Geometry of Two-Variable Associated 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.
As the two-variable Legendre polynomials  can defined by the generating function 
They reduce (Snapshot 1) to the two-variable 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 generalized two-variable Hermite–Kampé de Fériet (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
Applying similar simplification used in , equation 4 also implies that
Comparing 8 with 1, we find finally
By following the same criterion as in , we show the two-variable associated 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.
Also, the operators defined in  can be generalized in the same way; we end up with
Accordingly, the operator transforms an ordinary monomial (when is 0) into a associated 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.
 M. Artioli and G. Dattoli. "Geometry of Two-Variable Legendre Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometryOfTwoVariableLegendrePolynomials.
 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, 1985.
 M. Artioli and G. Dattoli. "Geometric Properties of Generalized Hermite Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometricPropertiesOfGeneralizedHermitePolynomials.