Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!)
By
Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
Posted: Nov. 20, 2015.
Last update of this web-page: Dec. 31, 2018.
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, as well as in arxiv.org.]
Thomas Edison said that genius is one percent
inspiration and ninety-nine percent perspiration.
Now that we have computers, they can do the perspiration part for us, but we need
meta-inspiration, meta-geniuses,
and meta-perspiration, to teach the human inspiration to our
silicon colleagues. Sooner or later, computers will also do the inspiration part, but
let humans enjoy the remaining fifty (or whatever) years left for them, and focus on
inspiration, meta-inspiration, and meta-perspiration,
and leave the actual perspiration part to their much faster-
and much more reliable- machine friends.
In this short article,
two recent beautiful proofs
of George Andrews, Aviezri Fraenkel, and James Sellers,
about the mod m characterization of the number of m-ary partitions
are simplified and streamlined, and then generalized to handle
many more cases, and prove much deeper theorems, with the help
of computers, of course.
Added Dec. 31, 2018
Giedrius Alkauskas
informed me that the Andrews-Fraenkel-Sellers theorem should be called the Alkauskas theorem. It was discovered, and
first proved (page 9, line 3) in his
1999 Third Course thesis
(DOI: 10.13140/RG.2.2.23219.12321), written in Lithuanian.
Maple Package
-
AFS.txt,
To discover and prove congruence theorems for a wide class of number-theoretical
integer sequences whose generating functions satisfy a certain type of Functional Equations, with applications to partitions
into "sparse" parts
Some Input and Output files for the Maple package AFS.txt
-
-
-
If you want to see the Capsules for the coefficients c(m*n+m-1) mod m of the functional equation
f(q)=1/((1-q)*(1-q^m))*f(q^m)
for m from 3 to 40 (except when mod 4 =2, when they do not exist)
the input yields
the output
-
If you want to see a verbose version of the above for the case m=20,
the input yields
the output
-
-
If you want to see the generalized capsules for the coefficients of c(m*n+1) mod m of the solutions to the Inhomogeneous Functional Equation
f(q)=1+1/(1-q)*f(q^m)
for m from 2 to 10
the input yields
the output
-
If you want to see the quasi-polynomial version of the above
the input yields
the output
-
If you want to see a verbose version of the above for m=10, and the googol-th through the googol+100-th terms (mod 10)
the input yields
the output
-
-
If you want to see the generalized capsules for the coefficients of c(m*n+m-1) mod m of the solutions to the Inhomogeneous Functional Equation
f(q)=1+q/((1-q)*(1-q^2))*f(q^m)
for m from 3 to 20 (except for the case where m mod 4=2, where they do not exist)
the input yields
the output
-
If you want to see the quasi-polynomial version of the above
the input yields
the output
-
If you want to see verbose versions of the above and the googol-th through the googol+100-th terms (mod 10)
the input yields
the output
-
-
If you want to see the generalized capsules for the coefficients of c(m*n+1) mod m of the solutions to the Inhomogeneous Functional Equation
f(q)=1+q/(1-q)^2*f(q^m)
for m from 3 to 20 (if sometimes FAILS, and then it says so)
the input yields
the output
-
If you want to see the quasi-polynomial version of the above
the input yields
the output
-
If you want to see verbose versions of the above and the googol-th through the googol+100-th terms (mod 10)
the input yields
the output
-
-
To take just one random (successful example), if you want to see a Generalized capsule for
the coefficient of q^(17*n+5) mod 17 of the functional equation
f(q)=1+q+ q/(1-q^2)^7 *f(q)
the input yields
the output
-
If you want to see the quasi-polynomial (in this case a pure polynomial) version of the above
the input yields
the output
-
If you want to see verbose versions of the above and the googol-th through the googol+100-th terms (mod 10)
the input yields
the output
-
-
To take just one random (successful example), if you want to see a Generalized capsule for
the coefficient of q^(17*n+16) mod 17 of the functional equation
f(q)=1+ q/((1-q)^5*(1-q^2)^3) *f(q^17))
the input yields
the output
-
If you want to see the quasi-polynomial (in this case a pure polynomial) version of the above
the input yields
the output
-
If you want to see verbose versions of the above and the googol-th through the googol+100-th terms (mod 10)
the input yields
the output
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