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) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul