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

• To: mathgroup at smc.vnet.net
• Subject: [mg46684] Factoring two-dimensional series expansions? (Ince polynomials again)
• From: AES/newspost <siegman at stanford.edu>
• Date: Sun, 29 Feb 2004 03:16:32 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```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] .

```

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