On integrality and asymptotic behavior of the $(k, l)$-G\"{o}bel sequences

Abstract

Recently, Matsuhira, Matsusaka, and Tsuchida revisited old studies on the integrality of $k$-G\"{o}bel sequences and showed that the first 19 terms are always integers for any integer $k\ge 2$. In this article, we further explore two topics: Ibstedt's $(k,l)$-G\"{o}bel sequences and Zagier's asymptotic formula for the $2$-G\"{o}bel sequence, and extend their results.