[seqfan] Re: Functional Equation -F(x)/F(-x) = exp(x)
Alexander P-sky
apovolot at gmail.com
Sat Jul 23 19:34:51 CEST 2011
It looks like A193341 is not even in the draft folder ready for review
yet - it is only listed as "allocated" so far - just FYI
I am curious (just curious) why 5! is not there in 3) while 1!, 2!,
3!, 4! and 6!,7!,8!,9!,10! and ... are
Regards,
ARP
On 7/23/11, Paul D Hanna <pauldhanna at juno.com> wrote:
> SeqFans,
> Consider the functional equation:
> (1) -F(x)/F(-x) = exp(x),
> for which there are an infinite number of solutions.
> I would like to know the functions that satisfy (1)!
>
> Also consider the infinite family of functions that satisfy:
> (2) G(-G(-x)) = x.
>
> I was pleasantly surprised to find that both of these conditions are
> satisfied by the e.g.f. of a new sequence A193341, defined by:
> (3) A(A(x)) = x*exp(A(x))
> where A(x) begins:
> A(x) = x + 2*x^2/(2!*2) + 6*x^3/(3!*4) + 16*x^4/(4!*8) - 144*x^6/(6!*32) +
> 5488*x^7/(7!*64) + 47104*x^8/(8!*128) - 2799360*x^9/(9!*256) -
> 29427200*x^10/(10!*512) +...
>
> Now, given any even function B(x) = B(-x), the product A(x)*B(x) will also
> satisfy (1) -F(x)/F(-x) = exp(x); this is trivial.
>
> Can anyone find another solution to (1) -F(x)/F(-x) = exp(x) that is not
> related to A(x) (defined by (3)) in a trivial manner?
>
> Thanks,
> Paul
>
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