Exploring General Apéry Limits via the Zudilin-Straub t-transform

Robert DOUGHERTY-BLISS and Doron ZEILBERGER

Abstract: Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that ‘tweaked’ the sum of the s^{th} powers of the n-th row of Pascal’s triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call “major Apéry miracles”, proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a “minor Apéry miracle” where an explicit constant, defined as an (extremely slowly-converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.

Preface: The Major and Minor (but still interesting!) Apéry Miracles

One way that Roger Apéry’s [A] [vdP] seminal proof of the irrationality of ζ(3) could have been discovered, in a counterfactual world, was to consider, out of the blue, the second-order linear recurrence

n^{3}u_{n} − (17n^{2} + 51n + 39) (2n + 3) u_{n−1} + (n − 1)^{3}u_{n−2} = 0 ,

and let a_{n} and b_{n} be the solutions of that recurrence with initial conditions

a_{0} = 0, a_{1} = 6 ; b_{0} = 1, b_{1} = 5,

then let the computer compute many terms, evaluate \frac{a_{1000}}{b_{1000}} to many decimals, and then use Maple’s identify, and lo and behold, get that it (most probably) equals ζ(3) (i.e. ∑_{i=1}^{∞} \frac{1}{i^{3}} ). Then, still empirically and numerically, after rewriting \frac{a_{n}}{b_{n}} as \frac{a′_{n}}{b′_{n}}, where now both numerator and denominators are integers (initially b_{n} were integers, but a_{n} were not), estimate that there exists a positive number δ (about 0.0805) such that

|\frac{a′_{n}}{b′_{n}} − ζ(3)| ≤ \frac{CONST ANT}{(b′_{n})^{1+δ}} ,

that immediately entails (see [vdP]) that ζ(3) is irrational.

Using the terminology that has now become standard (e.g. the titles of [CS] and [SZ]), we say that ζ(3) is the Apéry limit of the above recurrence, and initial conditions.

The reason that this was a major miracle, as explained so eloquently in [vdP], is that while any naturally occurring constant that is not obviously rational, e.g. the sum of the series ∑_{i=1}^{∞} \frac{1}{i(i+1)}, is definitely (in the everyday sense of the word) irrational (there are only ℵ_{0} rational numbers, while there are 2^{ℵ_{0}} real numbers, hence the ‘probability’ of a real number being rational is a (very small!) 0), it is extremely difficult to (rigorously) prove that a specific constant is irrational. Witness the fact that, in spite of many attempts, there are still no proofs of the irrationality of the Euler-Mascheroni constant γ (the limit of the partial sums of the harmonic series minus log(n), or equivalently − ∫_{0}^{∞} e^{−x} log x dx), the Catalan constant C := ∑_{i=0}^{∞} \frac{(−1)^{i}}{(2i+1)^{2}} , ζ(5) := ∑_{i=1}^{∞} \frac{1}{i^{5}} (and more generally ζ(2i + 1), for all i ≥ 2). The same is also true for e + π and e π (but they can’t be both rational [proof left to the reader]).

This theme is pursued in [Ze2], [Ze3] and much more recently, in [DKZ], [DZ] and [ZeZu], where the motivation was to discover irrationality proofs of other constants. In [DKZ] there were quite a few ‘Apéry miracles’, alas, most of them were reproved irrationality of constants that were already proved irrational (for example, algebraic numbers or logarithms of them) and the novelty was establishing explicit irrationality measures. This is still interesting, but not exciting. We also found few other ‘weird’ constants given in terms of products of Gamma values at rational numbers, that should yield fully rigorous first proofs of explicit constants, but since these constants were name-less, it gave us neither fame nor fortune.

