Introduction to the determinantal formulae for the Levin-type algorithms

There are two aims of this paper. Firstly, we shall introduce the determinantal formulae for the new Levin-type algorithms and secondly we shall demonstrate the similarities between the new Levin-type algorithms and the well established methods, namely, the Aitken Δ2 algorithm, the Brezinski theta algorithm and the Lubkin transformation. Actually, there are two groups of similarities. First of the group demonstrates the similarity between the super enhanced Levin algorithm, the super modified Levin algorithm, the Brezinski theta algorithm and the Lubkin transformation. The second group of similarity is between the modified Levin algorithm, the original Levin transformation and the Aitken Δ2 algorithm.