Date: 11 Apr 2010
Notes on Euler’swork on divergent factorial series and their associated continued fractions
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective.
L. Euler, De seriebus divergentibus, Opera Omnia, I, 14, 585–617.
V. S. Varadarajan, Euler Through Time: A New Look at Old Themes, American Mathematical Society, 2006.
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1973.
E. Borel, Lecons sur les Series Divergentes, Éditions Jacques Gabay, 1988 (reprinting of the original 1928 work).
S. Ramanujan, Collected Papers, pp. 350–351, Chelsea, 1962.
B. C. Berndt and R. A. Rankin, Ramanujan-Letters and Commentary, pp. 29–30, American Mathematical Society and London Mathematical Society, 1995.
G. N. Watson, Theorems stated by Ramanujan (VIII): Theorems on divergent series, J. London Math. Soc., 4 (1929), 82–86.CrossRef
O. Perron, Die Lehre von den Kettenbrüchen, Chelsea, 1950.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1948.
T. J. Stieltjes, Collected Papers, Springer, 1993.
B. C. Berndt Ramanujan’s Notebooks, Part II, Springer, 1989.
P. Deligne, Letter, April 19, 2007.
- Notes on Euler’swork on divergent factorial series and their associated continued fractions
Indian Journal of Pure and Applied Mathematics
Volume 41, Issue 1 , pp 39-66
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Divergent series
- factorial series
- continued fractions
- hypergeometric continued fractions
- Sturmian sequences