But the minor Apéry miracle was not number-theoretical but rather numerical-analytical. Here is an explicitly defined constant ζ(3) := ∑_{i=1}^{∞} \frac{1}{i^{3}} , in other words the limit of the sequence of rational numbers \{∑_{i=1}^{n} \frac{1}{i^{3}} \}, that converges very slowly to its limit. Realizing it as an Apéry limit, i.e. coming up with an explicit linear recurrence equation with polynomial coefficients, and two sets of initial conditions, for which the limit of the ratios (of the emerging two sequences) converges to that number with an exponentially decaying error. This enables one to compute the constant in question to many decimal digits. On the other hand, computing it to that accuracy using the definition would take zillions of years. The purpose of this article is to show how one can produce lots of other ‘minor Apéry miracles’ where one can express explicitly defined constants, whose definition entails very slow convergence, as Apéry limits of recurrences and initial conditions that enables computing these constants with exponentially decaying errors. The key idea is to introduce what we will call the Zudilin-Straub t-Transform, that generalizes a construction Armin Straub and Wadim Zudilin discovered [SZ] for the special case of sums of powers of binomial coefficients.

But before defining the Zudilin-Straub t-transform, let’s define Apéry limit more formally, and also introduce the new notion of Generalized Apéry limit.

Definition 1: A constant c is an Apéry limit if there exists a homogeneous linear recurrence with polynomial coefficients

∑_{i=0}^{L} p_{i}(n) X(n + i) = 0 ,

and two sets of initial conditions [a_{0}, a_{1}, …, a_{L−1}] and [b_{0}, b_{1}, …, b_{L−1}], such that if A(n) and B(n) are the solutions of that same recurrence with

A(0) = a_{0} , …, A(L − 1) = a_{L−1} ; B(0) = b_{0} , …, B(L − 1) = b_{L−1} ,

then

c = lim_{n→∞} \frac{A(n)}{B(n)} .

While not part of the definition, it turns out that often, and in all the naturally occurring cases, we also have the following nice feature.

Exponential decay of Error Property

There exist constants C > 0 and α > 1 such that

|\frac{A(n)}{B(n)} − c| ≤ \frac{C}{α^{n}} .

Given a recurrence and initial conditions, it is very fast to compute many terms. In fact one only needs constant memory (well, linear memory if you go by bit-size) and linear time to compute any specific approximation, \frac{A(n)}{B(n)}.

Let’s introduce a mild extension, that of a Generalized Apéry limit.

Definition 2: A constant c is a Generalized Apéry limit if there exist two sequences of rational numbers A(n) and B(n) such that

c = lim_{n→∞} \frac{A(n)}{B(n)} ,

where A(n) and B(n) are solutions of linear recurrences with polynomial coefficients (the first homogeneous, the second inhomogeneous)

∑_{i=0}^{L} p_{i}(n) B(n + i) = 0 ,

subject to initial conditions B(0) = b_{0}, …, B(L − 1) = b_{L−1}, and

∑_{i=0}^{L} p_{i}(n) A(n + i) = C(n) ,

subject to initial conditions A(0) = a_{0}, …, A(L − 1) = a_{L−1}, where the right side, C(n), in turn is a solution of another, (this time homogeneous) linear recurrence equation with polynomial coefficients

∑_{i=0}^{M} q_{i}(n) C(n + i) = 0 ,

subject to some initial conditions C(0) = c_{0}, …, C(M − 1) = c_{M−1}.

Note that by using recurrence operators, it is easy to express both A(n) and B(n) as solutions of the same homogeneous linear recurrence, so a generalized Apéry limit can always be expressed as an Apéry limit, alas, with the recurrences being of a much higher order, namely L + M.

