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On the Summability of Series by a Method of Valiron
Published online by Cambridge University Press: 20 January 2009
Extract
The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s if
This is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.
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- Copyright © Edinburgh Mathematical Society 1936
References
1 Summability (V, α) is usually defined by means of the limit
The definition which I have given makes for greater compactness throughout the paper.
2 Valiron, G., Rendiconti di Palermo, 42 (1917), 267–284.CrossRefGoogle Scholar
3 Hardy, G. H., Quarterly Journal, 35 (1904), 22–66.Google Scholar
4 Valiron, G., loc. cit.Google Scholar
5 Vijayaraghavan, T., Proc. London Math. Soc. (2), 27 (1927–1928), 316–326.Google Scholar
1 Whittaker, E. T. and Watson, G. N., Modern Analysis (1927), 475–476.Google Scholar For a proof of the particular case used above see MacRobert, T. M., Functions of a Complex Variable (1925), 116.Google Scholar
1 Hardy, G. H. and Littlewood, J. E., Annali di Pisa (2), 3 (1934), 54.Google Scholar
2 Vijayaraghavan, T., loc. cit.Google Scholar
3 Valiron, G., loc. cit.Google Scholar
4 Hardy, G. H. and Littlewood, J. E., Rendiconti di Palermo, 41 (1915), 1–18.Google Scholar
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