arXiv subjects
Publications and source records attributed to Hemant Bhate.
We establish some functional identities of theta functions, an elementary proof of classical fourth-order identities, Landen transformations, and q series from the eigenvectors of the discrete Fourier transform. Also, we derive connection between Rogers-Ramanujan type identity and theta function identity.
Eigenvectors of the discrete Fourier transform can be expressed using Ramanujan theta functions. New theta function identities, Ramanujan theta function identities, and generating functions for the quadratic numbers are a consequence.
We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and L functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki p group, Gupta-Fabrykowski group and BSV torsion-free group.
We investigate the Ihara zeta functions of finite Schreier graphs $\Gamma_n$ of the Basilica group. We show that $\Gamma_{1+n}$ is $2$ sheeted unramified normal covering of $\Gamma_n, ~\forall~ n \geq 1$ with Galois group $\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n > 1, r \geq 1$ the graph $\Gamma_{n+r}$ is $2^n$ sheeted unramified, non normal covering of $\Gamma_r.$ In order to do this we give the definition of the $generalized$ $replacement$ $product$ of Schreier graphs. We also show the corresponding results in zig zag product of Schreier graphs $\Gamma_n$ with a $4$ cycle.