S.C. Locke Abstracts

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Abstracts of Journal Articles

Expository paper:

MR3440477 Reviewed
Alvarez, Josefina(1-NMS); Locke, Stephen(1-FLAT)
Let's sum this series. (Spanish. English, Spanish summary)
Lect. Mat. 36 (2015), no. 2, 181 194.
40A05 (11B68 65B10)

This is a clearly written (in Spanish), rigorously detailed expository article on the basic properties of the real polylogarithm series at negative integer arguments, namely Σk=1 knck, for n = 0,1,2,3,..., and restricted to real values of c. Via exclusively elementary methods, suitable for discussion in an introductory course in analysis, the authors show how to compute the sum of this series and reveal its relation to the family of Eulerian polynomials. They derive some well-known closed-form expressions for the coefficients (the Eulerian numbers) and, in the last section, discuss some of their combinatorial interpretations and applications. In addition, they mention Euler's use of this series in relation to defining Riemann's zeta function at negative integers, which provides us with one of the early examples of how to "sum" a divergent series. All in all, there is sufficient material presented here to whet the interested reader's appetite and motivate him or her to travel further on some of the many paths that this series leads to. (Reviewed by Luis Manuel Navas Vicente.)

Note: It should be obvious that it was the first author who translated the paper into Spanish. Any inquiries about this paper directed at the second author would need to be in English.

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Last modified October 14, 2016, by S.C. Locke.