+p_n+k+1}{2}\right) }. \tag{2.10} \]

1. Analogously to how this was done in [1], we shall derive an integral equation for \(Q_m^q(x)\). Introduce functions of the second kind \(F_m^q(\rho_1)\), each of which is a solution of equation (1.13) and satisfies the condition

\[ F_m^q(\rho_1)\to 0 \quad \text{as } \rho_1\to\infty . \]

Up to a constant factor, from the general theory we have

\[ F_m^q(\rho_1)=E_m^q(\rho_1)\int_{\rho_1}^{\infty} \frac{d\sigma}{\bigl(E_m^q(\sigma)\bigr)^2\Delta\sigma}. \tag{3.1} \]

Let \(v(x,0)=Q_m^q(x)\) on the “disk” \(\rho_1=a_1\). Define the solution of equation (0.1) outside the “disk” by the formula

\[ v(x_1,x_2,\ldots,x_n,x_{n+1})= \frac{F_m^q(\rho_1)}{F_m^q(c_{n+1})} \prod_{\nu=2}^{n+1} E_m^q(\rho_\nu). \tag{3.2} \]

We shall now compute the derivative \(\dfrac{\partial v}{\partial z}\), considering its expression in ellipsoidal coordinates:

\[ \frac{\partial v}{\partial z} = 2z\sum_{\nu=1}^{n+1} \frac{ \displaystyle \prod_{\mu=1}^{n+1}(\rho_\nu^2-c_\mu^2) }{ \displaystyle \prod_{\mu=1}^{n+1}(\rho_\nu^2-\rho_\mu^2) } \frac{\partial v}{\partial \rho_\nu}\, \frac{1}{2\rho_\nu}. \tag{3.3} \]

Next we compute

\[ \lim_{\rho_1\to a_1} z^k \frac{\partial v}{\partial z} = \lim_{\rho_1\to a_1} \times \]

\[ \times \frac{ (\rho_1^2-c_{n+1}^2)^{\frac{k+1}{2}} \displaystyle\prod_{\nu=2}^{n+1}(c_{n+1}^2-\rho_\nu^2)^{\frac{k+1}{2}} \displaystyle\prod_{\nu=1}^{n}(\rho_1^2-c_\nu^2) }{ \displaystyle\prod_{\nu=1}^{n}(c_{n+1}^2-c_\nu^2)^{\frac{k+1}{2}} \displaystyle\prod_{\nu+2}^{n+1}(\rho_1^2-\rho_\nu^2) } \frac{\partial v}{\partial \rho_1} = \]

\[ = c_{n+1}^{-k} \frac{ \displaystyle\prod_{\nu=2}^{n+1}(c_{n+1}^2-\rho_\nu^2)^{\frac{k+1}{2}} }{ \displaystyle\prod_{\nu=2}^{n+1}(c_{n+1}^2-c_\nu^2)^{\frac{k+1}{2}} } \lim_{\rho_1\to c_{n+1}} (\rho_1^2-c_{n+1}^2)^{\frac{k+1}{2}} \frac{\partial v}{\partial \rho_1}. \tag{3.4} \]

From formulas (3.1) and (3.2) we obtain

\[ \frac{\partial v}{\partial \rho_1} = \frac{ \displaystyle\prod_{\nu=2}^{n+1} E_m^q(\rho_\nu) }{ F_m^q(c_{n+1}) } \left\{ \frac{dE_m^q(\rho_1)}{d\rho_1} \int_{\rho_1}^{\infty} \frac{d\sigma}{\bigl[E_m^q(\sigma)\bigr]^2\Delta(\sigma)} - \frac{1}{E_m^q(\rho_1)\Delta(\rho_1)} \right\}. \tag{3.5} \]

Consequently, from (3.4) and (3.5) we find

\[ \lim_{\rho_1\to a_1} z^k \frac{\partial v}{\partial z} = -\,c_{n+1}^{-k}\times \]

\[ \times \frac{ \displaystyle \prod_{\nu=2}^{n+1}\left(c_{n+1}^{2}-\rho_{\nu}^{2}\right)^{\frac{k-1}{2}} }{ \displaystyle \prod_{\nu=2}^{n}\left(c_{n+1}^{2}-c_{\nu}^{2}\right)^{\frac{k-1}{2}} } \, \frac{ \displaystyle \prod_{\nu=2}^{n+1} E_m^q(\rho_\nu) }{ \displaystyle E_m^q(c_{n+1})F_m^q(c_{n+1}) \prod_{\nu=2}^{n}\left(c_{n+1}^{2}-c_{\nu}^{2}\right)^{\frac{1}{2}} } = \]

\[ = - \frac{ \displaystyle \prod_{\nu=2}^{n+1} E_m^q(\rho_\nu) \prod_{\nu=2}^{n+1}\left(c_{n+1}^{2}-\rho_{\nu}^{2}\right)^{\frac{k-1}{2}} }{ \displaystyle \prod_{\nu=2}^{n}\left(c_{n+1}^{2}-c_{\nu}^{2}\right)^{\frac{k-1}{2}} E_m^q(c_{n+1})F_m^q(c_{n+1}) } = \]

\[ = - Q_m^q(x)\, \frac{ \left(1-\dfrac{x^2}{a^2}\right)^{\frac{k-1}{2}} }{ aE_m^q(a_1)F_m^q(a_1) }. \]

