[seqfan] Functional Equation -F(x)/F(-x) = exp(x)

Paul D Hanna pauldhanna at juno.com
Sat Jul 23 19:14:48 CEST 2011

     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? 

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