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+--- Thread: sandbox conjecture (/showthread.php?tid=662)
sandbox conjecture - tommy1729 - 06/13/2011
consider a(x) and b(x).
both functions are real-analytic.
both functions are entire.
both functions are strictly increasing on the reals.
a(0) > 0 , b(0) > 0
a(x) has exactly 2 positive real fixpoints : lower fixpoint A < 1 and upper fixpoint B > 1
b(x) has exactly 2 positive real fixpoints : lower fixpoints A < 1 and upper fixpoint B > 1
a(x) = b(x) has only 2 finite real solutions : their commen fixpoints.
** sandbox conjecture **
let c(x) = a(b(x)) and d(x) = b(a(x))
if sum k = 0 , oo (-1)^k a^k(1) has the same value as its corresponding matrix method - sum of elements in second line of 1/(1 + carleman(a(x))) ) -
and
if sum k = 0 , oo (-1)^k b^k(1) has the same value as its corresponding matrix method
and
if sum k = 0 , oo (-1)^k c^k(1) has the same value as its corresponding matrix method
then
sum k = 0 , oo (-1)^k d^k(1) has the same value as its corresponding matrix method.
tommy1729