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For n a positive integer, expressions of the form sin(nx), cos(nx), and tan(nx) can be expressed in terms of sinx and cosx only using the Euler formula and binomial theorem. ...

Solutions to the associated Laguerre differential equation with nu!=0 and k an integer are called associated Laguerre polynomials L_n^k(x) (Arfken 1985, p. 726) or, in older ...

A permutation, also called an "arrangement number" or "order," is a rearrangement of the elements of an ordered list S into a one-to-one correspondence with S itself. The ...

Let N steps of equal length be taken along a line. Let p be the probability of taking a step to the right, q the probability of taking a step to the left, n_1 the number of ...

A general type of statistical distribution which is related to the gamma distribution. Beta distributions have two free parameters, which are labeled according to one of two ...

Take x itself to be a bracketing, then recursively define a bracketing as a sequence B=(B_1,...,B_k) where k>=2 and each B_i is a bracketing. A bracketing can be represented ...

A C-matrix is a symmetric (C^(T)=C) or antisymmetric (C^(T)=-C) C_n (-1,0,1)-matrix with diagonal elements 0 and others +/-1 that satisfies CC^(T)=(n-1)I, (1) where I is the ...

A moment mu_n of a univariate probability density function P(x) taken about the mean mu=mu_1^', mu_n = <(x-<x>)^n> (1) = int(x-mu)^nP(x)dx, (2) where <X> denotes the ...

A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if n points on its circumference are joined by ...

An n×n matrix whose rows are composed of cyclically shifted versions of a length-n list l. For example, the 4×4 circulant matrix on the list l={1,2,3,4} is given by C=[4 1 2 ...