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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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Crawley WA 6009 mailto:paul at physics.uwa.edu.au
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