1. Introduction
Of the methods used today to test for the irrationality of a given number, we cite two separate approaches, one which seems to have overtaken the other recently. The first method is a direct consequence of Apéry’s landmark paper [
1], which uses two independent solutions of a specific three-term recurrence relation (see (
18) below) to generate a series of rationals whose limit at infinity is
. Many new proofs and surveys of such arguments have appeared since, e.g., Beukers [
2], Nesterenko [
3], Fischler [
4], Cohen [
5], Murty [
6], Badea [
7], and Zudilin [
8,
9,
10] to mention a few in a list that is far from exhaustive.
The idea and the methods used in Apéry’s work [
1] were since developed and have produced results such as André-Jeannin’s proof of the irrationality of the inverse sum of the Fibonacci numbers [
11], along with a special inverse sum of Lucas numbers [
12], and Zudilin’s derivation [
13] of a three-term recurrence relation for which there exists two rational valued solutions whose quotients approach Catalan’s constant. In addition, we cite Zudilin’s communication [
14] of a four-term recurrence relation (third-order difference equation) for which there exist solutions whose quotients converge to
, but no irrationality results are derived.
One approach involves consideration of the vector space
V over
spanned by 1,
,
,
…,
. Using a criterion by Nesterenko [
15] on the linear independence of a finite number of reals, Rivoal [
16] proved that
for all sufficiently large
n, from which it follows that the list
contains infinitely many irrationals. Rivoal [
17] complements this result by showing that at least one of the numbers
is irrational. In the same vein, Zudilin [
18] shows that at least one of
is irrational. Recently, Rival and Zudilin [
19] showed that among the odd zeta values
, at least two of them must be irrational. The latter result was complemented by Lai and Yu’s theorem [
20] that gives an effective lower bound on the number of irrational quantities among the numbers
.
We apply the theory of disconjugate or non-oscillatory three-, four-, and n-term linear recurrence relations on the real line to discuss some problems in number theory: specifically, to questions about the irrationality of various limits found as quotients of solutions at infinity and, in particular, to the irrationality and possible quadratic and higher algebraic irrationality of
where
is the classic Riemann zeta function. We recall that this classic number is defined classically as
The motivation here is two-fold. First, we can investigate the irrationality of a given number
L, say, by starting with an infinite series for
L, associating it to a three-term recurrence relation (and so possibly to a non-regular continued fraction expansion) whose form is determined by the form of the series in question, finding an independent solution of said recurrence relation whose values are rational and, if the conditions are right (cf. Theorem 1 below), deduce the irrationality of
L. We show that this abstract construction includes, as a special case, Apéry’s proof [
1] that
is irrational.
In trying to determine whether or not
is an algebraic irrational [
21], we address the question of whether
is algebraic of degree two or more over
. Although we cannot answer this claim unequivocally at this time, we find an equivalent criterion for the irrationality of the square of
, or for that matter, any other irrational that can be approximated by the quotient of two solutions of an appropriate three-term recurrence relation. In the case of
, the equivalent criterion (Theorem 4) referred to is a function of the asymptotic behaviour of solutions of a specific linear four-term disconjugate recurrence relation (Theorem 2) in which the products of the classic Apéry numbers play a prominent role and whose general solution is actually known in advance. We obtain as a result that appropriate products of the Apéry numbers satisfy a four-term recurrence relation, that is, (
32) below (indeed, given any
, there exists an
-term recurrence relation for which these numbers play a basic role). However, the products of these Apéry numbers are not sufficient in themselves to give us the irrationality of the square of
. Still, our results show that the irrationality of the square of
would imply the non-existence of linear combinations of appropriate products of Apéry sequences generating a principal solution of a certain type for this four-term linear recurrence relation. The converse is also true by our results, but we cannot show that such linear combinations do not exist. Hence, we cannot answer at this time whether
is irrational.
We extend said criterion for the irrationality of a square of limits obtained by means of Apéry-type constructions, or from continued fraction expansions to a criterion for algebraic irrationality (an irrational satisfying a polynomial equation of degree greater than two with rational coefficients) over
(Theorem 8). Here and everywhere else,
denotes the set of all rational numbers. It is then a simple matter to formulate a criterion for the transcendence of such limits. Loosely speaking, we show that an irrational number derived as the limit of a sequence of rationals associated with a basis for a linear three-term recurrence relation is transcendental if and only if there exists an infinite sequence of linear
m-term recurrence relations, one for each
, such that each one lacks a non-trivial rational valued solution with special asymptotics at infinity (cf., Theorem 9). Finally, motivated by the results on the four-term recurrences (Theorem 2), we present in the
Appendix A to this article accelerated series representations for
, for
, and similar series for
, where we display the cases
only leaving the remaining cases as examples that can be formulated by the reader. These results are independent of those in [
22] for
which use the generalized divisor function and Eisenstein series.
2. Preliminary Results
We present a series of lemmas useful in our later considerations. Recall that is the set of all real numbers, is the set of all non-negative integers (including zero), and is the set of all positive integers (excludes zero). Recall that denotes the set of all positive rational numbers.
Lemma 1. Let ,
, be two given infinite sequences such that the seriesconverges absolutely. Then, there exists a sequence satisfying (5) such thatwhere Proof of Lemma 1. For the given sequences
define the sequence
using (
3) below:
Then, by definition, the
satisfy the three-term recurrence relation
with
. Choosing the values
such that
, we solve the two-term recurrence relation
for a unique solution,
. Observe that these
values satisfy the same recurrence relation as the given
Since
by hypothesis, dividing both sides of (
5) by
gives (
2) upon summation and passage to the limit as
, since the resulting series on the left is a telescoping series. □
Lemma 2. Consider (4) where,
, for every, andLet,
, be two linearly independent solutions of (4). If, thenexists and is finite. Proof of Lemma 2. Since
,
, for every
, Equation (
4) is non-oscillatory at infinity [
23] or [
24]; that is, every solution
has a constant sign for all sufficiently large
n. From discrete Sturm theory, we deduce that every solution of (
4) has a finite number of nodes [
24]. As a result, the solution
, may, if modified by a constant, be assumed to be positive for all sufficiently large
n. Similarly, we may assume that
for all sufficiently large
n. Thus, write
for all
Once again, from standard results in the theory of three-term recurrence relations, there holds the Wronskian identity (
5) for these solutions. The proof of Lemma 1, viz. (
5), yields the identity
for each
. Summing both sides from
to infinity, we deduce the existence of the limit
L in (
6) (possibly infinite at this point), since the tail end of the series has only positive terms and the left side is telescoping.
We now show that the eventually positive solution
is bounded away from zero for all sufficiently large
n. This is basically a simple argument (see Olver and Sookne [
25] and Patula ([
24], Lemma 2) for early extensions). Indeed, the assumption
implies that
is increasing for all large
n. An induction argument provides the clue. Assuming that
for all
,
since
for all large
n. The result follows.
Now since
is bounded away from zero for large
n (i.e., bounded below uniformly in
n) and
by hypothesis, it follows that the series
that is,
L in (
6) is finite. □
Remark 1. The limit of the sequenceitself may be a priori finite. For applications to irrationality proofs, we need that this sequence with n. A sufficient condition for this is provided below.
Lemma 3. (Olver and Sookne [25], Patula ([24], Lemma 2)) Let,
for all sufficiently large n, where, anddiverges. Then, every increasing solutionof (4) grows without bound as. The notion of disconjugacy in its simplest form can be found in Patula [
24] or see Hartman [
23] for more general formulations. In our case, (
4) is a disconjugate recurrence relation on
if every non-trivial solution
has at most one sign change for all
. The following result is a consequence of Lemmas 2 and 3.
