Curious continued fractions

Abstract: 

There are relatively few transcendental numbers for which the continued fraction expansion is explicitly known. Here we present two new families of continued fractions for Engel series - sums of reciprocals - which arise from integer sequences generated by nonlinear recurrences with the Laurent property (one of the key features of Fomin and Zelevinsky's cluster algebras). Using the double exponential growth of the sequences, we show that the sum of such Engel series is transcendental. If time allows, we will make some remarks about continued fraction expansions in hyperelliptic function fields, and some related discrete integrable systems.