Bijective and Automated Approaches to Abel Sums

By Gil Kalai and Doron Zeilberger

.pdf    .tex

Written: May 1, 2024

Dedicated to Dominique Foata (b. Oct 12, 1934) on his forthcoming 90th birthday.

We present both bijective and automated approaches to Abel-type sums, dear to Dominique Foata.

Added May 15, 2024: read master blogger Gil Kalai's wonderful post

Maple packages

• AbelBijection.txt, a Maple package that implements the bijection in the paper, and its inverse

• AbelCeline.txt, a Maple package to automatically find functional recurrences to generalized Abel sum, using an algorithm due to John Majewicz, inspired by Sister Celine Fasenmyer

Sample Input and Output for AbelBijection.txt

• If you want to see 10 random examples of the bijection, and 10 of its inverse
the input file yields the output file.

Sample Input and Output for AbelCeline.txt

• If you want to see a few examples of statements (without proofs) of functional recurrences for a sampling of Abel sums
the input file yields the output file.

• If you want to see the above, with proof,
the input file yields the output file.

• If you want to see the statement (w/o proof) of functional recurrences for Abel Sums of the form

Sum(1/((n-k)!*k!^a)*x^k*(r+k)^(k-1+p)*(s-k+q)^(n-k)*x^k,k=0..n) for a from 1 to 6

the input file yields the output file.

• If you want to see the statement (with proof) of the above
the input file yields the output file.

• If you want to see the statement (w/o proof) of functional recurrences satisfied by the Abel sum (here p and q are any integers)

Sum(binomial(n,k)^2*x^k*(r+k)^(k-1+p)*(s-k+q)^(n-k)*x^k,k=0..n)

the input file yields the output file.

• If you want to see the above statement, with proof
the input file yields the output file.

• If you want to see the statement (w/o proof) of the (very complicated!) functional-recurrence equation satisfied by the Abel sum

Sum(binomial(n,k)*binomial(n+k,k)*(r+k)^(k-1+p)*(s-k+q)^(n-k)*x^k,k=0..n)

the input file yields the output file.

• If you want to see the statement (with proof) of the above
the input file yields the output file.

• If you want to see the statement (whose proof is routine) for differential recurrence relations relating Abel Sums of the from (for any number x, the equations are the same, only the initial conditions differ. Here p and q are arbitrary integers.

Sum(binomial(n,k)^a*x^k*(r+k)^(k-1+p)*(s-k+q)^(n-k)*x^k,k=0..n) for a from 1 to 6

the input file yields the output file.

• If you want to see the statement (whose proof is routine) for differential recurrence relations relating Abel Sums of the from (for any number x, the equations are the same, only the initial conditions differ. Here p and q are arbitrary integers.

Sum(binomial(n,k)^a*binomial(n+k,k)^b*x^k*(r+k)^(k-1+p)*(s-k+q)^(n-k)*x^k,k=0..n) for a and b from 1 to 4

the input file yields the output file.

• If you want to see the statement (whose proof is routine) for differential recurrence relations relating Abel Sums with more complicated kernels

the input file yields the output file.