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Multiplicative sequence

Concept in mathematics

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Source updated February 13, 2026

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.

01Definition

Let K_n be polynomials over a ring A in indeterminates p₁, ... weighted so that p_i has weight i (with p₀ = 1) and all the terms in K_n have weight n (in particular K_n is a polynomial in p₁, ..., p_n). The sequence K_n is multiplicative if the map

$K:\sum _{n=0}^{\infty }q_{n}z^{n}\mapsto \sum _{n=0}^{\infty }K_{n}(q_{1},\cdots ,q_{n})z^{n}$

is an endomorphism of the multiplicative monoid $(A[x_{1},x_{2},\cdots ][[z]],\cdot )$ , where $q_{n}\in A[x_{1},x_{2},\cdots ]$ .

The power series

$K(1+z)=\sum K_{n}(1,0,\ldots ,0)z^{n}$

is the characteristic power series of the K_n. 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

$\prod _{i=1}^{m}Q(\beta _{i}z)\$

for any m > j. This is symmetric in the β_i and homogeneous of weight j: so can be expressed as a polynomial K_j(p₁, ..., p_j) in the elementary symmetric functions p of the β. Then K_j defines a multiplicative sequence.

02Examples

As an example, the sequence K_n = p_n is multiplicative and has characteristic power series 1 + z.

Consider the power series

$Q(z)={\frac {\sqrt {z}}{\tanh {\sqrt {z}}}}=1-\sum _{k=1}^{\infty }(-1)^{k}{\frac {2^{2k}}{(2k)!}}B_{k}z^{k}\$

where B_k is the k-th Bernoulli number. The multiplicative sequence with Q as characteristic power series is denoted L_j(p₁, ..., p_j).

The multiplicative sequence with characteristic power series

$Q(z)={\frac {2{\sqrt {z}}}{\sinh 2{\sqrt {z}}}}\$

is denoted A_j(p₁,...,p_j).

The multiplicative sequence with characteristic power series

$Q(z)={\frac {z}{1-\exp(-z)}}=1+{\frac {x}{2}}-\sum _{k=1}^{\infty }(-1)^{k}{\frac {B_{k}}{(2k)!}}z^{2k}\$

is denoted T_j(p₁,...,p_j): these are the Todd polynomials.

03Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials with characteristic power series ${\frac {z}{1-\exp(-z)}}$ .

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Sources and credits

This article is adapted from the Wikipedia article “Multiplicative sequence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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