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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.
In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science. Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries.
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.
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 ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be ...
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 ...
The rising factorial is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for | | < by the power series (,;;) = = () ()! provided that ,,, …. Note, however, that the hypergeometric function literature typically uses the notation ( a ) n {\displaystyle (a)_{n}} for rising factorials.