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MathGroup Archive 2004

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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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