Searching for Apéry-Style Miracles
[Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]

By
Shalosh B. Ekhad and Doron Zeilberger

.pdf   .ps   .tex

First Written: May 17, 2014

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org]

Dedicated to Gert Almkvist (b. April 17, 1934), on becoming (at least, chronologically), an octogenerian (plus one month).
[Added Dec. 19, 2019: Sadly Gert Almkvist died on Nov. 24, 2018. He was one of the most original mathematicians that I have known]

Older mathematicians make beautiful provers! One of my great heroes is Roger Apéry who at the age of 64 astounded the mathematical world with a proof that

1/13+1/23+1/33+ 1/43+ ...

is not a ratio of two integers. Since very soon I will turn 64 myself, I was hoping to do the analogous thing for

1/15+1/25+1/35+ 1/45+ ...        .

Alas, since I am not nearly as smart as Apéry, my only hope was to teach to Shalosh Apéry's neat tricks, but to our disappointment we could not do it.

Nevertheless, once I taught Shalosh Roger's tricks, it reproduced in one second the original proof (see below), and found lots and lots of new irrationality proofs of many new constants, but none of them happened to be ζ(5), or any of the other `famous', yet-to-be-proven-irrational, constants.

But maybe YOU would be able to use the present Maple package, or an extension of it, to do it. Good luck!

This article is dedicated to yet another hero of mine, GERT ALMKVIST, who has just turned 80. Happy Birthday Gert, and may you continue to do great experimental math!

Input and Output files

• If you want to see all polynomials of the form
a/x+b+cx
where a,b,c, are integers between 1 and 15, that should (judging from empirical deltas) yield Apery-miracles for for contants that come from the sequence of constant terms of (a/x+b+cx)^n,

the input   yields the output

Note: In this output file the successful set is named Tovim.

• If you want to find out the disappointing fact that no polynomial of the form
a/x^2+b/x+c+dx+ex^2
for 1 ≤ a,b,c,d,e ≤ 5
yields (empirically) Apery-style miracles

the input   yields the output

Note that the output is the empty set.

• If you want to see sequences of empirical deltas, for various ratios of sequences arising from constant terms (using the amazing Almkvist-Zeilberger algorithm)

the input   yields the output

Note that for these examples they are all larger than 1, indicating miracles!

• If you want to see (using, again, the amazing Almkvist-Zeilberger algorithm), empirical deltas for sequences of rational numbers converging to various constants, given by Beukers-style integral representations, the

the input   yields the output

Note: only the last one, for arctan(1/5), produces an Apery-miracle, but the other ones are intersting too.

• If you want to see very verbose versions of the above, spelling out the actual rational approximations to the designated constants,

the input   yields the output

• If you want to see (using, now, the famous Zeilberger algorithm), empirical deltas for sequences of rational numbers converging to various constants, arising from binomial coefficients sum, like in the original Apéry proofs

the input   yields the output

Note: The first three are the famous Apery ones for log(2), Zeta(2), and Zeta(3), the fourth (disappointing) and fifth (good!) are random tries.

• If you want to see all polynomials of the form
a/x+b+cx
where a,b,c, are integers between 1 and 15, that PROVABLY (rigorously) yield Apery-miracles for for contants that come from the sequence of constant terms of (a/x+b+cx)^n,

the input   yields the output

Note: In this output file the successful set is named Tovim.

• If you want to see all polynomials of the form
a/x+b+cx+dx^2
where a,b,c,d are integers between 1 and 14, that PROVABLY (rigorously) yield Apery-miracles for for contants that come from the sequence of constant terms of (a/x+b+cx+dx^2)^n,

the input   yields the output

Note: In this output file the successful set is named Tovim.

• If you want to find out the disappointing fact that no polynomial of the form
a/x^2+b/x+c+dx+ex^2
for 1 ≤ a,b,c,d,e ≤ 8
yields (rigorously) Apery-style miracles

the input   yields the output

Note that the output is the empty set.

• If you want to see terse versions of fully-rigorous Apery-style proofs that come from constant-term expressions using the Amazing Almkvist-Zeilberger algorithm

the input   yields the output

• If you want to see VERBOSE versions (HUMANLY READABLE ARTICLES!) of fully-rigorous Apery-style proofs that come from constant-term expressions using the Amazing Almkvist-Zeilberger algorithm

the input   yields the output

Note: the last two sequences come from THIRD-order recurrences. Wow! So Apery-style miracles are not confined to sequences satisfying second-order recurrences.

• If you want to see fully-rigorous Beukers-style proofs using Integral represenations (but for single integrals rather than double or triple integrals), given in terse-style,

the input   yields the output

Note: only the last one yields irrationality, since there the (rigorous) delta is larger than 1.

• If you want to see verbose versions of these, with fully human-readable articles,

the input   yields the output

• If you want to see "virgin" data that only come from a linear recurrence with polynomial coefficients and initial values,

the input   yields the output

• If you want to see terse versions of Apery's famous proofs of the irrationality of log(2), Zeta(2), and Zeta(3)

the input   yields the output

• If you want to see verbose versions of Apery's famous proofs of the irrationality of log(2), Zeta(2), and Zeta(3), i.e. full-length humanly-readable proofs,

the input   yields the output

• If you want to see symbolic formulas for irrationality measure for families of constants, for various examples discussed in the article

the input   yields the output

• If you want to see symbolic formulas for irrationality measure for families of constants, for various examples discussed in the article

the input   yields the output

• If you want to see the input file for the symbolic formulas for irrationality measure for a 3-parameter constant, that yields a third-order recurrence equation. Here it is:
input
Unfortunately it took too long, so we don't have an output file.

• If you want to see 144 potential Apéry miracles

the input   yields the output

• If you want to see why the "first impressions" of Alf van der Poorten in his beautiful article: (Math. Intell. v. 1 (1979), 195-203), footnote 9, p. 202 were wrong`):

the input   yields the output

• If you want to see several examples of procedure RAaz NOT discussed in the present article (out of sheer laziness):

the input   yields the output

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger