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

• To: mathgroup at smc.vnet.net
• Subject: [mg46697] Re: Factoring two-dimensional series expansions? (Ince polynomials again)
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Tue, 2 Mar 2004 00:14:20 -0500 (EST)
• Organization: The University of Western Australia
• References: <c1s77b\$8vk\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <c1s77b\$8vk\$1 at smc.vnet.net>,
AES/newspost <siegman at stanford.edu> 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]

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 ...

> 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?

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.

> The question is motivated by some recent work in paraxial beam
> propagation

I assume you mean

Miguel A. Bandres and Julio C. Gutiérrez-Vega, 2004, OPTICS LETTERS
29(2):144-146

> 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 .

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} ]

> 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"

Not so mysterious. See

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

> 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] .

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)
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```

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