Vol. 19, No. 3, 1966

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The exponential analogue of a generalized Weierstrass series

George Joseph Kertz and Francis Regan

Vol. 19 (1966), No. 3, 461–466

The generalized Weierstrass series

∞∑    --zn---
an1 + z2n

has as its exponential analogue

∑∞      e−λnz

where {an} is a sequence of complex-valued constants and {λn} is any real-valued strictly monotone increasing unbounded sequence.

In this paper the λn will be chosen to be ln n. Then the above series becomes

      ∑∞      n−z
A(z) =   an 1+-n−2z,

hereafter called simply the A-series. In its region of absolute convergence an A-series can be expressed as a Dirichlet series; conversely, a Dirichlet series can be represented by an A-series. Under restrictions on the sequence {an}, the imaginary axis becomes a natural boundary of the function represented by the A-series.

Mathematical Subject Classification
Primary: 30.24
Received: 4 October 1965
Published: 1 December 1966
George Joseph Kertz
Francis Regan