Tommy's Gamma trick ?

[tommy1729]

Let f be continu and real and satisfy :

For x = Y


\( f(1) = 1 \) (1)

\( f ' (x) = \Pi_1^Y f(x) \) (2)

\( f(2) = e \)

For real \( 1 < Y < 2 \)

1) does that imply f is C^oo in \( x E [1,Y] \) ?

2) does that imply f is analytic in \( [1,Y] \) ?

3) is the solution Unique ? If not, how many variables are there ?

4) how to prove existance ?

5) does it follow \( f ' (2x) = \Pi_1^{2Y} f(x) \) ?

6) is the equation \( f ' (x) = exp[ \Sigma 1^Y f(x-1) ] \) equivalent ?

7) does it Neccessarily follow \( f(x+1) = exp {f(x)} \) for x>1 ?

8 ) If f is analytic on \( [1,Y] \) does that imply it is analytic in \( [2,1+Y] \) ?

Regards

Tommy1729

Some answers that I can prove


1) yes !

Proof sketch : C^0 -> C^1 because the product operator is C^0.
C^0^[2] = C^1.

C^1 -> C^oo by induction.

2) maybe. Need to consider the rate of the nth derivative.

3) if f(x) is a solution then f(x) + periodic is not.
The gamma method removes the + periodic part.

Partial answer

5) yes. 7) yes . 8 ) yes.
Easy.

Regards

Tommy1729