A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

In 1998, Allouche, Peyri\`ere, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.


Introduction
Let C(z) be a power series in one variable with rational coefficients, be the (p, k)-order Hankel determinant associated to C(z). For simplicity, we write H k (C) instead of H 0 k (C). The study of the non-vanishing of (p, k)-order Hankel determinants is an interesting question on its own, and this is the purpose of the present paper for the sequence (c k ) k≥0 being the Thue-Morse sequence. However, we start the introduction by pointing out a motivation coming from Diophantine approximation and concerning the study of rational approximation to the real numbers C(1/b), where b ≥ 2 is an integer.
Let ξ be an irrational, real number. The irrationality exponent µ(ξ) of ξ is the supremum of the real numbers µ such that the inequality has infinitely many solutions in rational numbers p/q. It follows from the theory of continued fractions that µ(ξ) is always greater than or equal to 2, and an easy covering argument shows that µ(ξ) is equal to 2 for almost all real numbers ξ (with respect to the Lebesgue measure). Furthermore, Roth's theorem asserts that the irrationality exponent of every algebraic irrational number is equal to 2. The reader is directed to the monograph [5] for proofs and refinements of these assertions. It is in general a very difficult problem to determine the irrationality exponent of a given transcendental real number ξ, unless ξ is given by its continued fraction expansion. Apart from more or less ad hoc constructions, there are only very few examples of transcendental numbers ξ whose irrationality exponent is known. When they can be applied, the current techniques allow us most often only to get an upper bound for µ(ξ).
Recently, Bugeaud [2] developed a method for computing µ(ξ) when ξ is a Thue-Morse-Mahler number. Let be the generating function of (t k ) k≥0 . It is proved in [2] that, for every integer b ≥ 2, the irrationality exponent of the real number is equal to 2. There are two main ingredients in the proof. A first one is the fact that T (z) satisfies a functional equation, namely a key tool in Mahler's proof [4] that T (1/b) is transcendental. A second one is the non-vanishing of Hankel determinants associated with t, a result established by Allouche, Peyrière, Wen and Wen [1].
Theorem APWW. For every positive integer k, the Hankel determinant H k (T ) is nonzero.
The proof given in [1] is long and difficult. It depends on a set of sixteen recurrence relations involving the (p, k)-order Hankel determinants and gives additional results on the values of the Hankel determinants H p k (T ). Subsequently, Coons [3] considered the functions (1.1) He proved that, for every integer b ≥ 2, the irrationality exponent of F (1/b) and G(1/b) is equal to 2. To this end, he followed the method of [2], replacing the use of Theorem APWW by that of the next result (Theorem 2 of [3]).
Theorem C. For every positive integer k, the Hankel determinants H 1 k (F ) and H 1 k (G) are nonzero.
Coons' proof of Theorem C is of the same level of difficulty as the one of Theorem APWW. It is long and hard to follow.
The aim of this note is to provide a unified, combinatorial proof of both Theorems APWW and C. We believe that our approach is much simpler than that of [1,3].
Our paper is organized as follows. The key combinatorial result, namely Theorem J, is stated in Section 2, along with three equivalent lemmas. Complete proofs of these lemmas and theorem are given in Section 3. We gather in Section 4 some additional statements, which follow from our approach. Then, in Section 5, we show how Theorems APWW and C can be easily derived from Theorem J.
When nothing else is specified, the notation a ≡ b means that the integers a and b are congruent modulo 2.

Permutations and involutions
Throughout this text, N denotes the set of non-negative integers. We introduce the sets (J1) For every integer m ≥ 1, the number of permutations σ ∈ Sym m such that i + σ(i) ∈ J for i = 0, 1, . . . , m − 1, is an odd number.
The proof is based on some combinatorial techniques. Since we want to enumerate permutations modulo 2, we can delete suitable pairs of permutations and the result will not be changed. The problem is then how to associate a given permutation with another to form a pair. Two methods are used in the present paper: (1) taking the inverse σ −1 of a given permutation σ; (2) exchanging two values by letting σ(i) := σ(j) and σ(j) := σ(i).
Those two methods are fully described in the proof of Theorem (J1).
We choose to separate the elements of a cycle by commas. Sometimes we write a list of sets under the two-lines representations. If the set A is under the index i, this means that i + σ(i) ∈ A. For example, the permutations in (J1) and (J2) are   0 1 2 3 4 5 6 7 8 · · · · · · · · · J J J J J J J J J   and   0 1 2 3 4 5 6 7 8 · · · · · · · · · J J J J J J J J N   respectively.
An involution is a permutation σ such that σ = σ −1 . In the cycle representation of an involution every cycle is either a fixed point (b) or a transposition (c, d). For every set A, a transposition (c, d) is said to be an A-transposition if c + d ∈ A and c + d is odd. This means also that there is an even number and an odd number in every A-transposition. Generally the order of the two numbers in a transposition does not matter. However, throughout this paper, we always write the even number before the odd number in every A-transposition. We state below three equivalent lemmas. The first (resp. second, third) assertion of any of these is equivalent to the first (resp. second, third) assertion of any of the other two lemmas.
We define three transformations: The transformation β is extended to the involutions σ on N | m such that all transpositions are K-transpositions, by applying β on every number in the cycle representation of σ. The transformation β for involutions is reversible, even though β on N is not reversible.
We do not know a priori whether the fixed point 3 is obtained from 6 or from 7. We must look at the transposition (3, 1) first. It is obtained from the permutation (6, 3) since we know that an even number is always before an odd number in the transposition. Thus, we can recover the K-transpositions (6, 3)(0, 5)(8, 1). All the other numbers are fixed points, so these are (7)(2)(4).
In the same way, the transformation γ is extended to the involutions σ on P | m such that all transpositions are J-transpositions, by applying γ on every number in the cycle representation of σ. Again, the transformation γ for involutions is reversible, even though γ on P | m is not reversible. For example In fact we can check that the image sets of β and γ are identical, thus the transformation γ −1 β is well defined. The above comments are still valid if J and K are exchanged. By the bijection γ −1 β, the following lemma is equivalent to Lemma N.

