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For |q|<1, the Rogers-Ramanujan identities are given by (Hardy 1999, pp. 13 and 90), sum_(n=0)^(infty)(q^(n^2))/((q)_n) = 1/(product_(n=1)^(infty)(1-q^(5n-4))(1-q^(5n-1))) ...

A rook polynomial is a polynomial R_(m,n)(x)=sum_(k=0)^(min(m,n))r_kx^k (1) whose number of ways k nonattacking rooks can be arranged on an m×n chessboard. The rook ...

Hoffman (1998, p. 90) calls the sum of the exponents in the prime factorization of a number its roundness. The first few values for n=1, 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, ...

The S distribution is defined in terms of its distribution function F(x) as the solution to the initial value problem (dF)/(dx)=alpha(F^g-F^h), where F(x_0)=F_0 (Savageau ...

In statistics, sampling is the selection and implementation of statistical observations in order to estimate properties of an underlying population. Sampling is a vital part ...

A Saunders graphic is a plot of the dth base-b digits of a function f(x,y) as a function of x and y. The plots above show Saunders graphics for the functions ...

A Lie group is called semisimple if its Lie algebra is semisimple. For example, the special linear group SL(n) and special orthogonal group SO(n) (over R or C) are ...

Consider the sum (1) where the x_js are nonnegative and the denominators are positive. Shapiro (1954) asked if f_n(x_1,x_2,...,x_n)>=1/2n (2) for all n. It turns out ...

Given a set P with |P|=p elements consisting of c_1 numbers 1, c_2 numbers 2, ..., and c_n numbers n and c_1+c_2+...+c_n=p, find the number of permutations with k-1 rises ...

The sinc function sinc(x), also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms. The full name ...