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

• To: mathgroup at smc.vnet.net
• Subject: [mg46693] Re: [mg46684] Factoring two-dimensional series expansions? (Ince polynomials again)
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 2 Mar 2004 00:14:16 -0500 (EST)
• References: <200402290816.DAA09127@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```AES/newspost wrote:
> 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] .

This is not quite what you want but may provide a good start.

Frederick W. Chapman. An elementary algorithm for the automatic
derivation and proof of tensor product identities via computer algebra.
Proceedings of the 2003 International Symposium on Symbolic and
Algebraic Computation (ISSAC 2003). J. R. Sendra, ed. 50-57. ACM Press.

It gives an algorithm for such factorizations when one begins with a
closed form function in two variables. The actual code used is fairly
short. There are numerous examples that may help to make clear how it
works. The main engine involves iterated limit computations.

The email for the author is listed as fwchapman at alumni.uwaterloo.ca.

A different approach might be to attempt an approxiamte numerical
factorization. I can give some pointers to the literature but so far as
I can discern this is a nontrivial area both from point of view of
coding and of getting reasonable results. But if the numerica
coefficients dies off very fast as your indices go to infinity then this
might be a viable direction to pursue.

Daniel Lichtblau
Wolfram Research

```

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