Role of functional equations in current panorama of pure mathematics

Question

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development of pure mathematics (also in relation to other branches).

Answer 1

The functional equation for the Riemann Zeta Function and its generalizations to $L$ functions is very important.

According to the Bohr-Mollerup theorem, a function $f$ which satisifes $f(1)=1$, $f(x+1)=xf(x)$ for $x>0$ and is logarithmically convex must be the Gamma function $\Gamma(x)$.

Somos Sequences are a set of sequence recurrence relations which seem to have profound relations to Jacobi theta functions, aztec diamonds and cluster algebras. They sometimes inexplicably produce integer valued sequences. Specifically, the octohedral recurrence:

$$f(n,i,j)f(n-2,i,j)=f(n-1,i-1,j)f(n-1,i+1,j)+f(n-1,i,j-1)f(n-1,i,j+1),$$

with appropriate boundary conditions comes up in a lot of places, one being Dodgson (Lewis Carroll) Condensation for determinants . Generalizations of this are sometimes called T-Systems.