Also note that in order to prove irrationality à la Apéry, exponential decay of error does not suffice. After writing the quotients of rational numbers A(n)/B(n) as \frac{a_{n}}{b_{n}}, where a_{n} and b_{n} are integers, one needs an inequality of the form,

|\frac{a_{n}}{b_{n}} − c| ≤ \frac{CONST ANT}{b_{n}^{1+δ}} ,

where δ > 0, yielding an irrationality measure 1 + \frac{1}{δ} (see [vdP]). The attempts in [Ze2],[Ze3], [DZ], and [DKZ], to find new (major) Apéry miracles, i.e. irrationality proofs of hopefully new constants, consisted of going backwards. Rather than trying to hit the “bull’s-eye”, one shoots first, and then draws the bull’s eye around the bullet hole. Using the Zeilberger, Almkvist-Zeilberger, and multi-Almkvist Zeilberger algorithms (see [DKZ] for references) we generated recurrences for known sequences B(n), then changed the initial conditions, getting a companion sequence A(n), then we computed (very fast, to high accuracy) approximations to the limit of A(n)/B(n). Then we used Maple’s identify (and our extensions) to conjecture an explicit value of the Apéry limit, and hoped to prove it later. But often, neither Maple, nor our extension, was able to identify the Apéry limit. In many cases we were able to identify the Apéry limit, but the δ turned out to be negative, so it was useless for proving irrationality. Nevertheless, since we always had 1 + δ > 0, we still got exponentially decaying error.

In this paper we will forget about our irrationality obsession and only enjoy the exponential decay of error property of Apéry limits (and the generalized version). However here the focus would be to introduce explicit constants, defined as a limit of very slowly-converging sequences, and express them as Apéry limits or generalized Apéry limits, enabling computing these constants very fast, to any desired accuracy. In other words, we will do what numerical analysts call convergence acceleration, with very dramatic acceleration.

The work of Straub and Zudilin that motivated the Zudilin-Straub t-transform

In a recent beautiful article [SZ] (Theorem 1.3 there) (that brilliantly proved some conjectures in the equally beautiful article [CS]) they expressed ζ(2j) j = 1, 2, 3 … (or equivalently π^{2j}) as Apéry limits with explicit recurrences and explicit initial conditions. The recurrences in question were obtained from the Zeilberger [Ze1] (see also [PWZ]) algorithm applied to sums of binomial powers, also known as Franel numbers (here s is any positive integer):

F_{n}^{(s)} := ∑_{k=0}^{n} \binom{n}{k}^{s} = ∑_{k=0}^{n} \frac{n!^{s}}{k!^{s}(n − k)!^{s}} ,

with s ≥ 2j + 1. The beauty is that they did it for infinitely many cases.

Let’s describe what they did. Recall that the rising factorial, also called Pochhammer symbol (that features in the definition of a hypergeometric series), is defined by

(x)_{n} := x (x + 1) ⋯ (x + n − 1) .

(Note that (1)_{n} = n!.)

The starting point in [SZ] was to consider, instead of the sequence of integers {F_{n}^{(s)}}_{n=1}^{∞}, the sequence of rational functions, let’s call them {f_{n}^{(s)}(t)}_{n=1}^{∞}, defined by

f_{n}^{(s)}(t) := ∑_{k=0}^{n} \frac{n!^{s}}{(1 + t)_{k}^{s}(1 − t)_{n−k}^{s}} ,

(note that f_{n}^{(s)}(0) = F_{n}^{(s)}).

Their key idea was to apply the Zeilberger algorithm to the modified sum rather than the original sum, and see what happens. Let’s recall some basic definition from Wilf-Zeilberger algorithmic proof theory [PWZ] that would also be needed later on, when we will describe our generalization of the Straub-Zudilin [SZ] work.

Definition 3 ([PWZ], p. 64): A discrete function F (n, k) (defined on {(n, k) | 0 ≤ n, k < ∞}) is said to be a proper hypergeometric term if it can be written in the form

F (n, k) = P (n, k) \frac{∏_{i=1}^{uu}(a_{i}n + b_{i}k + c_{i})!}{∏_{i=1}^{vv}(u_{i}n + v_{i}k + w_{i})!}x^{k} ,  (1)

in which x is an indeterminate over, say, the complex numbers, and

• P is a polynomial

• the a’s, b’s, u’s, and v’s are specific integers, that is to say, they do not contain additional parameters, and

• the quantities uu and vv are finite non-negative, specific integers.

