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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 simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
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 .
Serre defined a p-adic modular form to be a formal power series with p-adic coefficients that is a p-adic limit of classical modular forms with integer coefficients.The weights of these classical modular forms need not be the same; in fact, if they are then the p-adic modular form is nothing more than a linear combination of classical modular forms.
There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere.Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit.
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.
If is an ordinary point, a fundamental system is formed by the linearly independent formal Frobenius series solutions ,, …,, where [[]] denotes a formal power series in with (), for {, …,}. Due to the reason that the starting exponents are integers, the Frobenius series are power series.
In the mathematical field of infinite group theory, the Nottingham group is the group J(F p) or N(F p) consisting of formal power series t + a 2 t 2 +... with coefficients in F p. The group multiplication is given by formal composition also called substitution. That is, if = + =