Written: May 1, 2024

Appeared in the Seminaire Lotharingien de Combinatoire v. 91, paper B91a.

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.

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  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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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