Lemma 4. Letin (4) and. Letsatisfyfor. Then - (a)
Equation (4) is a disconjugate three-term recurrence relation on - (b)
There exists a solutionwithfor all,
increasing and such that for any other linearly independent solution, we have the relation for some suitable constant β, for all sufficiently large m, where L is the limit.
- (c)
If, in addition, we have (8) satisfied for some sequenceetc., then the solutionin item (2)
grows without bound, that is, as .
Item (b) of the preceding lemma is recognizable by anyone working with continued fractions [
21]. Of course, continued fractions have convergents (such as
above) that satisfy linear three-term recurrence relations and their quotients, when they converge, converge to the particular number (here represented by
L) represented by the continued fraction. In this article, we view the limits of these quotients in terms of asymptotics of solutions of disconjugate recurrence relations with a particular emphasis on principal solutions.
4. A Criterion for Algebraic Irrationality and Transcendence
In this section, we generalize the results of the previous sections. Thus, given an irrational number L whose rational approximations are found by either an Apéry-type argument on a three-term recurrence relation, or perhaps using a continued fraction expansion of L, we prove that if L is not algebraic of degree less than or equal to , then L is algebraic of degree m (over ) if and only if there exists a disconjugate -term linear recurrence relation having a specific type of rational valued principal solution. This will lead to Theorem 5 above as a special case. This will lead to, of course, a necessary and sufficient condition for the transcendence of such quantities. We outline the construction of this characteristic recurrence relation by pointing out the first two important special cases first as motivation: The first case is Theorem 5 as mentioned above. The second case is a “degree 3” version of Theorem 5.
In relation to (
29) can be found a higher order analog of Theorem 2. We start with the five-term recurrence relation
where the leading term
for all
n, and
Note that the hypothesis,
for all
n, is equivalent to
for all
n. Then, for any given pair of linearly independent solutions
of (
29), the sequences
,
,
, and
+
+
form a linearly independent set of solutions for (
37). Given that we know how to test for degree 2 irrationality of limits
L via Theorem 5, we can formulate an analogous result for degree 3 irrationality next.
Note: In the sequel, we always assume that the
in question are positive for all
n (as they arise from a disconjugate Equation (
29)). There is no loss of generality in assuming this, since the proofs involve limiting arguments. In addition, unless otherwise specified, we assume that
.
Theorem 6. Letandfor all n. Letbe two independent rational valued solutions of (29) such thatwhere L is irrational and L is not algebraic of degree 2. Then, L is algebraic of degree three overif and only if (37) has a non-trivial rational valued solutionsuch that The proof is similar to that of Theorem 4 and so is omitted.
Remark 5. A re-examination of the proof of Theorem 4 which serves as a template for all other such proofs to follow shows that the tacit assumptions on L can be waived to some extent. The previous result may then be re-formulated as follows.
Let
, a solution of (
37), have the basis representation
where
and the subscript
i in
for the basis coordinates is determined by counting the number of
A values in the basis vector immediately following it.
Theorem 7. Letandfor all n. Letbe two independent rational valued solutions of (29) such that Then, L is algebraic of degree at most 3 if and only if there exists a non-trivial rational valued solutionof (37) satisfying (38). Proof of Theorem 7. Idea: Using (
39), we see that since
is rational, then so are the
,
, not all of which are zero. Next, as
,
and so
L is algebraic of degree no greater than 3. Conversely, let
L be algebraic of degree no greater than 3 and let
be its defining polynomial where not all
are zero. Then, choosing the solution
of (
37) in the form (
39) with the same quantities
that appear as the coefficients of
p, we see that since
, (
38) is satisfied. □
Remark 6. In order to improve on Theorem 6, we need to add more to the solutionappearing therein. For example, it is easy to see that under the same basic conditions on the, if there exists a non-trivial rational valued solutionof (37) withsatisfying (38), then L is algebraic of degree no greater than 3. On the other hand, if L is algebraic of degree 3, then there exists a non-trivial rational valued solutionof (37) with satisfying (38). Theorems 5 and 6 give us an idea on how to proceed next. In essence, we now have some way of determining whether or not the limit L is algebraic of degree 3 based on the fact that it is not algebraic of lower degree. The general result is similar, but first, we describe the construction of the required linear higher-order recurrence relations. In order to test whether the limit L in Theorem 6 is algebraic of degree m, , we will require a linear recurrence relation containing -terms or equivalently an -th order linear difference equation (an equation involving “finite differences” in the traditional sense). This new equation is found from a prior knowledge of the kernel of the associated operator.
