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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).
By successively integrating by parts, the formal power series is recovered as an asymptotic approximation to this expression for x(t). Euler argues (more or less) that since the formal series and the integral both describe the same solution to the differential equations, they should equal each other at t = 1 {\displaystyle t=1} , giving
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 ...
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.
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.
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 = =,
This article uses the coefficient extraction operator [] for formal power series, as well as the (labelled) operators (for cycles) and (for sets) on combinatorial classes, which are explained on the page for symbolic combinatorics. Given a combinatorial class, the cycle operator creates the class obtained by placing objects from the source ...
In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions , as a concise relation between the generating series for elementary , complete ...
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