# Factoring two-dimensional series expansions? (Ince polynomials again)

Discussion in 'Mathematica' started by AES/newspost, Feb 29, 2004.

1. ### AES/newspostGuest

This is a math question rather than a Mathematica question, but anyway:

Suppose I have a function f[x,y] that's a power series expansion in
factors x^m y^n , that is,

(1) f[x, y] = Sum[ a[m,n] x^m y^n, {m, 0, mm}, {n, 0, mm} ]

with known a[m,n] coefficients

Are there algorithmic procedures for factoring this function
(analytically or numerically) into a simple product of power series or
simple folynomials in x and y separately, i.e.,

(2) f[x ,y] = fx[x] fy[x]

or maybe

(3) f[z1, z2] = fz1[z1] fz2[z2]

where z1 and z2 are linear combinations of x and y ?

Or more realistically there tests for *when* or whether the original
function can be so factored?

The question is motivated by some recent work in paraxial beam
propagation in which the function f[x,y] is actually the sum of
Hermitian polynomials, call 'em h[m,x] and h[n,y] for brevity, with
expansion coefficients b[m,n], i.e.

(4) f[x, y] = Sum[ b[m,n] h[m,x] h[n,y], {m, 0, mm}, {n, 0, mm} ]

where the coefficients b[m,n] can be arbitrary but there is a special
constraint that m + n = a constant integer p .

Apparently this expansion can be factored into a product like (3) where
the functions fz1{z1} and fz2[z2] are both some kind of mysterious
"Ince polynomials" and the variables z1 and z2 are elliptical
coordinates in the x,y plane, with the elliptical coordinate system
vasrying with the choice of the coefficients b[m,n] .

AES/newspost, Feb 29, 2004

2. ### Paul AbbottGuest

Well, given a[m,n] and mm, Factor will do this.

Clearly, if a[m,n] = b[m] c[n], then the sum is separable. Further if
(2) holds then you can work out for yourself the relationship between
the a[m,n] and the expansion coefficients in fx and fy ...
In this context, I think you are asking a group theoretical question.
The papers by Miller et al. <http://www.ima.umn.edu/~miller/bibli.html>,
especially

Lie theory and separation of variables. VII. The Harmonic oscillator
in elliptic coordinates and Ince polynomials, with C.P. Boyer and E.G.
Kalnins. J. Math. Phys. 16 (1975), pp. 512-517.

is relevant.
I assume you mean

Miguel A. Bandres and Julio C. Gutiérrez-Vega, 2004, OPTICS LETTERS
29(2):144-146
Then why don't you just reduce the double sum to a single sum,

(4) f[x, y] = Sum[ b[m] h[m,x] h[p-m,y], {m, 0, mm} ]
Not so mysterious. See

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964)
Cheers,
Paul

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Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
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Paul Abbott, Mar 2, 2004