Recall that for any proper hypergeometric term F (n, k) as defined above, the Zeilberger algorithm [Ze1] [PWZ] furnishes, for some non-negative integer L (called the order) , polynomials in n, p_{0}(n), …, p_{L}(n), as well as another proper hypergeometric term, G(n, k) called the certificate (that furthermore has the property that G(n, k)/F (n, k) is a rational function of (n, k)), such that

p_{0}(n)F (n, k) + p_{1}(n)F (n + 1, k) + … + p_{L}(n)F (n + L, k) = G(n, k + 1) − G(n, k) .

By summing with respect to k from k = 0 to k = n, we have the immediate corollary that the hypergeometric sum

f(n) := ∑_{k=0}^{n} F (n, k) ,

satisfies the linear recurrence equation with polynomial coefficients

p_{0}(n)f(n) + p_{1}(n)f(n + 1) + … + p_{L}(n)f(n + L) = G(n, n + 1) − G(n, 0) .  (2)

Note that in general, the right side is not zero, so one gets an inhomogeneous linear recurrence, but whenever the summand is natural, for example any binomial coefficient sum that contains \binom{n}{k} in it (in particular, of course, for the Franel sequences) the right side vanishes, and one gets that the sum f(n) satisfies a homogeneous linear recurrence equation with polynomial coefficients.

Since (1 + t)_{k} = (t + k)!/t! and (1 − t)_{n−k} = (n − (t + k))!/(−t)!, the generalized Franel sum above, namely f_{n}^{(s)}(t) := ∑_{k=0}^{n} \frac{n!^{s}}{(1+t)_{k}^{s}(1−t)_{n−k}^{s}}, yields the same recurrence, but of course not the same certificate. In particular the right side G(n, n + 1) − G(n, 0) is no longer 0 but some multiple of the summand by a rational function of t and n. But then came a nice surprise, indeed, yet-another-miracle. For any specific s, this miracle follows immediately from the output of the Zeilberger algorithm. It is proved in general in [SZ] (using human ingenuity).

The Straub-Zudilin-Franel miracle

For any s > 2, the right side of the inhomogeneous recurrence satisfied by f_{n}^{(s)}(t) is divisible by t^{s+1} if s is odd and by t^{s} if s is even.

It follows that for any r < s, the sequence obtained by extracting the coefficient of t^{r} from the summand \frac{n!^{s}}{(1+t)_{k}^{s}(1−t)_{n−k}^{s}} of f_{n}^{(s)}(t) automatically satisfies the very same homogeneous linear recur-

rence satisfied by the Franel sequence F_{n}^{(s)}, but, of course, with different initial conditions. Hence the limit of the ratios of the later sequence with the Franel sequence F_{n}^{(s)} is an Apéry limit of some constant. Can we describe this constant directly? Yes we can!

Let’s rewrite the sum f_{n}^{(s)}(t) as follows

f_{n}^{(s)}(t) := ∑_{k=0}^{n} \frac{n!^{s}}{(1 + t)_{k}^{s}(1 − t)_{n−k}^{s}} = ∑_{k=0}^{n} \frac{n!^{s}}{k!^{s}(n − k)!^{s}} · (\frac{k!^{s}(n − k)!^{s}}{(1 + t)_{k}^{s}(1 − t)_{n−k}^{s}} ) .

Let’s give the coefficient of t^{r} in the Taylor expansion (about t = 0) of \frac{k!^{s}(n−k)!^{s}}{(1+t)_{k}^{s}(1−t)_{n−k}^{s}} a name.

Definition 4: c_{r}^{(s)}(n, k) is the coefficient of t^{r} in the Taylor expansion of the rational function of t \frac{k!^{s}(n−k)!^{s}}{(1+t)_{k}^{s}(1−t)_{n−k}^{s}}.

