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A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
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.
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 is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case ϕ ( q ) = ( q ; q ) ∞ = ∏ k = 1 ∞ ( 1 − q k ) {\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})} is known as Euler's function , and is important in combinatorics , number ...
There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer. Throughout let A(z) denote a formal power series = =,
A formal series is a R-valued function c, on the free monoid A *, which may be written as ∑ w ∈ A ∗ c ( w ) w . {\displaystyle \sum _{w\in A^{*}}c(w)w.} The set of formal series is denoted R A {\displaystyle R\langle \langle A\rangle \rangle } and becomes a semiring under the operations
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
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