In [2] it was noted that the function

\[ w(x,z)= \underbrace{\int \cdots \int}_{n\ \text{times}} \frac{f(\xi)\,d\xi}{ \left[(x-\xi)^2+z^2\right]^{\frac{n+k-1}{2}} }, \tag{3.6} \]

\[ 1-\frac{x^2}{a^2}\ge 0 \]

is continuous in the domain \(z\ge 0\), analytic in the domain \(z>0\), decreases at infinity as \(R^{-(n+k-1)}\), where \(R=\sqrt{x^2+z^2}\), and is a solution of the Darboux equation (0.1).

A relation expressing the source function was proved:

\[ f(x)=-\frac{1}{B}\lim_{z\to 0} z^k\,\frac{\partial v}{\partial z}, \qquad \text{where } B=2\pi^{\frac{n}{2}}\, \frac{ \Gamma\left(\dfrac{k+1}{2}\right) }{ \Gamma\left(\dfrac{n+k-1}{2}\right) }. \]

By virtue of the uniqueness of the solution of equation (0.1) possessing the indicated properties, we obtain \(w(x,z)=v(x,z)\). Therefore

\[ f(x)=\frac{1}{B}\,\frac{Q_m^q(x)}{a}\, \frac{ \left(1-\dfrac{x^2}{a^2}\right)^{\frac{k-1}{2}} }{ E_m^q(a_1)F_m^q(a_1) }. \]

Expressing the value \(v(x,0)\) in terms of \(f(x)\), we obtain an integral equation for \(Q_m^q(x)\):

\[ Q_m^q(x)= \frac{ \Gamma\left(\dfrac{n+k-1}{2}\right) }{ 2\pi^{\frac{n}{2}}\Gamma\left(\dfrac{k+1}{2}\right) E_m^q(a_1)F_m^q(a_1) } \times \]

\[ \times \underset{1-\frac{x^2}{a^2}\geqslant 0}{\underbrace{\int\cdots\int}_{n\ \text{times}}} \frac{\left(1-\frac{\xi^2}{a^2}\right)^{\frac{k-1}{2}}Q_m^q(\xi)\,d\xi} {\left[(x-\xi)^2\right]^{\frac{n+k-1}{2}}}. \tag{3.7} \]

Formula (3.7) gives, for example, the possibility of expanding the kernel \(r^{-\frac12(n+k-1)}\) in (3.6) in the polynomials \(Q_m^q(x)\).

4. For the Dirichlet problem for the Darboux equation posed at the beginning we have the following. Let the boundary value on the “disc” \(\rho_1=a_1\) be a polynomial \(f(x)\) of some degree \(m\) in the variables \(x_1,x_2,\ldots,x_n\); it can always be represented in the form of a finite linear combination of polynomials

\[ f(x)=\sum_{s=0}^{m}\sum_{i=1}^{N} h_s^{q_i} Q_s^{q_i}(x), \tag{4.1} \]

where \(N=C_{s+n-1}^{\,n-1}\). Each coefficient \(h_s^{q_i}\) can be found by multiplying both sides of (4.1) by

\[ Q_s^{q_i}(x)\left(1-\frac{x^2}{a^2}\right)^{\frac{k-1}{2}} \]

and subsequently integrating over the “disc” \(1-\frac{x^2}{a^2}\geqslant 0,\ x_{n+1}=0\), taking into account the orthogonality relation (2.9) and the normalization (2.10). If now we pass to ellipsoidal coordinates, using formulas (2.8) and (3.2), then it is not difficult to see that the formula

\[ V=\sum_{s=0}^{m}\sum_{i=1}^{N} h_s^{q_i} \frac{F_s^{q_i}(\rho_1)}{F_s^{q_i}(a_1)} \prod_{\nu=2}^{n+1} L_s^{q_i}(\rho_\nu) \tag{4.2} \]

gives an effective solution of the Dirichlet problem posed above.

In conclusion I express my deep gratitude to N. A. Rostovtsev for his guidance.

References

1. Rostovtsev N. A. PMM, 28, No. 1, 111–127, 1964.

2. Rostovtsev N. A.; Khranevskaya I. E. Evaluation of one \(n\)-fold integral. Theoretical Conference of Teachers of the Mathematical and Physical Departments of the Pedagogical Institute of the Far East. Ussuriisk, 1963.

3. Hobson E. V. The Theory of Spherical and Ellipsoidal Harmonics. IL, 1952, pp. 434–451.

4. Fichtenholz G. M. A Course of Differential and Integral Calculus, 3. Fizmatgiz, 1960, p. 397.

5. Szegő G. Orthogonal Polynomials. Fizmatgiz, 1962, p. 158.

Received by the editors
March 1, 1966

Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR