# Some new Families of Tasoevian- and Hurwitzian Continued Fractions

## TL;DR

This paper introduces new classes of Hurwitzian and Tasoevian continued fractions, providing closed-form expressions, methods for constructing long quasi-periodic fractions, and formulas for finite fractions with arithmetic progression partial quotients.

## Contribution

It presents novel closed-form formulas for several new families of Hurwitzian and Tasoevian continued fractions, including methods for generating long quasi-periodic fractions with arbitrary parameters.

## Key findings

- Derived closed-form expressions for new continued fraction classes.
- Developed iterative constructions for long quasi-periodic fractions.
- Provided formulas for finite fractions with arithmetic progression partial quotients.

## Abstract

We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and $[0; \overline{e u^{n}, f v^{ n}}]_{n=1}^\infty$. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzian- and Tasoevian continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. We also derive expressions for arbitrarily long \emph{finite} continued fractions whose partial quotients lie in arithmetic progressions.