Experimenting with Apéry Limits and WZ pairs

By Robert Dougherty-Bliss and Doron Zeilberger

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Fisrt Written: Sept. 10, 2021. This version: Sept. 20, 2021.

published in Maple Transactions, v.1, issue 2 (2021), Article 14359.

In fond memory of the amazing Borwein brothers: Jonathan (20 May 1951 - 2 August 2016) and Peter (10 May, 1953 - 23 August 2020), great pioneers of Experimental Mathematics

This article, dedicated with admiration in memory of Jon and Peter Borwein, illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered, experimentally, an infinite family of cubic irrationalities, that suggests very good effective irrationality measures (lower than Liouville's generic 3), and that seemed to go down to the optimal 2. Paul Voutier (see the postscript by him) pointed out that this family was already known.

Maple packages

• AperyWZnew.txt, a Maple package to investigate generalizations of Apery's proof with other WZ pairs

• WZpairs.txt, a Maple package to find WZ pairs

• AperyLimits.txt, to invesitage Apery limits given by binomial coefficients sums (using the Zeilberger algorithm) , and contour integrals (using the Almkvist-Zeilberger algorithm).

for the sake of history here is the original version

Sample Input and Output for AperyWZnew.txt

• If you want to see rigorous proofs of a weaker forms of the conjetures of Marc Chamberland and Armin Straub, and empirical proofs of the full conjectures, the
the input gives the output .

• If you want to see, in a few seconds a computerzied redux of the Alladi-Robinson irrationality proof of log(2) and the Apery proofs for Zeta(2) and Zeta(3),
the input gives the output .

• If you want to see, conjectured Diophantine approximations to weired constants, enegenered by , what our article called the WZ-pair Chu(r,r,0,n) for small rational r, none of them, unfortunately leading to irrationality proof, but that are nevertheless interesting,
the input gives the output .

Sample Input and Output for WZpairs.txt

• If you want to see WZ-pairs that come from the kernels that lead to the Apery irrationality proofs of log(2), Zeta(2), and Zeta(3), from scratch
the input gives the output .

Sample Input and Output for AperyLimits.txt

• If you want to see lots of Apery Limits, generated by binomial coefficients sum and the Zeilberger algorithm, together with their empirical deltas
the input gives the output .

• If you want to see even more such Apery Limits (still using the Zeilberger algorithm, but with a more coplicated summands)
the input gives the output .

• If you want to see lots of Apery Limits, (handpicked such that they all have positive deltas, hence yielding irrationality proofs, alas, mostly to constants already proved to be irrational a long time ago, and those that Maple was unable to identify, probably also are such, but who knows?, generated by constant term expressions, using the Almkvist-Zeilberger algorithm
the input gives the output .

• If you want to see lots of Apery Limits, (handpicked such that they all have positive deltas, hence yielding irrationality proofs, alas, mostly to constants already proved to be irrational a long time ago, and those that Maple was unable to identify, probably also are such, but who knows?, generated by binomial coefficients sum where the summand is a product of the form binomial(n,k)*binomial(a1*n+b1*k,n)*x^k for small positive integres a1,b1,x
the input gives the output .

• If you want to see very fast, and good diopantine approximations to the cubic root of

64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c)=0 (c pos. integer, larger than 2)

that as c gets larger, seem to imply (modulo filling-in the details) EFFECTIVE rational measures better than Liouville's 3, and as c gets larger, it seems to converge to the best-possible 2 (as proved, non-effectively, by Klaus Roth)
the input gives the output .

• If you want to see optimal diopantine approximations to the quadratic irrationality

-3*c-3/2-3*(c^2+c)^(1/2)

that gives the same, optimal irrationality measure, namely 2, via a second-order recurrence that is NOT constant coefficients
the input gives the output .