Interesting consequences of the Straub-Zudilin-Franel miracle

Now fix a power s > 2 and r < s + 1, and let B(n) be the Franel sequence and A(n) the sequence of the coefficients of t^{r} in the Taylor expansion of f_{n}^{(s)}(t). Then

B(n) = ∑_{k=0}^{n} \binom{n}{k}^{s} ,

A(n) = ∑_{k=0}^{n} \binom{n}{k}^{s} c_{r}^{(s)}(n, k) .

Now consider the Apéry limit

lim_{n→∞} \frac{A(n)}{B(n)} = \frac{∑_{k=0}^{n} \binom{n}{k}^{s}c_{r}^{(s)}(n, k)}{∑{n \atop k=0} \binom{n}{k}s} .

Note that the left side is a certain Apéry limit with the Franel recurrence and appropriate initial conditions (hence the limit can be computed very fast, with exponentially-decaying error). On the other hand the right side is a weighted-average of the n + 1 numbers

{c_{r}^{(s)}(n, k) | 0 ≤ k ≤ n} ,

with weights \{\binom{n}{k}^{s} | k = 0, …, n\}. Since (recall the Central Limit Theorem, see De Bruijn’s classic, section 3.4, for details) most of the weight is in the middle, we have that the above Apéry limit has an explicit description, namely

lim_{n→∞} c_{r}^{(s)}(n, ⌊n/2⌋) .

For any specific r, one can express c_{r}^{(s)}(n, k) in terms of partial sums of the harmonic series or partial sums of ζ(i) (i.e. ∑_{k=1}^{n} \frac{1}{k^{i}}). As r gets larger, things get complicated but our Maple package Zudilin.txt can handle it easily (it is implemented by procedure CrL(A,r,L)). In particular, for r = 2, things are still fairly easy. We have the following lemma, whose proof is omitted. The proof involves elementary manipulations with power series, fully automated in our Maple package, and is too boring to reproduce here.

Lemma 1:

c_{2}^{(s)}(n, k) = \frac{s}{2} ( ∑_{i=1}^{k} \frac{1}{i^{2}} + ∑_{i=1}^{n−k} \frac{1}{i^{2}} ) + \frac{s^{2}}{2} ( − ∑_{i=1}^{k} \frac{1}{i} + ∑_{i=1}^{n−k} \frac{1}{i})^{2} .

Note that in particular, the Apéry limit for the case r = 2 in the Straub-Zudilin t-version of Franel equals s ζ(2).

For each specific integer s, (and also for each specific even integer r) we can prove that (for sufficiently large s) you get ζ(r), as is proved, in general, using human ingenuity, in [SZ].

Generalized Franel Sums

The Zeilberger algorithm, just as easily, can find a homogeneous linear recurrence equation with polynomial coefficients for the generalized Franel sum

∑_{k=0}^{n} \binom{n}{k}^{s}a^{k} ,

for any positive rational number a (or for that matter even for symbolic a). It turns out that for all the s that we tried, the same miracle, i.e. that the right side of the linear inhomogeneous recurrence satisfied by

∑_{k=0}^{n} \frac{n!^{s}}{(1 + t)_{k}^{s}(1 − t)_{n−k}^{s}} · a^{k} ,

is divisible by t^{r} for r ≤ s. It follows that the first s Taylor coefficients vanish. We are sure that this is provable in general using the method of [SZ], but we leave it to the interested reader.

Let us now state the following result.

