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

Discussion in 'Mathematica' started by AES/newspost, Feb 29, 2004.

  1. AES/newspost

    AES/newspost Guest

    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] .
    AES/newspost, Feb 29, 2004
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  2. AES/newspost

    Paul Abbott Guest

    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 ...
    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>,

    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.
    I assume you mean

    Miguel A. Bandres and Julio C. Gutiérrez-Vega, 2004, OPTICS LETTERS
    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} ]
    Not so mysterious. See

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

    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)
    35 Stirling Highway
    Crawley WA 6009 mailto:p
    AUSTRALIA http://physics.uwa.edu.au/~paul
    Paul Abbott, Mar 2, 2004
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