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) mod m of the functional equation

f(q)=1/(1-q)*f(q^m)

for m from 2 to 10

the input   yields the output

• If you want to see a verbose version (with a humanly readable proof) of the scheme for the coefficients c(19*n) mod 19 of the functional equation

f(q)=1/(1-q)*f(q^19)

the input   yields the output

• 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

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger