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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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