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If X is a discrete random variable taking values x in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1] G ( z ) = E ( z X ) = ∑ x = 0 ∞ p ( x ) z x , {\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},} where p {\displaystyle p} is the probability mass ...
Let α be a multi-index for a power series f(x 1, x 2, …, x n). The order of the power series f is defined to be the least value r {\displaystyle r} such that there is a α ≠ 0 with r = | α | = α 1 + α 2 + ⋯ + α n {\displaystyle r=|\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} , or ∞ {\displaystyle \infty } if f ≡ 0.
When n is an integer, the solution P n (x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of Sturm–Liouville theory .
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell . Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)} , called the Bell 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).
Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.
An orthogonal basis for L 2 (R, w(x) dx) is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function f ∈ L 2 (R, w(x) dx) orthogonal to all functions in the system.
[2] We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye X Y −1. The next key result is this one: