Some new families of Tasoevian and Hurwitzian continued fractions.

*(English)*Zbl 1158.11004Hurwitz continued fractions have been investigated by [D. N. Lehmer, Am. J. Math. 40, 375–390 (1918; JFM 46.0336.05), Scripta Math. 29, 17–24 (1973; Zbl 0263.10012)] and more recently by the reviewer in [Acta Arith. 107, 161–177 (2003; Zbl 1026.11012), Monatsh. Math. 145, 47–60 (2005; Zbl 1095.11008), Math. Pannon. 17, 91–110 (2006; Zbl 1121.11009), Czech. Math. J. 57, 919–932 (2007; Zbl 1163.11009), and Sarajevo J. Math. 4, 155–180 (2008)]. Tasoev continued fractions have been investigated by the reviewer in [Math. Proc. Cambridge Philos. Soc. 134, 1–12 (2003; Zbl 1053.11006), J. Number Theory 109, 27–40 (2004; Zbl 1082.11005)] and the above cited papers.

The author and Wyshinski derived several variations of both continued fractions from known results about \(q\)-continued fractions.

In this paper the author derives closed form expressions for several new classes of Hurwitz and Tasoev continued fractions including \[ [0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1}]_{n=0}^\infty, \] \([0;\overline{c+dm^n}]_{n=1}^\infty\) 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 Hurwitz and Tasoev continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. The author also derives expressions for arbitrarily long finite continued fractions whose partial quotients lie in arithmetic progressions.

The author and Wyshinski derived several variations of both continued fractions from known results about \(q\)-continued fractions.

In this paper the author derives closed form expressions for several new classes of Hurwitz and Tasoev continued fractions including \[ [0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1}]_{n=0}^\infty, \] \([0;\overline{c+dm^n}]_{n=1}^\infty\) 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 Hurwitz and Tasoev continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. The author also derives expressions for arbitrarily long finite continued fractions whose partial quotients lie in arithmetic progressions.

Reviewer: Takao Komatsu (Hirosaki)

##### MSC:

11A55 | Continued fractions |