Arithmetic properties of coefficients of power series expansion of \(\prod _{n=0}^{\infty }\left( 1-x^{2^{n}}\right) ^{t}\) (with an appendix by Andrzej Schinzel)
Arithmetic properties of coefficients of power series expansion of ?n=0 1 ? x2n t (with an appendix by Andrzej Schinzel)
Maciej Gawron 0
Piotr Miska 0
Maciej Ulas 0
0 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University , ? ojasiewicza 6, 30-348 Krako?w , Poland
Let F (x ) = n?=0(1 ? x 2n ) be the generating function for the ProuhetThue-Morse sequence ((?1)s2(n))n?N. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function For t ? N+ the sequence ( fn (t ))n?N is the Cauchy convolution of t copies of the Prouhet-Thue-Morse sequence. For t ? Z<0 the n-th term of the sequence ( fn (t ))n?N counts the number of representations of the number n as a sum of powers of 2 where each summand can have one among ?t colors. Among other things, we present a characterization of the solutions of the equations fn (2k ) = 0, where k ? N, and fn (3) = 0. Next, we present the exact value of the 2-adic valuation of the number fn (1 ? 2m )-a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.
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Ft (x ) = F (x )t =
The research of the Maciej Gawron and Maciej Ulas was supported by the grant of the Polish National
Science Centre No. UMO-2012/07/E/ST1/00185.
B Maciej Ulas
Mathematics Subject Classification 11P81 ? 11P83 ? 11B50
1 Introduction
Let n ? N and by s2(n) denote the sum of (binary) digits function of n, i.e., if
n = km=0 ?k 2k with ?k ? {0, 1}, is the unique expansion of n in base 2 then s2(n) =
m
k=0 ?k . Next, let us define the Prouhet?Thue?Morse sequence (the PTM sequence
for short) on the alphabet {?1, +1} as t = (tn)n?N, where tn = (?1)s2(n). The
sequence t satisfies the following recurrence relations: t0 = 1 and
for n ? N. The PTM sequence has many remarkable properties and found applications
in combinatorics on words, analysis on manifolds, number theory and even physics [1].
One of the striking properties of the sequence t is the simple shape of the generating
function F (x ) = n?=0 tn x n ? Z[[x ]]. Indeed, from the recurrence relations we easily
deduce the functional equation F (x ) = (1 ? x )F (x 2) and in consequence the identity
F (x ) =
n=0
1 ? x 2n
n=0
1 ? x 2n =
n=0
has also a strong combinatorial property. Indeed, the number bn counts the number of
representations of a non-negative integer n in the form
n =
i=0
where k ? N and ?i ? N. One can easily prove that the sequence b satisfies: b0 =
b1 = 1 and
b2n = b2n?1 + bn, b2n+1 = b2n
for n ? 1. The above sequence is called the sequence of the binary partition function.
It was introduced by Euler and was studied by Churchhouse [5] (one can also consult
the papers [6,11,13]).
From the discussion above we see that both t and b are sequences of coefficients
of the power series expansion of Ft (x ) = F (x )t for t = 1 and t = ?1, respectively.
It is quite natural to ask: what can be proved about sequences of coefficients of Ft (x )
for other integer values of t ? This question was our main motivation for writing this
paper.
Let t be a variable and consider the sequence f (t ) = ( fn(t ))n?N of coefficients of
the power series expansion of the function Ft (x ) = F (x )t , i.e.,
Ft (x ) =
n=0
n=0
From the definition of f (t ) we see that for any given t ? Z the sequence f (t ) is
a sequence of integers. In the sequel we will study three closely related sequences.
More precisely, in Sect. 2 we present some properties of the sequence f (t ) treated
as a sequence of polynomials with rational coefficients. This is only a prelude to our
research devoted to the values of the polynomials fn at integer arguments. Section 3
is devoted to the study of the sequence
tm = (tm (n))n?N,
where m ? N+ is fixed and tm (n) = fn(m), i.e., tm (n) is just the value of the
polynomial fn at t = m. We prove several properties of the sequence tm for certain
values of m. In particular, in Theorem 3.3 we characterize the 2-adic valuation of the
sequence tm for m being a power of 2 and m = 3. In the second part of this section
we concentrate on the study of arithmetic properties of the sequence tm for m = 2
and m = 3. It is a simple observation that the sequence t2 is closely related to the
values of the Stern polynomials at ?2. Moreover, we prove that the set of values of
t2 is just Z\{0}, which is the statement of Theorem 3.17 and that our sequence is
log-concave, i.e., for each n ? N+ we have t2(n)2 > t2(n ? 1)t2(n + 1) (Theorem 14).
We also characterize the set of those n ? N+ such that t3(n) = 0 (Theorem 3.13). This
allows us to prove that there are infinitely many values of n such that the polynomial
fn(t )/t is reducible (Corollary 3.14). Section 4 is devoted to the study of the sequence
bm = (bm (n))n?N, where m ? N+ is fixed and bm (n) = fn(?m), i.e., bm (n) is just the
value of the polynomial fn at t = ?m. The sequence bm (...truncated)