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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).

Martin Sleziak
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Dal
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  • From my youth, when I dealt with functional equations at math competitions, I still have a humble, but experienced and opposite opinion: in most cases they are not deep and can be easily solved by some manipulations. Exclusions are very rare (see, for instance, Discrete Liouville's Theorem). – Alex Ravsky Feb 01 '15 at 08:26

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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.

Alex R.
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  • For Somos sequences, the word "bizarre" is not what I would use. They came to me after years of studying addition theorems of Jacobi theta functions. They exhibit simple examples of the important "Laurent phenomenon". – Somos Jul 19 '18 at 19:28
  • @Somos: Point taken! I really only meant to emphasize the "profound" part, as the sequences seem to be appearing more and more in tiling combinatorics and integrable systems. – Alex R. Jul 19 '18 at 21:24