Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences
Let \(\{R_k\}_{k\geq 1}\) be a linear recurrence of positive integers satisfying \[ R_{k+n}=c_1 R_{k+n-1}+\dots+c_n R_k\quad(k\geq 1), \] where \(n\geq 2\) and \(c_1,\dots,c_n\) are nonnegative integers with \(c_n\neq 0\). Let \[ \Theta(x,a,q)=\sum_{k=1}^\infty\frac{x^k q^{R_1+R_2+\dots+R_k}}{(1-a q^{R_1})(1-a q^{R_2})\dots(1-a q^{R_k})}=\sum_{k=1}^\infty\prod_{l=1}^k\frac{x q^{R_l}}{1-a q^{R_l}}, \] which is an analogue of a sort of \(q\)-hypergeometric series \[ \sum_{k=1}^\infty\frac{(-1)^k q^{k(k-1)/2}x^k}{(1-a)(1-a q)\dots(1-a q^{k-1})}. \] In this paper, the author gives a necessary and sufficient condition for the values \(\Theta(x,a,q)\) at any distinct algebraic points to be algebraically independent, using Mahler's method. In addition, if \(\gcd(R_{k+1}-R_k,R_{k+2}-R_{k+1},\dots,R_{k+n}-R_{k+n-1})=1\), then \(\Theta(x,a,q)\) takes algebraically independent values for any distinct triples \((x,a,q)\) of nonzero algebraic numbers with \(|a|\leq 1\) and \(|q|<1\). Moreover, \(\Theta(a,a,q)\) is expressed as an nonregular continued fraction \[ \frac{a q^{R_1}}{1-a q^{R_1}}{\atop+}\frac{-a q^{R_2}(1-a q^{R_1})}{1}{\atop+}\frac{-a q^{R_3}(1-a q^{R_2})}{1}{\atop+\dots+}\frac{-a q^{R_n}(1-a q^{R_{n-1}})}{1}{\atop+\dots} \] and \(\Theta(x,1,q)\) is an analogue of \(q\)-exponential function \[ 1+\sum_{k=1}^\infty\frac{x^k q^{1+2+\dots+k}}{(1-q)(1-q^2)\dots(1-q^k)}. \]
scientific article; zbMATH DE number 5624862