In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.
Definition Let Kn be polynomials over a ring A in indeterminates p1, ... weighted so that pi has weight i (with p0 = 1) and all the terms in Kn have weight n (in particular Kn is a polynomial in p1, ..., pn). The sequence Kn is multiplicative if the map
K : ∑ n = 0 ∞ q n z n ↦ ∑ n = 0 ∞ K n ( q 1 , ⋯ , q n ) z n {\displaystyle K:\sum _{n=0}^{\infty }q_{n}z^{n}\mapsto \sum _{n=0}^{\infty }K_{n}(q_{1},\cdots ,q_{n})z^{n}}
is an endomorphism of the multiplicative monoid ( A [ x 1 , x 2 , ⋯ ] [ [ z ] ] , ⋅ ) {\displaystyle (A[x_{1},x_{2},\cdots ][[z]],\cdot )} , where q n ∈ A [ x 1 , x 2 , ⋯ ] {\displaystyle q_{n}\in A[x_{1},x_{2},\cdots ]} . The power series
K ( 1 + z ) = ∑ K n ( 1 , 0 , … , 0 ) z n {\displaystyle K(1+z)=\sum K_{n}(1,0,\ldots ,0)z^{n}}
is the characteristic power series of the Kn. A multiplicative sequence is determined by its characteristic power series Q(z), and every power series with constant term 1 gives rise to a multiplicative sequence. To recover a multiplicative sequence from a characteristic power series Q(z) we consider the coefficient of z j in the product
∏ i = 1 m Q ( β i z ) {\displaystyle \prod _{i=1}^{m}Q(\beta _{i}z)\ }
for any m > j. This is symmetric in the βi and homogeneous of weight j: so can be expressed as a polynomial Kj(p1, ..., pj) in the elementary symmetric functions p of the β. Then Kj defines a multiplicative sequence.
Examples As an example, the sequence Kn = pn is multiplicative and has characteristic power series 1 + z. Consider the power series
Q ( z ) = z tanh z = 1 − ∑ k = 1 ∞ ( − 1 ) k 2 2 k ( 2 k ) ! B k z k {\displaystyle Q(z)={\frac {\sqrt {z}}{\tanh {\sqrt {z}}}}=1-\sum _{k=1}^{\infty }(-1)^{k}{\frac {2^{2k}}{(2k)!}}B_{k}z^{k}\ }
where Bk is the k-th Bernoulli number. The multiplicative sequence with Q as characteristic power series is denoted Lj(p1, ..., pj). The multiplicative sequence with characteristic power series
Q ( z ) = 2 z sinh 2 z {\displaystyle Q(z)={\frac {2{\sqrt {z}}}{\sinh 2{\sqrt {z}}}}\ }
is denoted Aj(p1,...,pj). The multiplicative sequence with characteristic power series
Q ( z ) = z 1 − exp ( − z ) = 1 + x 2 − ∑ k = 1 ∞ ( − 1 ) k B k ( 2 k ) ! z 2 k {\displaystyle Q(z)={\frac {z}{1-\exp(-z)}}=1+{\frac {x}{2}}-\sum _{k=1}^{\infty }(-1)^{k}{\frac {B_{k}}{(2k)!}}z^{2k}\ }
… excerpt ends here. Continue reading the full article.
