Board index Computation of Series A products' partial fraction decomposition

## A products' partial fraction decomposition

Post your questions related to Computation of Series here.

Moderators: galactus, sos440, zaidalyafey

### A products' partial fraction decomposition

Sun Aug 14, 2016 9:10 am

Posts: 105
How to obtian the coefficients $a_i$
$\frac{1}{{\left( {1 - {q^{x + 1}}} \right)\left( {1 - {q^{x + 2}}} \right) \cdots \left( {1 - {q^{x + m}}} \right)}} = \frac{{{a_1}}}{{1 - {q^{x + 1}}}} + \frac{{{a_2}}}{{1 - {q^{x + 2}}}} + \cdots + \frac{{{a_m}}}{{1 - {q^{x + m}}}}$

### Re: A products' partial fraction decomposition

Mon Aug 15, 2016 10:28 am

Posts: 23
the coefficient is

a_i = q^( (1 - m) x) (Product[(q^i - q^j), j=1...m; j not equal i])^(-1)

### Re: A products' partial fraction decomposition

Thu Aug 18, 2016 1:56 pm

Posts: 105
andreas wrote:
the coefficient is

a_i = q^( (1 - m) x) (Product[(q^i - q^j), j=1...m; j not equal i])^(-1)

Thank you very much. But, I don't konw that how to obtain the coefficients a_i, can you give me the specific process?

### Re: A products' partial fraction decomposition

Mon Aug 22, 2016 2:12 pm

Posts: 23
I found it by trial and error, but a proof by induction should be possible.