Re: Known recursion link to Hermite polynomials not solved in Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg69583] Re: Known recursion link to Hermite polynomials not solved in Mathematica*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Sat, 16 Sep 2006 03:50:22 -0400 (EDT)*Organization*: The University of Western Australia*References*: <ee647q$5nb$1@smc.vnet.net> <eebd4g$l8m$1@smc.vnet.net> <eee0d6$6f3$1@smc.vnet.net>

In article <eee0d6$6f3$1 at smc.vnet.net>, Roger Bagula <rlbagula at sbcglobal.net> wrote: > With Bessel function recurrences you are allowed negative and rational > quantum > numbers, but not with HermiteH. What appears to happen is that the > normal integral function which is called the error function > behaves in a way to make this up for these negative or integral states. > The two types ( Bessel and Hermite-error function) seem to be mirrors > of each other except for this. This is not clear to me -- nor I expect to most readers of this group. > 1) The Hermite-error function type > a[n]=a0* a[n - 1] + (b0*n +c0) a[n - 2]: a0,b0,c0 Integers > 2) The Bessel type is: > a[n]=(a0*n+b0) a[n - 1] + c0 a[n - 2]: a0,b0,c0 Integers To be consistent, surely you should have a[n]=(b0*n+a0) a[n - 1] + c0 a[n - 2]: a0,b0,c0 Integers so that the relationship between the recurrences is clear. > The result is more a reflection of the limitations of the current > mathematical notation in Mathematica than the actual sequential > recursions. What limitation of mathematical notation in Mathematica? > It seems strange that the mathematics of probability ( which is widely > used in many fields) should be less solvable ( Using the RSolve level of > discrete functions) in Mathematica than that of > the Bessel function which isn't all that used in the real world. What on earth do you mean? The Bessel function appears everywhere in the real world! One nice example: the natural shape of a horn is a bessel function. See http://www-ccrma.stanford.edu/~jos/tiirts/Horn_Reflectance_Filter.html And what about waves in 2D, and normal modes of a drum? > As far as I know no one ( or can search) has noticed or reported this > type of recurrence or this symmetry property before. If you can more clearly describe the symmetry property you see, I expect that you will find it is well known ... Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul