Chromatic Calculations for the S1 × S1 Transfer

where on the left hand side the coefficient of XrY s is βrβs. If n is a natural number then β(X) = β([n]X), where [n]X = [n]F X is the n-series. We alse have β(X)−1 = β([−1]X), where F ([−1]X,X) = 0 = F (X, [−1]X). In fact, for any non-zero q ∈ Z(p), we can make sense of [q]X and β(X) = β([q]X). Now recall the Araki and Hazewinkel generators vn, wn ∈ BP2pn−2 for which v0 = w0 = p and vn ≡ wn mod In, where In = (v0, . . . , vn−1) C BP∗. The Araki generators are defined by the identity