Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-24T02:48:10.144Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of an identity of Euler

Published online by Cambridge University Press:  18 June 2020

Geoffrey Brown  and
Narayanaswamy Balakrishnan
Geoffrey Brown
Affiliation:
Queen's University, Department of Mathematics and Statistics, 48 University Avenue, Jeffery Hall, Kingston, Ontario, Canada, K7L 3N6 e-mail: 1geoffrey.brown@gmail.com
Narayanaswamy Balakrishnan
Affiliation:
McMaster University, Department of Mathematics and Statistics, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4K1 e-mail: bala@mcmaster.ca
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we state the original identity proved by Euler, and we provide an alternative proof of this result. We then extend the approach to derive a further identity for the sum of reciprocals of squares, along with two other related identities. We conclude by obtaining yet another extension of the identity, and this is accomplished by means of a probability density function, or pdf.

Type
Articles
Information
The Mathematical Gazette , Volume 104 , Issue 560 , July 2020 , pp. 241 - 246
Check for updates
Copyright
© Mathematical Association 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Harmonic number, Wikipedia (2019), available at https://en.wikipedia.org/wiki/Harmonic_numberGoogle Scholar
Finite Mathematics, Holt, Rinehart and Winston of Canada, Limited, Toronto, 1988. Exercise 4(c) p. 194.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N., Continuous univariate distributions – Vol. 2, (2nd edn.) John Wiley & Sons, New York, (1995).Google Scholar