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

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

2010-01-26

arxiv.org/abs/1001.4695v2

American Mathematical Monthly Vol. 118, No. 2, pp. 136-152 (2011)

Mathematics

Classical Analysis and ODEs

19+ pages, 2 figures

Scientific paper

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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