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2010

How to Add a Noninteger Number of Terms: From Axioms to New Identities

Mueller, Markus, Schleicher, Dierk

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Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends.

These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.
We illustrate how fractional sums can be used to derive infinite sum and special functions identities; the corresponding proofs turn out to be particularly simple and intuitive.

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R. W. Gosper, M. E. H. Ismail, and R. Zhang, On some strange summation formulas, Illinois J. Math
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V. S. Adamchik, Multiple gamma function and its application to computation of series, Ramanujan J
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M. Müller and D. Schleicher, How to add a non-integer number of terms, and how to produce unusual infinite summations, J. Comput. Appl. Math
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L. Euler, Dilucidationes in capita postrema calculi mei differentialis de functionibus inexplicabilibus [ E613
Cited in the paper.

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K. B. Oldham and J. Spanier, The Fractional Calculus
2007
Later among the works it cites.
———, Fractional sums and Euler-like identities, Ramanujan J
2010
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