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This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations 
$af(z^{r})=f(z)+R(z)$ , where 
$a$ is a nonzero complex number, 
$r$ an integer greater than 1, and 
$R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.
We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field 
$\mathbb{I}$ , in particular of the values of 
$q$ -exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincaré-type equations and the connection between the solutions of these functional equations and the generalized Heine series.
Using Padé approximations of the q-logarithmic series 
we obtain new approximation measures for values of q-logarithms and for the series 
, k = 1, 2, …, where (un) is a recurrence sequence, satisfying
Let
where λ is rational and not an integer. The author investigates lower estimates for example for
where the αi are distinct rational numbers not 0, and where x1, …, xk, are integers and 
