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On the irrationality of generalized \(q\)-logarithm. (English) Zbl 1415.11099
Summary: For integer \(p, | p| >1\), and generic rational \(x\) and \(z\), we establish the irrationality of the series \[ \ell _p(x,z)=x\sum _{n=1}^\infty \frac{z^n}{p^n-x}. \] It is a symmetric (\(\ell _p(x,z)=\ell _p(z,x)\)) generalization of the \(q\)-logarithmic function (\(x=1\) and \(p=1/q\) where \(| q|<1\)), which in turn generalizes the \(q\)-harmonic series (\(x=z=1\)). Our proof makes use of the Hankel determinants built on the Padé approximations to \(\ell _p(x,z)\).

MSC:
11J72 Irrationality; linear independence over a field
11C20 Matrices, determinants in number theory
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
41A20 Approximation by rational functions
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