On the unboundedness of the coefficients of power series ∏∞n=0(1-x2n)m
Let ∏∞n=0(1-x2n)be the generating function of the Prouhet-Thue-Morse sequence.Let Fm(x)=(F(x))m=(∏∞n=0(1-x2n))m≔∑∞n=0tm(n)xn.In 2018,Gawron,Miska and Ulas proposed a conjecture on the unboundedness of the sequence{tm(n)}∞n=1 for m≥2.They also proved this conjecture for m=3 and m=2k by studying the 2-adic of tm(n),where k is a positive integer.In this paper,we intro-duce a new method for this conjecture.In this method,a class of anti-centrosymmetric matrices are firstly ob-tained by studying the recursive relation of tm(n).Then the conjecture may be proved by calculating the ei-genvalues of the matrices.In particular,we prove the conjecture for m=5 and 6 by presenting unbounded sub-sequences of{t5(n)}∞n=1 and{t6(n)}∞n=1.Meanwhile,we also partially prove another conjecture on the 2-adic values of t5(n)by calculating the 2-adic value of a sub-sequence of{t5(n)}∞n=1.
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