Cohen-Macaulay local rings with e<inf>2</inf> = e<inf>1</inf> − e + 1

In this paper we study Cohen-Macaulay local rings of dimension d, multiplicity e and second Hilbert coefficient e2 in the case e2=e1−e+1. Let h=μ(m)−d. If e2≠0 then in our case we can prove that type(A)≥e−h−1. If type(A)=e−h−1 then we show that the associated graded ring G(A) is Cohen-Macaulay. In the next case when type(A)=e−h we determine all possible Hilbert series of A. In this case we show that depthG(A) completely determines the Hilbert Series of A.