Consider the sequence un defined as follows: un=+1 if the sum of the base b digits of n is even, and un=−1 otherwise, where we take b=2. Recall that the Woods-Robbins infinite product involves a rational function in n and the sequence un. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in n and the sequence un was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of b. We illustrate the approach by evaluating a large class of similar infinite products.
- radix representations
- digit sums
- Prouhet-Thue-Morse sequence
- Woods and Robbins product
- closed formulas for infinite products
Throughout this chapter will denote a non-negative integer. Let denote the sum of the base digits of , and put . We study infinite products of the form
(We show in Section 2 that converges for ).
Several infinite products inspired by it were discovered afterwards (see, e.g., [3, 4]). But none of them involve the sequence . Moreover, almost nothing is known (see, e.g., [5, 6]) about the similar product
Our goal is to study these infinite products in detail. This will allow us to gain a deeper understanding of such products as well as evaluate more products like the Woods-Robbins identity.
2. General properties of the function
First we establish a general result from  on convergence.
Lemma 1.1 Let be a rational function such that the values are defined and nonzero for . Then, the infinite product converges if and only if the numerator and the denominator of have the same degree and same leading coefficient.
Proof. If the infinite product converges, then must tend to when tends to infinity. Thus the numerator and the denominator of have the same degree and the same leading coefficient.
Now suppose that the numerator and the denominator of have the same leading coefficient and the same degree. Decomposing them in factors of degree , it suffices, for proving that the infinite product converges, to show that infinite products of the form
converge for complex numbers and such that and do not vanish for any . Since the general factor of such a product tends to , it is equivalent, grouping the factors pairwise, to proving that the product
converges. Since and , we only need to prove that the infinite product
converges. Taking (the principal determination of) logarithms, we see that
which gives the convergence result.
Hence converges for any . Using the definition of , it follows that for any ,
One can ask the natural question: is the unique function satisfying these properties?
2.1 A new function
Properties 1 and 2 above give
Hence we can rewrite property 3 as
Thus can be computed using only the quantities , via
So understanding is equivalent to understanding , in the sense that each can be completely evaluated in terms of the other.
Taking in Eq. (10) gives the functional equation:
Similar questions can be asked for : is it the unique solution to Eq. (11)? What about monotonic/continuous/smooth solutions?
3. Infinite products
3.1 Direct approach
Theorem 1.1 The following relations hold.
This follows immediately using properties 1–3 in Section 2.
Take in Eq. (12).
Take in Eq. (12).
Corollary 1.1 For any positive rational number , there exist monic polynomials , both at most cubic, such that
Furthermore, if is an integer, then and can be chosen to be at most quadratic.
We still do not know exactly which numbers are given by such infinite products.
3.2 Functional equation approach
Recall the functional Eq. (11):
Taking in Eq. (11) gives
i.e., . This shows that
Next, taking in Eq. (11) gives
hence , and we recover the Woods-Robbins identity
Similarly, taking in Eq. (11) gives
Taking in Eq. (11) gives
hence , and this gives
Taking in Eq. (11) and using the previous result gives
which is equivalent to
These identities can be also combined in pairs to obtain other identities.
4. Some analytical results
We saw in the previous section that some of the infinite products we evaluated were integers, some were rational, and some were quadratic irrational. In the hope of further understanding their nature, we now study the analytical behaviors of and .
Lemma 1.2 Let .
1. If , then .
2. If , then
3. If , then
Proof. Using properties 1–3 from Section 2, it suffices to prove 2.
Let , and put
Note that is positive and strictly decreasing to . Using , it follows that and , for each . Using summation by parts,
So for . Exponentiating and taking gives the desired result.
Lemma 1.2 together with Eq. (10) implies the following results.
Theorem 1.2is strictly decreasing on , and is strictly increasing on .
from which the result follows.
Theorem 1.3 For , is strictly decreasing in and strictly increasing in .
Proof. By Eq. (10),
hence the result follows from Theorem 1.2.
Theorem 1.4 For ,
We now prove some results on differentiability.
Theorem 1.5is smooth on .
Proof. Take and in Eq. (30). Then the sequence of smooth functions on converges pointwise to .
Differentiating with respect to gives
as , for any and . Thus converges uniformly on , which shows that , hence , is differentiable on .
Now suppose that derivatives of up to order exist for some . Note that
as , for any and . Hence converges uniformly on , i.e., is differentiable on .
Therefore, by induction, has derivatives of all orders on .
Theorem 1.6 Let . Then
Proof. Let . By Theorem 1.5,
for . So by Taylor’s inequality, the remainder for the Taylor polynomial for of degree is absolutely bounded above by
which tends to as , since and . Therefore equals its Taylor expansion about for in the given range.
5. Further remarks on
As mentioned in Section 1, not much is known about the quantity . We give the following explanation as to why might behave specially in a sense.
Note that the only way nontrivial cancelation occurs in Eq. (11) is when . Likewise, nontrivial cancelation occurs in Eq. (10) or property 3 in Section 2 only for and . That is, the victim of any such cancelation is always or . So we must look for other ways to study .
Using the two known values and , the following expressions for are obtained from Theorem 1.6 by choosing various values for and .
The Dirichlet series appearing in the above expressions were studied in . We think that these identities and the results from Section 4 might help in shedding some light on the nature of .
6. Conclusions and future developments
We evaluated infinite products involving the digit sum function by splitting the product based on the congruence classes modulo . We illustrated two approaches for doing so, one by direct computation and another using functional equations. For we proved some analytical results to aid us in understanding the behavior of these products. Many open questions still remain.
Although we only considered the base , many of the results above easily generalize to other bases. One possible direction toward a generalization is to take . Another is , where is a primitive -th root of unity. We leave these as work to be done in the future.