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  2. Formal power series - Wikipedia

    en.wikipedia.org/wiki/Formal_power_series

    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).

  3. Lagrange inversion theorem - Wikipedia

    en.wikipedia.org/wiki/Lagrange_inversion_theorem

    Faà di Bruno's formula gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the nth derivative of a composite function. Lagrange reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...

  4. Power series solution of differential equations - Wikipedia

    en.wikipedia.org/wiki/Power_series_solution_of...

    In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

  5. Power series - Wikipedia

    en.wikipedia.org/wiki/Power_series

    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 ...

  6. Fuchsian theory - Wikipedia

    en.wikipedia.org/wiki/Fuchsian_theory

    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.

  7. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_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.

  8. Plethystic exponential - Wikipedia

    en.wikipedia.org/wiki/Plethystic_exponential

    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 ...

  9. Formal calculation - Wikipedia

    en.wikipedia.org/wiki/Formal_calculation

    In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for ...