Theorem 1: For any positive rational number a, and for s ≥ 3, and for 1 ≤ r < s if s is odd, and for 1 ≤ r ≤ s − 1 if s is even. Let α be the such that, as n → ∞, k = αn maximizes the summand

\binom{n}{k}^{s}a^{k}

(this can be easily found by computing the ratio of consecutive terms, setting it equal to 1 and solving for α as n → ∞, it is implemented in our Maple package by procedure FindMaxk(F,n,k)). (Note that α is always an algebraic number, that our Maple package can always find in each case.) Let B(n) be ∑_{k=0}^{n} \binom{n}{k}^{s}a^{k} and let A(n) be solution of the same recurrence, of order L, say, that is satisfied by B(n) by the Zeilberger algorithm, but with the initial conditions extracted from the coefficients of t^{r} in ∑_{k=0}^{n} (\frac{n!}{(1+t)_{k}(1−t)_{n−k}} )^{s} a^{k} , for n = 0, …, n = L − 1, then the sequence \frac{A(n)}{B(n)} converges with exponentially decaying error to the constant

lim_{n→∞} c_{r}^{(s)}(n, ⌊αn⌋) .

Disclaimer: We took the liberty of calling the above statement ”Theorem”, and we are sure that it is correct, but we did not work out the full details of the proof. Wadim Zudilin informed us that indeed he strongly believes that the full proof should follow along the lines described in [deB], section 3.4. and [SZ].

The proof of the exponential decay of error property, in general for all such binomial coefficients sums, and more generally, for any binomial coefficient sum with positive summand, follows the same lines as the special case of Apéry’s original proof in [vdP] (p. 196), and is omitted here. Note that for any specific binomial coefficient sum, it is proved fully rigorously, ab initio, by the computer.

Note that trying to evaluate this limit from the definition would take for ever, since the convergence is so slow. So we can get lots of minor Apéry miracles.

The output files https://sites.math.rutgers.edu/~zeilberg/tokhniot/oZudilin1.txt and
https://sites.math.rutgers.edu/~zeilberg/tokhniot/oZudilin2.txt contain many exam-

ples.

Let’s explain why α is algebraic. The summand \binom{n}{k}^{s}a^{k} is clearly unimodal, hence it is maximal for that value of k when the ratio of consecutive terms satisfy

(\frac{n − k + 1}{k})^{s} a ≤ 1 < (\frac{n − k}{k + 1} )^{s} a

Writing k = αn, and remembering that n → ∞, we have

(1 − α)^{s}a = 1 ,

and this an algebraic equation for α for any specific positive integer s and positive a.

The Zudilin-Straub t-Transform

The beautiful construction in [SZ] was obtained by replacing k! by (1+t)_{k} and (n−k)! by (1−t)_{n−k}. This naturally leads to

Definition 5: The Zudilin-Straub t-transform of the proper hypergeometric term given in (1) is

F̂(n, k; t) = P (n, k + t) \frac{∏_{i=1}^{uu}(b_{i}t + 1)_{a_{i}n+b_{i}k+c}}{∏_{i=1}^{vv}(v_{i}t + 1)_{u_{i}n+v_{i}k+w_{i}}} x^{k} .

It is immediate to see that the recurrence obtained via the Zeilberger algorithm applied to the Zudilin-Straub t-transform of any proper hypergeometric term is the same as the original, except that the right side is not zero, i.e., since the right side of Equation (2) is no longer 0, the linear recurrence is inhomogeneous Unfortunately, in general, the Franel-Straub-Zudilin miracle does not occur, but it is easy to see that for any specific positive integer r the coefficient of t^{r} in the Taylor expansion of the right side is still P -recursive in n (i.e. satisfies a linear recurrence equation with polynomial coefficients), and a recurrence for it can be algorithmically obtained, either by the ‘holonomic machine’ ([ApaZ] [K]), or by ‘guessing’ that can be made fully rigorous using the general theorems of [ApaZ]. In this way we can get lots of generalized Apéry limits describing constants defined as limits of explicit sequences that converge very slowly. Rather than stating the formal theorem we refer the reader to the output files given in the front of this article

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/zudilin.html .

