Arithmetic-geometric mean sequences over finite fields ${\mathbb{F}}_q$, where $q\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$

Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields ${\mathbb{F}}_q$ such that $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. In this paper, we extend the definition of arithmetic-geometric mean sequences over ${\mathbb{F}}_q$ such that $q\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$. We explain the connection of these sequences with graphs and show the properties of the corresponding graphs in the case $q\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$.