Sequence transformations in proofs of irrationality of some fundamental constants

Transformation of number sequences (convergence acceleration) is one of the classical chapters of numerical analysis. These algorithms are used both for solution of practical problems and for the development of more advanced numerical methods. At the same time, numerical methods have found numerous applications in the number theory. One of the classical problems of number theory is the proof of irrationality of some fundamental constants, where the high rate of convergence of sequences of rational numbers plays a crucial role. However, as far as we know, the applications of (classical) convergence acceleration algorithms to the proofs of irrationality do not exist. This study is an attempt to fill this gap and to draw attention to this direction of research.