In particular, we recommend the following output files

https://sites.math.rutgers.edu/~zeilberg/tokhniot/oZudilin1.txt

for many examples of Theorem 1, and the output file

https://sites.math.rutgers.edu/~zeilberg/tokhniot/oZudilin3a.txt , and
https://sites.math.rutgers.edu/~ zeilberg/tokhniot/oZudilin5.txt ,

for examples where the recurrence equation for B(n) is inhomogeneous

The Maple package Zudilin.txt implements everything. Once you download it to your computer (that has Maple), load it to a Maple session with read `Zudilin.txt`. To get a list of the main functions, type ezra();, and to get help with a specific function, type ezra(FunctionName). For example, to get help with procedure ZT (that implements the Zudilin-Straub t-transform) type

ezra(ZT);. Enjoy!

Conclusion

While major Apéry miracles are few and far between, thanks to the Zudilin-Straub t-transform, we can obtain many minor Apéry miracles. The front of this article contains numerous examples, but using our Maple package, readers can generate many new ones.

Acknowledgment: Many thanks are due to Wadim Zudilin and the two referees, whose insightful and useful remarks considerably improved the exposition.

References

[A] Roger Apéry, “Interpolation de fractions continues et irrationalité de certaine constantes” Bulletin de la section des sciences du C.T.H.S. #3 p. 37-53, 1981.

[ApaZ] Moa Apagodu and Doron Zeilberger, Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger Theory , Adv. Appl. Math. 37 (2006)(Special Regev issue), 139-152.

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/multiZ.html .

[CS] Marc Chamberland and Armin Straub, Apéry limits: experiments and proofs, Amer. Math. Monthly 128 (2021), 811-824.

https://arxiv.org/abs/2011.03400 .

[deB] N.G. De Bruijn, Asymptotic methods in Analysis, Second Edition. North-Holland, 1961.

[DKZ] Robert Dougherty-Bliss, Christoph Koutschan, and Doron Zeilberger, Tweaking the Beukers integrals in search of more miraculous irrationality proofs á la Apéry, Ramanujan Journal, published-on-line before print, volume and page number tbd.

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/beukers.html

[DZ] Robert Dougherty-Bliss and Doron Zeilberger, Experimenting with Apéry limits and WZ pairs, Maple Transactions, v.1, issue 2 (2021).

https://mapletransactions.org/index.php/maple/article/view/14359
https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/wzp.html .

[K] Christoph Koutschan, Advanced applications of the holonomic systems approach, PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009.

http://www.koutschan.de/publ/Koutschan09/thesisKoutschan.pdf .

[PWZ] M. Petkovsek, H. S. Wilf and D. Zeilberger, A=B, AK Peters, Wellesley, (1996).

https://sites.math.rutgers.edu/~zeilberg/AeqB.pdf .

[SZ] Armin Straub and Wadim Zudilin, Sums of powers of binomials, their Apéry limits, and Franel’s suspicions, Intern. Math. Research Notices (to appear), 19 pages.

https://arxiv.org/abs/2112.09576 .

[vdP] Alf van der Poorten, A proof that Euler missed... Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer 1 (1979), 195-203.

[Ze1] Doron Zeilberger, The method of creative telescoping, J. Symbolic Computation 11 (1991), 195-204.

https://sites.math.rutgers.edu/~zeilberg/mamarimY/creativeT.pdf .

[Ze2] Doron Zeilberger, Closed form (pun intended!), in: “Special volume in memory of Emil Grosswald”, M. Knopp and M. Sheingorn, eds., Contemporary Mathematics 143, 579-607, AMS, Providence (1993).

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pun.html ,

[Ze3] Doron Zeilberger, Computerized deconstruction, Advances in Applied Mathematics 30 (2003), 633-654.

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/derrida.html .

[ZeZu] Doron Zeilberger and Wadim Zudilin, Automatic Discovery of Irrationality Proofs and Irrationality Measures, International Journal of Number Theory 17 (2021), 815-825.

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gat.html

Robert Dougherty-Bliss , Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: robert dot w dot bliss at gmail dot com .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .