fibonacci like

[tommy1729]

\( f(z) = f(z - 1) + f(z + i) \)

If we assume a solution of the form \( f(z) = r^z \) exists, like in the Fibonacci numbers, we can get the equation

\( e^{(1+i)u} - e^u = -1 \)

and then the solutions for the functional equation are given by \( f(z) = e^{uz} \) for any \( u \)-value satisfying the above exponential equation. The functional equation is linear, so any linear combination of such solutions will be another solution, and since there are infinitely many such \( u \)-values, we can even consider infinite sums

\( f(z) = \sum_{n=0}^{\infty} C_n e^{u_n z} \)

with arbitrary \( C_n \), provided this sum converges. Since there are infinitely many constants \( C_n \), one could say the equation is like it has "infinitely many initial conditions".

Note that these may not be the only possible solutions -- remember that the very simple case \( f(z) = f(z-1) \) has all 1-periodic functions as solutions. Not sure what the appropriate analogy is here. But the above could be thought of as a sort of "canonical" solution like how Binet's formula solves the Fibonacci numbers.

nice post mike !

i conjecture that all non-periodic entire function solutions are of this form.

note that i didnt say anything about the periodic solutions , not even sure they exist.

i wonder if elliptic solutions exist. (once again flirting with double periodic functions )

beautiful memories ; as an early teenager i defined 2 classes of functions as

\( f(z) = \sum_{n=0}^{\infty} C_n e^{G z} \)

\( f(z) = \sum_{n=0}^{\infty} D_n e^{E z} \)

and assumed them to be equivalent , where G are the gaussian integers and E are the eisenstein integers.

these were my " pre - taylor " series before i learned about taylor or laurent or even kahn series.

similarly my gaussian / eisenstein polynomials were :

\( f(z) = \sum_{n=0}^{A} C_n e^{G z} \)

\( f(z) = \sum_{n=0}^{B} D_n e^{E z} \)

for positive integer A and B , before i learned about polynomials or signomials.

i did some investigations which could be considered pre-galois theory , pre-abelian variety , multisections and searching for zero's.

and a lot of modular arithmetic , which seemed related.

( and abelian groups of order p^2 of course )

even today , i still find all that intresting.

maybe mike has a similar history ?

later i switched to number theory , but partially never forgot that.

( 'partially' because apparantly i forgot the relation to " fibonacci like " , i did find that fibo equation in my old papers of " gaussian polynomials " )

sorry for the emo.

regards

tommy1729