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On The Product of Two Power Series

Published online by Cambridge University Press:  20 November 2018

H. Davenport  and
G. Pôlya
H. Davenport
Affiliation:
University College, London
G. Pôlya
Affiliation:
Stanford University
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We consider the product of two power series with positive coefficients: .

What conditions will ensure that the coefficients wn shall be either (i) monotonic, or (ii) logarithmically convex? By the latter, we mean that wn2 ≤ Wn-1Wn+1 for n = 1, 2, … In investigating this question, which was suggested by a special example, we have found it convenient to express the conditions in terms of the ratios of un, vn to certain binomial coefficients, rather than in terms of un, vn themselves.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 1 - 5
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 lMath. ZeiL, vol. 28 (1928), 161-170.