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Re: Simplify by Recurrence Relations

• To: mathgroup at smc.vnet.net
• Subject: [mg75788] Re: Simplify by Recurrence Relations
• From: Norbert Marxer <marxer at mec.li>
• Date: Thu, 10 May 2007 05:23:08 -0400 (EDT)
• References: <f1s3n1$h2g$1@smc.vnet.net>

On 9 Mai, 11:19, Mr Ajit Sen <senr... at yahoo.co.uk> wrote:
> Dear Mathgroup,
>
>   Could someone please show me how to simplify a
>   function by using its recurrence relations.
>
>   As a simple example, let's take the Bessel
> recurrence
>   relation
>
>      BesselJ[n+1,z]= 2n/z BesselJ[n,z]-BesselJ[n-1,z].
>
>   How do I get Mathematica (5.2 !) to evaluate
>
>                D[BesselJ[2,x],x]
>
>   as   (1-4/x^2)BesselJ[1,x]+ 2/x BesselJ[0,x]
>
>   instead of (BesselJ[1,x]-BesselJ[3,x])/2 ?
>
>   [Basically, reduce the order to 0 &/or 1, so that
> all
>   J0 and J1 can be factored out later.]
>
>   Thanking you in advance.
>
>   Ajit.
>
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Hello

You could apply repeatedly (i.e. //.) the substitution for n>1

BesselJ[n_, z_] :> (2(n - 1)/z BesselJ[n - 1, z] - BesselJ[n - 2,
z]) /; n > 1

The following

D[BesselJ[2, x], x] //. BesselJ[n_, z_] :> (2(
    n - 1)/z BesselJ[n - 1, z] - BesselJ[n - 2, z]) /; n > 1 // Expand

will give (except factoring out) what you want, i.e.

(2*BesselJ[0, x])/x + BesselJ[1, x] - (4*BesselJ[1, x])/x^2

Best Regards
Norbert Marxer