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Gosper’s Algorithm

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Hypergeometric Summation

Part of the book series: Universitext ((UTX))

Abstract

For a moment, let’s have a break from searching for hypergeometric term solutions and recurrence equations of infinite series. Instead, we will deal with sums with variable limits of summation, an interesting topic in itself. Later, this will prove to be a useful tool in discovering an algorithmic method for infinite sums.

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Notes

  1. 1.

    Of course this is only so if \(f\) is absolutely continuous. Otherwise the Fundamental Theorem of Calculus might not be applicable.

  2. 2.

    Relation \(\gcd (f,g)=1\) means that there is no nontrivial, hence nonconstant, common divisor.

  3. 3.

    We use the same functions \(p_k, q_k\), and \(r_k\) as Gosper did, even though he worked with the backward antidifference. It may mean that we have some more shifts here than necessary.

  4. 4.

    For the sake of completeness let the maximum of the empty set equal \(-\infty \).

  5. 5.

    For convenience, we set \(\deg (0)\,{:=}\,-1\). Then (5.21) remains valid for non-zero constant \(f_k\).

  6. 6.

    The current author did the implementation of Version V.4 which is available in the sumtools package. However, this version is now superseded by the SumTools package.

  7. 7.

    It might be inefficient to use the modified implementation in the general case, though.

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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). Gosper’s Algorithm. In: Hypergeometric Summation. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6464-7_5

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