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On certain infinite products II

Published online by Cambridge University Press:  26 February 2010

Masao Toyoizumi
Masao Toyoizumi
Affiliation:
Department of Mathematics, Toyo University, Kawagoe-Shi, Saitama 350, Japan.
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Extract

Given a sequence a(l), a(2), a(3), … of complex numbers such that a(n) ≤ 0(nc) for some c > 0, we define, for Im(z) > 0,

where q(λ) = exp (2πiz/λ), λ > 0 and a is a real number. Throughout this paper, for complex numbers x, w with x ≠ 0, xw = exp (w log x) and the principal branch is taken for the logarithm. Then it is easily verified that the infinite product converges absolutely and uniformly in every compact subset of the upper half plane H. Hence f(z) is holomorphic in H. The aim of this paper is to determine holomorphic functions in H defined by (1) which satisfy the special transformation formula

for some real number k under certain assumptions.

Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 1 - 11
Copyright
Copyright © University College London 1984

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References

1.Hecke, E.. Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung. Math. Ann., 112 (1936), 664–699.CrossRefGoogle Scholar
2.Toyoizumi, M.. On certain infinite products. Muthematika, 30 (1983), 4–10.CrossRefGoogle Scholar