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Let ( R, m, k ) be an artinian Gorenstein ring with      dim k m / m 2 ⩾  2 {\dim _k}m/{m^2} \geqslant 2      . If      P R {P^R}       denotes the Poincaré-series and      ϕ  R {\phi _R}       denotes the Bass-series of  R , then      ϕ  R = ( P R −  1 ) x −  1 ( 1 −  P R x 2 ) −  1 {\phi _R} = ({P^R} - 1){x^{ - 1}}{(1 - {P^R}{x^2})^{ - 1}}       with      R ¯  = R / 0 : m \bar R = R/0:m      , see Proposition 1. With similar methods as in the proof of Proposition 1 we calculate the Poincaré-series of a certain class of local rings: Let ( R, m, k ) be a local ring,      ( α  i j ) ({\alpha _{ij}})       a      n −  1 ×  n n - 1 \times n      -matrix with entries in  m ,      n ⩾  3 n \geqslant 3      . Let      x 1 , …  , x n {x_1}, \ldots ,{x_n}       be elements in  m  and      A \mathfrak {A}       the ideal generated by the maximal minors of      ( α  i j ) ({\alpha _{ij}})       and by      Σ  j α  i j x j ( i = 1 , …  , n −  1 ) {\Sigma _j}{\alpha _{ij}}{x_j}(i = 1, \ldots ,n - 1)      . If depth      A = n \mathfrak {A} = n      , then      P R / A ⋅  ( ( 1 −  X ) n −  1 −  X ) ( 1 + X ) n = P R {P^{R/\mathfrak {A}}} \cdot ({(1 - X)^{n - 1}} - X){(1 + X)^n} = {P^R}      .

Two applications of change of rings theorems for Poincaré series