Geometry of Two-Variable Legendre Polynomials
On the left is a three-dimensional plot of a Legendre polynomial in two variables and ; on the right is a two-dimensional plot of the surface cut by a plane perpendicular to the axis. The exponential operator transform is defined in the Details.
Two-variable Legendre polynomials  are defined by the generating function
They reduce to the ordinary Legendre polynomials  after the substitution
The Laplace transform identity
casts the RHS of equation 1 into the form
by setting and in equation 2. Recalling  that the two-variable Hermite–Kampé dé Fériét (H-KdF) polynomials
are generated through
we find that
and thus from equations 3 and 5 we get
From the following property of the H-KdF polynomials,
equation 6 becomes
Equation 5 also implies that
and the previous expression can be replaced in equation 7, giving
With the following simplifications of the RHS of equation 8,
and the definition of Euler's gamma function
Comparing 9 with 1, we find finally
By following the same criterion as in , we show the two-variable Legendre polynomials in a three-dimensional plot, displaying the relevant geometrical structure, and we have specified the polynomials determined by the intersection with a plane moving along the axis and parallel to the , plane.
Let us now note that the use of the operational identity
Also by recalling  that
we end up with
Accordingly, the operator transforms an ordinary monomial (when is 0) into a Legendre polynomial (when is not 0), and the plots represent the relevant geometrical interpretation. The monomial-polynomial evolution is shown by moving the cutting plane orthogonal to the axis: for a specific value of the polynomial degree, the polynomials lie on the cutting plane, as shown in the snapshots.
 D. Babusci, G. Dattoli, and M. Del Franco, Lectures on Mathematical Methods for Physics, Rome: ENEA, 2011. opac22.bologna.enea.it/RT/2010/2010_ 58_ENEA.pdf.
 L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, New York: MacMillan and Co., 1985.
 M. Artioli and G. Dattoli, "The Geometry of Hermite Polynomials," Wolfram Demonstrations Project, 2015. demonstrations.wolfram.com/TheGeometryOfHermitePolynomials.
 M. Artioli and G. Dattoli, "Geometric Properties of Generalized Hermite Polynomials," Wolfram Demonstrations Project, 2015. demonstrations.wolfram.com/GeometricPropertiesOfGeneralizedHermitePolynomials.