Some recognizable forms of simple continued fractions

Abstract

An explicit form of Hurwitzian continued fraction whose quasi‐period is determined by polynomials of degree larger than 1 had been unknown. Here, we show a reduction formula that yields some recognizable expressions given the sequence of partial quotients in the simple continued fraction expansion. As an illustrations, we can obtain [0;12,22,32,42,…] = 1+124(1−1144+17259200+25706341150592000000−…)1+14+136 (1−1144+17259200−149233280000+25706341150592000000−…) and a number of other interesting results.