Hankel Determinants Of Cantor Sequence And Its Difference Sequences

In mathematics and other fields, substitution sequences is playing a more and moreimportant role. This makes people giving more interest on them and getting lots of results. Inthe field of fractal geometry, physics, number theory, portfolio analysis and ergodic theory,the application of the substitution sequences have achieved great success. After Allouche,Peyrie`re, Wen and Wen[30]got the Hankel determinants of the Thue-Morse sequence is2-automatic,Kamae, Tamura and Wen[31]calculated the Hankel determinants of the Fibonaccisequence.Wen and Wu[33]got the Hankel determinants of the Cantor sequence. And Wenand Guo[34]got the Hankel determinants of all the difference sequences of the Thue-Morsesequence.In this paper, we will study the Hankel determinants of the fixed point of the sub-stitution σ: aâ†'aba, bâ†'bbb and its difference sequences,which is also called Cantorsequence. After introduced the Cantor sequence, its difference sequences and the Hankeldeterminants of a sequence, we estimate the Hankel determinant of these sequences step bystep.Firstly, we get the Hankel determinants of0,1,2order difference sequences, as abasis for calculating Hankel determinants of higher order difference sequences; and thenwe derive the iterative formulas of the high order difference and low order differencesequences;finally, by using the Hankel determinants of low order difference sequencesand the iterative formulas of the high order difference and low order difference, we getall Hankel determinants of the difference sequences. After getting these results, wegive some discussions on the periodicity of the Hankel determinants and get that as athree-dimensional sequences,{|Γk,pn|}n≥0,p≥0,k≥0(mod3) is periodic.