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An important example of the unilateral Z-transform is the probability-generating function, ... In this method, the Z-transform is expanded into a power series. This ...
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
Conversely, every polynomial is a power series with only finitely many non-zero terms. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform).
Z-transform – the special case where the Laurent series is taken about zero has much use in time-series analysis. Fourier series – the substitution z = e π i w {\displaystyle z=e^{\pi iw}} transforms a Laurent series into a Fourier series, or conversely.
Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is =, has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function f(z) of Example 1 is the derivative of g(z). Example 3: The power series
Additional series representations for the r-order harmonic number exponential generating functions for integers are formed as special cases of these negative-order derivative-based series transformation results. For example, the second-order harmonic numbers have a corresponding exponential generating function expanded by the series
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
This expression is found by writing 2 s Li s (−z) / (−z) = Φ(z 2, s, 1 ⁄ 2) − z Φ(z 2, s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e 2πt + 1) in place of 1 / (e 2πt − 1) to the second Φ series.