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I would like use Python to verify the following identities:

  • $ \frac{1}{1-z} = 1 + z + z^2 + z^3 + \dots $

  • $ \frac{1}{1-z - z^2} = 1 + z + 2z^2 + 3z^3 + \dots $

  • $ q \prod_{n \geq 1} (1 - q^n)^{24} = q - 24 q^2 + 252 q^3 + \dots$

The last one could be evaluated by checking the poly1d class in numpy.

Python doesn't seem to have a Laurent series or rational functions capability for the first two examples.

Perhaps there are numerical methods for generating these coefficients very quickly. I am not sure what a good data type for storing series with potentially negative exponents.

Another possibility might be to manipulate generators.

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  • $\begingroup$ Do you mean that you want to verify the above equalities for any values of $z$ and $q$ or for specific values? $\endgroup$ – Dominique Dec 28 '13 at 0:00
  • $\begingroup$ try the web site Wolfram alpha. use the series command. very easy. $\endgroup$ – Nasser Dec 28 '13 at 6:38
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I think you are looking for capabilities not available in Python itself nor Numpy, but that are available in SymPy. Sympy is a computer algebra package developed in Python. You can try it "live" in its on-line shell. Documentation on series development can be found here.

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