← back to Research## Self-similar polynomials

### A personal history of the thesis work

Why are the polynomials called self-similar polynomials? Because they are self-similar: Compare seed functions F1,F2, F3 and generated functions F4, F5, F6 on a different scale. I uncovered this mathematics from a messy electronic problem of the Fibonacci quasicristal, which is self-similar. The polynomials are the quasiperiodic extension of the Chebyshev polynomials of the first kind.

When I was a freshman and a sophomore, I studied parts of Landau's the classical theory of fields with Prof. K. Kwarabayashi, again a part of Pauli's theory of relativity with Prof. T. Izuyama, and a small part of Whittaker's a treatise on the analytical dynamics of particles and rigid bodies with Prof. Y. Takahashi, the director of Research Institute for Mathematical Sciences, Kyoto Univ. 2003. I learned many things such as the motion of an electron in both electric and magnetic fields, the group calculation of the Lorentz transformations, and a solution using Frenet-Serret's formula. When I was a junior, I studied Helgason's differential geometry and symmetric spaces with Prof. K. Yajima and London's macroscopic theory of superconductivity with Prof. T. Masumi and a number of good textbooks with good friends. I cleary found that I do hate abstract math. Now I use a part of differential geometry things, but all things are practical.

When I was a graduate student seeking something to do, Prof. Takahashi gave me preprints about a one-dimensional Fibonacci quasicrystal and taught me the concept of the Fibonacci numbers on Lie groups, which led to my thesis work. His contribution to M. Kohmoto and Y. Ohno's paper is essential and the paper contains every story, and therefore I would say that other numirical papers by many authors at that time seemed to me nowhere to go. I did not like numerical studies and escaped from just numerical sort of things. I wanted to create something, but had no hope and any confidence to become a resercher. Therefore my thesis work was intended to survive in history, not in my lifetime (Recent development as dynamical systems such as Prof. K. Niizeki and co-waorkers appears to be nice).

Dr. T. k-.Suzuki found the self-similar polynomials of the second kind and got Doctor's degree by the work in 1992. I was happy to hear that he became an associate professor at Center for Nano Materials and Technology, Japan Advanced Institute of Science and Technology. He was my great disciple and I was his little master.