Cauchy product

From Academic Kids

In mathematics, the Cauchy product of two sequences of real or complex numbers, named in honor of Augustin Louis Cauchy, is a discrete convolution given as follows. The Cauchy product of <math> (a_n)_{n=0}^\infty<math> and <math>(b_n)_{n=0}^\infty<math> is defined by

<math>c_n=\sum_{k=0}^n a_k b_{n-k}<math>

for each n=0, 1, 2, ...

Formal power series

This concept captures the essence of multiplication of formal power series. Given two such series,

<math>\sum_{n=0}^\infty a_n x^n<math>

and

<math>\sum_{n=0}^\infty b_n x^n,<math>

their product is

<math>\sum_{n=0}^\infty c_n x^n=\sum_{n=0}^\infty \sum_{k=0}^n a_k b_{n-k} x^n.<math>

One multiplies just as if one were working with algebra of finite sums without worrying about questions of convergence. It has the same "form" as multiplication of finite sums, and is therefore called "formal" multiplication.

Convergence

If one works with convergent power series rather than formal power series, does Cauchy multiplication give correct results, i.e., does the product of the two series converge, and is the scalar to which it converges equal to the product of the sums of the other two series? A partial answer is this: If one series of complex numbers converges, and the other converges absolutely, then the answer to both questions is "yes".

A variant

Often one writes formal power series in the form

<math>\sum_{n=0}^\infty {a_n \over n!}x^n,<math>

so that the coefficients may be thought of as values of the derivative of the formal power series at 0. (However, since formal power series are not in general convergent power series, to speak of their "values" may be problematic.)

The product of that series with

<math>\sum_{n=0}^\infty {b_n \over n!}x^n<math>

is

<math>\sum_{n=0}^\infty {c_n \over n!}x^n<math>

where

<math>{c_n \over n!}=\sum_{k=0}^n{a_k \over k!}{b_{n-k} \over (n-k)!}.<math>

One may see this relation written in the form

<math>c_n=\sum_{k=0}^n{n \choose k} a_k b_{n-k},<math>

and some writers on the subject may call the resulting sequence <math>(c_n)_{n=0}^\infty<math> the Cauchy product.

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