06/13/2011, 05:47 AM

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

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