a1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay – 5, India.
Let an be an increasing sequence of real numbers and bn a sequence of positive real numbers. We deal here with the Dirichlet series and its Laurent expansion at the abscissa of convergence, λ, say. When an and bn behave like
as N → ∞, where P2(x) is a certain polynomial, we obtain the Laurent expansion of f (s) at s = λ, namely
where P1(x) is a polynomial connected with P2(x) above. Also, the connection between P1 and P2 is made intuitively transparent in the proof.
(Received July 26 1985)