The proofs
We begin with several comments. The proofs of all the theorems and lemmas are based on induction on the lengths of the permutations. Small values of m can be easily checked by hand. The proof of (N 2) uses (P 1), the proof of (P 1) uses (J2), and the proof of (J2) uses again (N 2). This is not a circular reasoning, because the length of permutations is smaller than the length of the original permutations. For every permutation σ, we say that σ contains a column odd even (in the two-line representation) if there is some odd number j such that σ(j) is even. We make similar sentence pour even even , even odd and odd odd . For short we say that a permutation σ is "in (J1)" if σ satisfies the conditions described in the statement of Theorem (J1), and that an involution The two parts composed by numbers from P and from Q are "independent". The cardinalities of the two parts and the number of fixed points in each side are characterized by m. If m = 2k + 1 is odd, then every involution σ in (N 2) has exactly one fixed point, in P | k or Q| k+1 , according to the parity of k. Hence The last equality follows from Lemmas (P 1) and (Q1). If m = 2k is even and k is odd, then the cardinalities of P | k and Q| k are odd. So that there is one fixed point in P | k and one in Q| k . Hence ν(N | 2k , 0, J) = 0 and Again, the last equality follows from Lemmas (P 1) and (Q1). If m = 2k is even and k is even, then the cardinalities of P | k and Q| k are even. Three situations may occur: (i) no fixed point neither in P | k nor in Q| k ; (ii) two fixed points in P | k and no fixed point in Q| k ; (iii) two fixed points in Q| k and no fixed point in P | k . Hence, we get The first equality follows from Lemmas (P 1) and (Q1). The second equality is proved by using the bijection δ described in Section 2.
Proof of (J2). -For every permutation σ in (J2), i + σ(i) ∈ J for i ≤ m − 2. If σ(m − 1) is odd and σ contains a odd odd column, let j σ(j) be the first odd odd column. We then define another permutation τ obtained from σ by exchanging σ(j) and σ(m − 1). This procedure is reversible. We can delete the pair σ and τ . Similarly we can delete every permutation σ such that σ(m − 1) is even and σ contains an even even column. There only remain two types of permutations. If σ(m − 1) is odd, then every number under an odd number in the two-line representation is even. If σ(m − 1) is even, then every number under an even number is odd, as shown here: The letter "e" and "o" represent an even and odd numbers respectively. There are two cases to be considered. (I.2) If σ(m − 1) is odd, then there are as many even numbers as odd numbers. Since the last column is odd odd , there is necessarily a unique column even even , which will be called "intruder", and marked by the * sign. The permutations in (J2) are of type * even where the last equality follows from (N3).

Further results
For subsets A, B, C, D of N let A B C D m denote the number of permutations σ in Sym m such that ) is a transformation reversible. Hence the total number of involutions is equal to ν(N | m−1 , 1, 0) + ν(N | m−1 , 1, 0) plus an even number, which is 0 modulo 2. The number of permutations of the type given by the first factor is equal to ν(N | m−1 , 0, K) ≡ 1 and that of the type given by the second one is equal to ν(N | m , 1, K) ≡ 1 by (N1). Hence Proof of (K3). -We need only count the involutions without fixed points, so that the number of such permutations is 0 when m is odd. If m = 2k is even, all transpositions are necessarily K-transpositions, except the one which contains m − 1. Removing m − 1 yields an involution with one fixed point. The number of such involutions is ν(N | m−1 , 1, K) ≡ 1 by (N1).
Furthermore, in Lemma (N3) we have a mixed formula for 0 or 2 fixed points. From the proof of (N 2) and (N 3) we can separate it into two more precise formulas.
The proofs of Theorems APWW and C combine Theorem J with the next two lemmas.
Lemma 1. For k ≥ 0, the integer g k+1 is odd if, and only if, k is in J.

Proof.
Let k be a non-negative integer. It follows from (1.1) that g k+1 is equal to the number of pairs (j, m) of integers j ≥ 1, m ≥ 0 such that j2 m = k + 1. In particular, writing k + 1 = j 0 2 m 0 with j 0 odd, we see that showing that g k+1 = m 0 + 1. Consequently, g k+1 is odd if, and only if, m 0 is even, that is, if and only if, k is in J.

Lemma 2.
For k ≥ 0, the integer δ k := (t k+1 − t k )/2 is odd if, and only if, k is in J.