Multivariate polynomials

While doing research, I came across some interesting multivariate versions of familiar orthogonal polynomials like Legendre or Chebyshev. In the end, they appear to not be very useful for my particular research problem. Still, I want to write a bit about what I found because they are kind of cute.

The generating function for the Legendre polynomials is \[\frac{1}{\sqrt{1-2x \beta + x^{2}}} = \sum_{n = 0}^{\infty} x^{n} P_{n}(\beta)\] while for Chebyshev polynomials we have \[\frac{1}{1-2x \beta + x^{2}} = \sum_{n = 0}^{\infty} x^{n} U_{n}(\beta)\] One can see that these are special cases of \[\frac{1}{\left(1 - 2 x \beta + x^{2}\right)^{\alpha}} = \sum_{n = 0}^{\infty} x^{n}C_{n}^{\hphantom{n}\alpha}(\beta)\] The $C_{n}^{\hphantom{n}\alpha}(\beta)$ are known as Gegenbauer or ultraspherical polynomials. All of these polynomials are defined for $|\beta| \leq 1$. The infinite sums converge only for $|x| < 1$.

The polynomials introduced above are univariate. That is, they depend on a single variable $\beta$. Legendre polynomials arise when one considers the inverse magnitude of the distance between two points in space: \[\frac{1}{|x - y|} = \frac{1}{\sqrt{x^{2} + y^{2} - 2 x \cdot y}} = \frac{1}{|x|}\sum_{n = 0}^{\infty} \xi^{n}P_{n}(\beta)\] where \[\xi \equiv \frac{|y|}{|x|} < 1 \qquad \beta \equiv \frac{x \cdot y}{|x| |y|} = \cos{\theta}\] Another instance when Legendre polynomials appear is when one considers the eikonal Coulomb potential. The Coulomb potential has the form: \[V(q) = \frac{\kappa}{|q|}\] and the eikonal trajectory is: \[q(u) = uX + x \quad \Longrightarrow \quad |q(u)| = \sqrt{u^{2} X^{2} + 2 u x \cdot X + x^{2}}\] So the eikonal Coulomb potential is \[V_{\mathcal{E}} = \frac{\kappa}{\sqrt{u^{2}X^{2} + 2 u x \cdot X + x^{2}}} = \frac{\kappa}{|x|}\sum_{n = 0} \xi^{n} P_{n}(\beta)\] where now \[\xi \equiv \frac{|X|}{|x|} u \qquad \beta = - \frac{x \cdot X}{|x| |X|}\qquad |u| \leq \frac{1}{2}\] During my research my adviser and I consider the magnitude of a non-local coordinate interval: \[d(t_{1}, \, t_{2}) \equiv |q_{1}(t_{1}) - q_{2}(t_{2})|\] In the eikonal approximation, this leads to a bi-variate polynomial of order two with three parameters. Each position function has the form \[q_{j}(u_{j}) = u_{j} X_{j} + x_{j}\] So then, \[d^{2}(u_{1}, \, u_{2}) = (u_{1}X_{1} - u_{2}X_{2})^{2} + 2 (u_{1}X_{1} - u_{2}X_{2}) \cdot (x_{1} - x_{2}) + (x_{1} - x_{2})^{2}\] The inverse of this non-local coordinate interval can be written as \[\frac{1}{d(u_{1}, \, u_{2})}= \frac{1}{|x_{1} - x_{2}|} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \xi_{1}^{m}\xi_{2}^{n} P_{mn}(\beta_{1}, \, \beta_{2}|\beta_{12})\] where \[\xi_{j} \equiv \frac{|X_{j}|}{|x_{1} - x_{2}|} u_{j} \qquad \beta_{1} \equiv - \frac{X_{1} \cdot (x_{1} - x_{2})}{|X_{1}| |x_{1} - x_{2}|} \qquad \beta_{2} \equiv \frac{X_{2} \cdot (x_{1} - x_{2})}{|X_{2}| |x_{1} - x_{2}|} \qquad \beta_{12} \equiv -\frac{X_{1} \cdot X_{2}}{|X_{1}||X_{2}|}\] It is easy to see that the $P_{mn}$ satisfy \[P_{m0}(\beta_{1}, \, \beta_{2}|\beta_{12}) = P_{m}(\beta_{1}) \qquad P_{0n}(\beta_{1}, \, \beta_{2}|\beta_{12}) = P_{n}(\beta_{2})\] \[m! n!P_{mn}(\beta, \, \beta|1) = (m+n)!P_{m+n}(\beta)\] \[P_{mn}(\beta_{1}, \, \beta_{2}|\beta_{12}) = P_{nm}(\beta_{2}, \, \beta_{1}|\beta_{12})\] Some non-trivial cases are \[P_{11} = -\beta_{12} + 3 \beta_{1} \beta_{2}\] \[P_{12} = \frac{3}{2} \left[-2 \beta_{12} \beta_{2} + \beta_{1}(-1 + 5 \beta_{2}^{2}) \right]\] \[P_{13} = \frac{1}{2} \left[-3 \beta_{12} (-1+5\beta_{2}^{2}) + 5 \beta_{1} \beta_{2} (-3 + 7\beta_{2}^{2}) \right]\] \[P_{22} = \frac{3}{4} \left[1 + 2 \beta_{12}^{2} - 20 \beta_{12} \beta_{1} \beta_{2} - 5(\beta_{1}^{2} + \beta_{2}^{2}) + 35 \beta_{1}^{2}\beta_{2}^{2} \right]\] \[P_{14} = \frac{5}{8} \left[ -4 \beta_{12} \beta_{2} (-3 + 7 \beta_{2}^{2}) + \beta_{1} (3 - 42 \beta_{2}^{2} + 63 \beta_{2}^{4}) \right]\] \[P_{23} = \frac{5}{4} \left[\beta_{2} (3 + 6 \beta_{12}^{2} - 7 \beta_{2}^{2})  + 21 \beta_{1}^{2}\beta_{2}(-1 + 3 \beta_{2}^{2}) - 6 \beta_{1} \beta_{12}(-1 + \beta_{2}^{2})\right]\] The $P_{mn}$ are non-trivial generalization of the univariate Legendre polynomials. We can study the multivariate analog of Gegenbauer polynomials by looking at the generating function \[\frac{1}{|q_{1}(t_{1}) - q_{2} (t_{2})|^{2\alpha}} = \frac{1}{|x_{1} - x_{2}|^{2\alpha}} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \xi_{1}^{m}\xi_{2}^{n} C_{mn}^{\hphantom{mn}\alpha}(\beta_{1}, \, \beta_{2}|\beta_{12})\] I would like to get some recursion relations for these things. Perhaps it is useful to work with \[L_{mn} = \frac{m! n!}{(m+n)!}P_{mn}\] and track down some sort of pattern.