Exploring General Apéry Limits via the ZudilinStraub ttransform
By Robert DoughertyBliss and Doron Zeilberger
.pdf
.tex
published in Journal of Difference Equations and Applications, v. 29(1) (2023), 3442.
Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that `tweaked' the sum of the s^{th} powers of
the nth 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 socalled Apéry limit)
is very rare, we do get, every time, at least a "minor Apéry miracle" where an explicit constant, defined as an (extremely slowlyconverging)
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 t^{2} of the ZudilinStraub ttransform 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 ZudilinStraub ttransform of Sum(binomial(n,k)^L*a^k,k=0..n), taking the coefficient of t^{r}, for
for L from 3 to 8, for r from 2 to L1, 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 ZudilinStraub ttransform of Sum(binomial(n,k)^3,k=0..n), taking the coefficient of t^{4},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^3*2^k,k=0..n), taking the coefficient of t^{4},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^3*3^k,k=0..n), taking the coefficient of t^{4},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^4,k=0..n), taking the coefficient of t^{4},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^4*2^k,k=0..n), taking the coefficient of t^{4},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^5,k=0..n), taking the coefficient of t^{6},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)^8,k=0..n), taking the coefficient of t^{8},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)*binomial(n+k,k),k=0..n), taking the coefficient of t^{2},
the input file
yields the output file

If you want to see a theorem with interesting (generalized) Apéry limit arising from the ZudilinStraub ttransform of Sum(binomial(n,k)*binomial(n+k,k),k=0..n), taking the coefficient of t^{2},
the input file
yields the output file
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