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==Definition==
Let ''K''<sub>''n''</sub> be polynomials over a ring ''A'' in indeterminates ''p''<sub>1</sub>,... weighted so that ''p''<sub>''i''</sub> has weight ''i'' (with ''p''<sub>0</sub> = 1) and all the terms in ''K''<sub>''n''</sub> have weight ''n'' (so that ''K''<sub>''n''</sub> is a polynomial in ''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>. The sequence ''K''<sub>''n''</sub> is ''multiplicative'' if an identity
:<math>\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i </math>
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:<math>\sum K_n(1,0,\ldots,0) z^n </math>
is the ''characteristic power series'' of the
To recover a multiplicative sequence from a characteristic power series ''Q''(''z'') we consider the coefficient of ''z''<sup>''j''</sup> in the product
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:<math> \prod_{i=1}^m Q(\beta_i z) \ </math>
for any ''m'' > ''j''. This is symmetric in the ''β''<sub>''i''</sub> and homogeneous of weight ''j'': so can be expressed as a polynomial ''K''<sub>''j''</sub>(''p''<sub>1</sub>,...''p''<sub>''j''</sub>) in the [[elementary symmetric function]]s ''p'' of the
==Examples==
As an example, the sequence ''K''<sub>''n''</sub> = ''p''<sub>''n''</sub> is multiplicative and has characteristic power series
Consider the power series
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:<math>Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k \ </math>
where ''B''<sub>''k''</sub> is the ''k''-th [[Bernoulli number]]. The multiplicative sequence with ''Q'' as characteristic power series is denoted ''L''<sub>''j''</sub>(''p''<sub>1</sub>, ..., ''p''<sub>''j''</sub>).
The multiplicative sequence with characteristic power series
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For example, the [[Todd genus]] is associated to the Todd polynomials with characteristic power series <math>\frac{z}{1-\exp(-z)}</math>.
==References==
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