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From: Axel Vogt <&noreply@axelvogt.de>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: A simplification challenge.
Date: Fri, 12 Sep 2014 19:30:07 +0200
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On 11.09.2014 19:53, peter.luschny@gmail.com wrote:
> Once upon the time an old friend of Maple from the Crimea
> used to post simplification challenges to this group.
>
> In recollection of Vladimir today's simplification challenge:
>
> bi:=n->BesselI(4+n,-2); bk:=n->BesselK(4+n,2);
> i:=n->BesselI(n,2); k:=n->BesselK(n,2);
>
> Simplify
> a := n ->(bi(n)*(5*k(5)-k(6))+bk(n)*(5*i(5)+i(6)))/(i(6)*k(5)+i(5)*k(6));
>
> Peter
>
> P. S. Note seq(round(evalf(a(n), 99)), n=0..20);

Hm ... http://oeis.org/A058309 due to your hint: using the package 'gfun'
finds a recursion rec := {-u(n)+(-n-5)*u(n+1)+u(n+2), u(0)=0, u(1)=1},
feeding a for u and calling 'simplify' confirms it.

Maple can not solve it. Could it be that Janson's article (in the link)
gives a more simple expression for your a(n) ?