A simple continued fraction test for the irrationality of functions (Q1206741)

The following theorem is proved: Suppose that the continued fraction \(f(x)={\overset \infty {\underset {n=1} K}} {{a_ n(k)} \over {b_ n(x)}}\) converges to a finite value. The elements in \(f(x)\) are nonzero polynomials in a variable \(x>0\) with nonnegative coefficients. Suppose further that \(\{p_{2k-1}\}_{k\geq 1}\) is a sequence of integers such that \(\deg b_{2n} > p_{2n+1}\) and \(\deg b_{2n} > \deg a_{2n}+p_{2n-1}\) for \(n\geq 1\) with the existence of \(\lim_{x\to\infty} T_{2n-1}(x)x^{-p_{2n-1}}\neq 0\) for \(n\geq 1\), where \(T_ k(x)={\overset \infty {\underset {n=k} K}} {{a_ n(x)}\over {b_ n(x)}}\). Then \(f(x)\) is an irrational function of \(x>0\). An application of the theorem is also shown.