Geometry of Two-Variable Legendre Polynomials

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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.

Contributed by: Marcello Artioli and Giuseppe Dattoli (January 2016)
Open content licensed under CC BY-NC-SA



Two-variable Legendre polynomials [1] are defined by the generating function

They reduce to the ordinary Legendre polynomials [2] after the substitution

The Laplace transform identity

casts the RHS of equation 1 into the form

by setting and in equation 2. Recalling [3] 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

we get

Comparing 9 with 1, we find finally

By following the same criterion as in [3], 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 [3] 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.


[1] D. Babusci, G. Dattoli, and M. Del Franco, Lectures on Mathematical Methods for Physics, Rome: ENEA, 2011. 58_ENEA.pdf.

[2] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, New York: MacMillan and Co., 1985.

[3] M. Artioli and G. Dattoli, "The Geometry of Hermite Polynomials," Wolfram Demonstrations Project, 2015.

[4] M. Artioli and G. Dattoli, "Geometric Properties of Generalized Hermite Polynomials," Wolfram Demonstrations Project, 2015.

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