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

By Robert Dougherty-Bliss and Doron Zeilberger

.pdf    .tex

published in Journal of Difference Equations and Applications, v. 29(1) (2023), 34-42.

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that `tweaked' the sum of the sth 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 an arbitrary binomial coefficients sum, gettings a practically unlimited supply of Apéry limits. While getting what we call "major Apéry miracles", proving irrationality of the associated constant (i.e. the so-called Apéry limit) 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.

Written: May 2022.

# Maple package

• Zudilin.txt, a Maple package to generate lots of Apéry limits

# Sample Input and Output for Zudilin.txt

• If you want to see 30 theorems with interesting Apéry limits arising from the coeficient of t2 of the Zudilin-Straub t-transform of Sum(binomial(n,k)^L*a^k,k=0..n) for L from 3 to 7 and a from 1 to 6,
the input file yields the output file

• If you want to see 144 theorems with interesting Apéry limits arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^L*a^k,k=0..n), taking the coefficient of tr, for for L from 3 to 8, for r from 2 to L-1, and a from 1 to 6,
the input file yields the output file

The theorems below involve generalized Apéry limits where the numerator sequence no longer satisfies a homogenous linear recurrence (like the denominator sequence that came from the binomial coefficient sum), but an inhomog. one, with, nevertheless, a holonomic right hand side. It takes much longer to discover, but once discoverd it gives again very fast converging sequences to constants whose definition involves extremely slow convergence]

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^3,k=0..n), taking the coefficient of t4,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^3*2^k,k=0..n), taking the coefficient of t4,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^3*3^k,k=0..n), taking the coefficient of t4,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^4,k=0..n), taking the coefficient of t4,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^4*2^k,k=0..n), taking the coefficient of t4,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^5,k=0..n), taking the coefficient of t6,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)^8,k=0..n), taking the coefficient of t8,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)*binomial(n+k,k),k=0..n), taking the coefficient of t2,
the input file yields the output file

• If you want to see a theorem with interesting (generalized) Apéry limit arising from the Zudilin-Straub t-transform of Sum(binomial(n,k)*binomial(n+k,k),k=0..n), taking the coefficient of t2,
the input file yields the output file

Articles of Doron Zeilberger