I have a complicated recursion replation, which I'm sure is unsolvable. (By "unsolvable" I mean that there is no closed form solution expressing [tex]\xi_1[/tex], [tex]\xi_2[/tex], [tex]\xi_3[/tex], etc. in terms of [tex]\xi_0[/tex].) It goes(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{(k+4)!}{k!}\xi_{k+4} +K_1 (k+2)(k+1)\xi_{k+2}+ [ K_2 k(k-1) +K_3] \xi_{k} +K_4 \xi_{k-2} =0, [/tex]

for [tex]k=0,1,2,3...[/tex], where [tex]K_1[/tex], [tex]K_2[/tex], [tex]K_3[/tex] and [tex]K_4[/tex] are constants. However, what I do know about the [tex]\xi_k[/tex]'s is that

[tex]\sum_{k_{even}} \xi_{k}=\sum_{k_{odd}} \xi_k=1,[/tex]

and that

[tex]\sum_{k_{even}}k \xi_{k}=\sum_{k_{odd}}k \xi_k=0.[/tex]

[tex]\xi_k[/tex] is also zero for all [tex]k<0[/tex]. What I want to do is produce an approximation of the sum for

[tex]S(z) =\sum_{k=0}^\infty \xi_k z^k.[/tex]

Does anyone have any idea how to do this?

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# Approximating unsolvable recursion relations

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