
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/((nk)!*k!^a)*x^k*(r+k)^(k1+p)*(sk+q)^(nk)*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)^(k1+p)*(sk+q)^(nk)*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!) functionalrecurrence equation satisfied by the Abel sum
Sum(binomial(n,k)*binomial(n+k,k)*(r+k)^(k1+p)*(sk+q)^(nk)*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)^(k1+p)*(sk+q)^(nk)*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)^(k1+p)*(sk+q)^(nk)*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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