Mahler's method gives algebraic independence results for the values of functions of several variables satisfying certain functional equations under the transformations of the variables represented as a kind of the multiplicative action of matrices with integral entries. In the Mahler's method, the entries of those matrices must be nonnegative; however, in the special case stated in this paper, one can admit those matrices to have a negative entry. We show the algebraic independence of the values of certain functions satisfying functional equations under the transformation represented by such matrices, expressing those values as linear combinations of the values of ordinary Mahler functions.
"Arithmetic Properties of Solutions of Certain Functional Equations with Transformations Represented by Matrices Including a Negative Entry." Tokyo J. Math. 37 (1) 211 - 223, June 2014. https://doi.org/10.3836/tjm/1406552440