Computerizing the AndrewsFraenkelSellers Proofs on the Number of mary 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 webpage: 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 ninetynine percent perspiration.
Now that we have computers, they can do the perspiration part for us, but we need
metainspiration, metageniuses,
and metaperspiration, 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, metainspiration, and metaperspiration,
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 mary 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 AndrewsFraenkelSellers 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 numbertheoretical
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+m1) mod m of the functional equation
f(q)=1/((1q)*(1q^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/(1q)*f(q^m)
for m from 2 to 10
the input yields
the output

If you want to see the quasipolynomial 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 googolth through the googol+100th terms (mod 10)
the input yields
the output


If you want to see the generalized capsules for the coefficients of c(m*n+m1) mod m of the solutions to the Inhomogeneous Functional Equation
f(q)=1+q/((1q)*(1q^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 quasipolynomial version of the above
the input yields
the output

If you want to see verbose versions of the above and the googolth through the googol+100th 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/(1q)^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 quasipolynomial version of the above
the input yields
the output

If you want to see verbose versions of the above and the googolth through the googol+100th 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/(1q^2)^7 *f(q)
the input yields
the output

If you want to see the quasipolynomial (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 googolth through the googol+100th 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/((1q)^5*(1q^2)^3) *f(q^17))
the input yields
the output

If you want to see the quasipolynomial (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 googolth through the googol+100th terms (mod 10)
the input yields
the output
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
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