As usual, we let
be two linearly independent solutions of (
29). We seek a homogeneous linear
-term recurrence relation whose basis (consisting of
terms) is described as follows: Two basic elements are given by
along with a corresponding term with all these
A values replaced by
B values. For each given
k,
, consider
where the sum contains
distinct terms. Each summand is found by enumerating all the ways of choosing
k-terms from the full product of
A values and then replacing each such
A with a
B (keeping the subscripts the same).
As an example, consider the case
and
. There is then such a sum of
terms, the totality of which looks like
As
k varies from 0 to
m, the collection of all such “sums of products” gives us a collection of
terms of the form
. That this specific set of elements is a linearly independent set may depend on the nature of the interaction of the
in (
29) as we saw above (e.g.,
in (
37)). At any rate, since every solution
of this new recurrence relation must be a linear combination of
, we see that the compatibility relation is found by setting the determinant of this matrix,
equal to zero, for every
n. This and the repeated use of the recurrence relation (
29) gives the required
-term recurrence relation of which (
37) and (
30) are but special cases.
Note: In the sequel we always assume that the set consisting of the “sums of products” described above is a linearly independent set of elements. This is equivalent to various conditions to be imposed upon the coefficient of the leading and trailing terms of the ensuing -term recurrence relation whose construction is presented above.
Theorem 8. Let,
andin addition to other conditions enunciated in the note above. Let. Consider two independent rational valued solutionsof (29) such thatwhere L is not algebraic of degree less than or equal to. Then, L is algebraic of degree m overif and only if the-term linear recurrence relation described above has a non-trivial rational valued solutionsuch that An analog of Theorem 7 can also be formulated, which is perhaps easier to use in practice.
Theorem 9. Let,
andin addition to other conditions enunciated in the note above. Let. Letbe two independent rational valued solutions of (29) such that Then, L is algebraic of degree at most m overif and only if the-term linear recurrence relation described above has a non-trivial rational valued solutionsuch that Remark 7. Since the condition in Theorem 9 puts a bound on the degree m of algebraic irrationality over, it also gives an equivalent criterion for the transcendence of numbers L whose limits are found by using quotients of solutions of three-term recurrence relations. In particular, associated to the special numberis an infinite sequence of specific linear recurrence relations of every order, as constructed above, involving sums of products of both sets of Apéry numbers. The transcendence ofis then equivalent to the statement that none of the infinite number of (disconjugate) recurrence relations constructed has a non-trivial rational valued principal solution of the type described.
Example 2. In this final example, we interpret Apéry’s construction [1], for the irrationality ofin the context of the non-existence of rational valued solutions of recurrence relations with predetermined asymptotics. Recall that Apéry’s three-term recurrence relation for the proof of the irrationality ofis given by [1] In order to apply Theorem 2, we need to express this equation in self-adjoint form; that is, we simply multiply both sides byresulting in the equivalent Equation (29) with,
. The Apéry solutions of this equation (e.g., [26]) are given byand We know, from Apéry’s paper, that these solutions are such thatas, and we already know thatis irrational. It follows from the above considerations that the four-term recurrence relationwhere and whose solution space is spanned by the elements,
andcannot have a non-trivial rational valued solutionsatisfying However, we also know thatis actually transcendental (as it is a rational multiple of) and so cannot be algebraic of any finite degree. Hence, for each m, none of the-term recurrence relations that can be constructed as described above has a non-trivial rational valued solution satisfying (38).