Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs À La Apéry
By Robert DoughertyBliss, Christoph Koutschan, and Doron Zeilberger
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Appendix(.pdf)
Posted: Jan. 13, 2021
In honor of Wadim Zudilin, on the occaison of his trunc(50*Zeta(5))th birthday
There are only aleph_{0} rational numbers, while there are 2^{aleph0} real numbers.
Hence the probability that a randomly chosen real number would be rational is 0.
Yet proving rigorously that any specific, natural, real constant, is irrational
is usually very hard, witness that there are still no proofs of the irrationality of the EulerMascheroni
constant, the Catalan constant, or ζ(5).
Inspired by Frits Beukers' elegant rendition of Apéry's seminal proofs of the irrationality of
ζ(2) and ζ(3), and heavily using WilfZeilberger algorithmic proof theory and Koutschan's efficient
Holnomic Functions programs, we systematically searched for other similar integrals, that lead to
irrationality proofs. We found quite a few candidates for such proofs, including π^{1/2}Γ(7/3)/Γ(1/6)
and π^{1/2}Γ(19/6)/Γ(8/3)
Maple packages

GenBeukersLog.txt,
a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of the
AlladiRobinson integrals that occurred in their elegant treatment of irrationality measures of log(2) and many other logs.

GenBeukersZeta2.txt,
a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral
that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(2) is irrational.

GenBeukersZeta3.txt,
a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral
that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(3) is irrational.
Sample Input and Output for GenBeukersLog.txt

If you want to see many candidates for Apérystyle proofs of generalized AlladiRobinson integrals,
many of them identifified (and since there either logs of rational numbers or explicit algebraic numbers will not get us famous)
but many of them not yet identified, so potentially not yet proved to be rational
The input file
yields the output file

If you want to see numerous sketches (modulo some fillingindetails for the divisibility lemmas) of
Apérystyle proofs similar to the AlladiRobinson irrationality proofs of log(c) for many rationals c,
including some that may be not yet proved to be irrational (at least Maple can't identify them)
The input file
yields the output file

If you want to see many candidates for Apérystyle proofs of supergeneralized AlladiRobinson integrals,
(i.e. the 4paramter family of constants
int(x^a*(1x)^b)/(1+c*x)^(d+1),x=0..1)/int(x^a*(1x)^b)/(1+c*x)^d,x=0..1)
many of them identified (and since there either logs of rational numbers or explicit algebraic numbers will not get us famous)
but many of them not yet identified, so potentially not yet proved to be rational
The input file
yields the output file
Sample Input and Output for GenBeukersZeta2.txt

If you want to see numerous sketches (modulo some fillingindetails for the divisibility lemmas) of
Apérystyle proofs similar to Beukers' irrationality proof of Zeta(2), most of them for constants
that are probably already proved to be irrational (e.g. log(2), sqrt(2), Pi*sqrt(3)) but a few that
Maple can't identify, and are potentially the first proof of their irrationality
The input file
yields the output file

If you want to see synopsises of irrationality proofs of some constants, some identified, some not yet
(hence giving hope that they are not yet proved to be irrational)
The input file
yields the output file

If you want to see lots of challenges for Wadim Zudilin, giving conjectured 3F2 evaluations.
[Not related to irrationality]
The input file
yields the output file
Sample Input and Output for GenBeukersZeta3.txt

If you want to see numerous sketches (modulo some fillingindetails for the divisibility lemmas) of
Apérystyle proofs similar to Beukers' irrationality proof of Zeta(3), most of them for constants
that are probably already proved to be irrational (e.g. log(2), sqrt(2), Pi*sqrt(3)) but a few that
Maple can't identify, and are potentially the first proof of their irrationality
The input file
yields the output file
Articles of Doron Zeilberger
Doron Zeilberger's Home Page
Robert DoughertyBliss 's Home Page
Christoph Koutschan's Home Page