Specializability of continued fractions of infinite series involving some recurrence sequences

Abstract

In this paper, we describe the continued fraction expansion for the series $\sum_{k}T^{-a(k)}$, where $(a(n))_{n}$ is the A248098 sequence, or the A213967 sequence in the OEIS, defined by the same recurrence formula $a(n)=1+ a(n-1)+a(n-2) +a(n-3)$ for $n\geq3$ with different initial conditions. We prove that these two series are specializable and converge to transcendental numbers when we specialize $T$ by an even integer $\geq4$ for the first one and by any integer $\geq2$ for the second one. We also describe the continued fraction expansion for the series $\sum_{k}T^{-\mathcal{L}_{2,k}}$, where $(\mathcal{L}_{2,k})_{k}$ is the sequence of modified Leonardo $2$-number. We prove that it is specializable and converges to a transcendental number when we specialize $T$ by an integer $\geq2$. We then show that the arguments suggested by van der Poorten and Shallit in \cite{PR-SH}, that the phenomenon of specializability for series of the kind that they studied, is not reserved to just this kind, but it is included in a large class of series.