By Marko Petkovsek, Herbert S. Wilf

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**Example text**

You will now observe that the equation F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k) is true. Sum that equation over all integers k, and note that the right side telescopes to 0. , is constant. 5. Verify that the constant is 1 by checking that k F (0, k) = 1. ✷ The rational function R(n, k) is the key that turns the lock. The lock is the proof outlined above. If you want to prove an identity, and you have the key, then just put it into the lock and watch the proof come out. We’re going to illustrate the method now with a few examples.

3. Completely factor the polynomials P and Q into linear factors, and write the term ratio in the form P (k) (k + a1 )(k + a2 ) · · · (k + ap ) = x Q(k) (k + b1 )(k + b2 ) · · · (k + bq )(k + 1) If the factor k + 1 in the denominator wasn’t there, put it in, and compensate by inserting an extra factor of k + 1 in the numerator. Notice that all of the coefficients of k, in numerator and denominator, are +1. Whatever numerical factors are needed to achieve this are absorbed into the factor x. 4. You have now identified the input series.

4, −n ;1 . 1F1 1 3 Read in DiscreteMath‘RSolve‘ before attempting to FactorialSimplify something. 4 Software that identifies hypergeometric series 41 To finish on a positive note, we’ll ask Mathematica to identify quite a tricky sum for us, by entering Sum[(−1)ˆkBinomial[r − s − k, k]Binomial[r − 2k, n − k]/(r − n − k + 1), {k, 0, n}]. 2) which is extremely helpful. Next let’s try a session with Maple. The capability in Maple to identify a series as a pFq [· · · ] rests with the function convert/hypergeom.