Notes on Euler’swork on divergent factorial series and their associated continued fractions Authors
First Online: 11 April 2010 DOI:
Cite this article as: Digernes, T. & Varadarajan, V.S. Indian J Pure Appl Math (2010) 41: 39. doi:10.1007/s13226-010-0019-8
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.
hypergeometric continued fractions
L. Euler, De seriebus divergentibus,
Opera Omnia, I, 14, 585–617.
E. J. Barbeau and P. J. Leah, Euler’s 1760 paper on divergent series,
MATH CrossRef MathSciNet
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.
Lecons sur les Series Divergentes, Éditions Jacques Gabay, 1988 (reprinting of the original 1928 work).
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.
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.
K. G. Ramanathan, Hypergeometric series and continued fractions,
Proc. Indian Acad. Sci. (Math. Sci).
Letter, April 19, 2007.
M. F. Smiley, A proof of Sturm’s theorem,
Amer. Math. Monthly
CrossRef MathSciNet Copyright information
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