Even and odd parts of limit periodic continued fractions
The limit periodic continued fraction \(K(a_ n/1)\), \(a_ n\to a\in {\hat {\mathbb{C}}}={\mathbb{C}}\cup \{\infty \}\) converges if \(a\not\in [-\infty,- (1/4]\) in which case the even and odd parts of \(K(a_ n/1)\) also converge and to the same value as does \(K(a_ n/1)\). It is shown that if \(a_ n\to \infty\) so that \(a_{n+1}/a_ n\to 1\) and \(1+a_ 2\neq 0\neq 1+a_ n+a_{n+1}\), \(n=2,3,4,...\), then in the even and odd parts \(K(c_ n/1)\) and \(K(d_ n/1)\), one has \(c_ n\to -(1/4)\) and \(d_ n\to -(1/4)\). Applying convergence or divergence criteria for \(K(c_ n/1)\) or \(K(d_ n/1)\) one can conclude convergence or divergence of \(K(a_ n/1).\) Continued fractions of these types appear in connection with hypergeometric functions and are therefore of interest. A number of examples are considered.
scientific article; zbMATH DE number 3975290
scientific article