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Zeilberger’s Algorithm

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Book cover Hypergeometric Summation

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Abstract

In this chapter, we introduce Zeilberger’s extension of Gosper’s algorithm, using which one can not only prove hypergeometric identities but also sum definite series in many cases, if they represent hypergeometric terms.

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Notes

  1. 1.

    The sum can also have bounds \([-n,\ldots ,n]\) as the Dixon example has. We need mainly \(s_0=F(0,0)\ne 0\).

  2. 2.

    “To convince ourselves of the validity of Apéry’s proof we need only complete the following exercise: show that (7.9) is valid for (7.8). Neither Cohen nor I had been able to prove this in the intervening two months...”.

  3. 3.

    This can fail only if some of the zeros of the highest or lowest coefficient polynomials are integers.

  4. 4.

    Closedform is the same as closedform, with hyperterm replaced by the inert form Hyperterm in the output, preventing evaluation, and hence emphasizing the hypergeometric structure.

  5. 5.

    Without the option recursion=up, the procedure gives the recurrence equation in terms of downward shifts.

  6. 6.

    There is no recurrence w.r.t. \(x\), but w.r.t. \(y=ix\).

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  1. Fachbereich 10 Mathematik und Naturwissenschaften, Universität Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany

    Wolfram Koepf

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Koepf, W. (2014). Zeilberger’s Algorithm. In: Hypergeometric Summation. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6464-7_7

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