Abstract
Glaisher’s formulas for \({\dfrac{1} {\pi }^{2}}\) are reviewed. Two generalized formulas are proved by using the WZ-method (named after Wilf and Zeilberger). Also an improvement of Fritz Carlson’s theorem (proved in an Appendix by Arne Meurman) is used.
In memory of Herb Wilf
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Bauer, Von den Coefficienten der Kugelfunctionen einer Variablen, J. Reine Angew. Math. 56 (1859), 101–129.
W. Chu, Dougall’s bilateral 2 H 2 − series and Ramanujan-like π − formulae, Math. of Comp. 80 (2010), 2253–2251.
J. W. L. Glaisher, On series for \(\dfrac{1} {\pi }\) and \({\dfrac{1} {\pi }^{2}}\), Quart. J. Math. 37 (1905), 173–198.
P. Levrie, Using Fourier-Legendre expansions to derive series for \(\dfrac{1} {\pi }\) and \({\dfrac{1} {\pi }^{2}}\), Ramanujan J. 22 (2010), 221–230.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Almkvist, G. (2013). Glaisher’s Formulas for \({\frac{1} {{\pi }^{2}}}\) and Some Generalizations. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-30979-3_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30978-6
Online ISBN: 978-3-642-30979-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)