Multiplicative sequence

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

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

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

The power series

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

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

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

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.

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

Consider the power series

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

where 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

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

is denoted A_j(p₁,...,p_j).

The multiplicative sequence with characteristic power series

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

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

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 ${\displaystyle {\frac {z}{1-\exp(-z)}}}$.

• Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.