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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).
This series was first considered by Euler, who applied summability methods to assign a finite value to the series. [1] The series is a sum of factorials that are alternately added or subtracted. One way to assign a value to this divergent series is by using Borel summation , where one formally writes
Recurrence relation, telescoping series; Generating functions as an application of formal power series. Cyclic sieving; Schrödinger method; Exponential generating function; Stanley's reciprocity theorem; Binomial coefficients and their properties; Combinatorial proof. Double counting (proof technique) Bijective proof; Inclusion–exclusion ...
The next formulas for powers, logarithms, and compositions of formal power series are expanded by these polynomials with variables in the coefficients of the original generating functions. [ 4 ] [ 5 ] The formula for the exponential of a generating function is given implicitly through the Bell polynomials by the EGF for these polynomials ...
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.
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 ...
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
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.