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A statistic w on the symmetric group S_n is called a weighted inversion statistic if there exists an upper triangular matrix W=(w_(ij)) such that ...

Let H=l^2, (alpha_n) be a bounded sequence of complex numbers, and (xi_n) be the (usual) standard orthonormal basis of H, that is, (xi_n)(m)=delta_(nm), n,m in N, where ...

The Whittaker functions arise as solutions to the Whittaker differential equation. The linearly independent solutions to this equation are M_(k,m)(z) = ...

Let a piecewise smooth function f with only finitely many discontinuities (which are all jumps) be defined on [-pi,pi] with Fourier series a_k = 1/piint_(-pi)^pif(t)cos(kt)dt ...

Willans' formula is a prime-generating formula due to Willan (1964) that is defined as follows. Let F(j) = |_cos^2[pi((j-1)!+1)/j]_| (1) = {1 for j=1 or j prime; 0 otherwise ...

The Zak transform is a signal transform relevant to time-continuous signals sampled at a uniform rate and an arbitrary clock phase (Janssen 1988). The Zak transform of a ...

Zeno's paradoxes are a set of four paradoxes dealing with counterintuitive aspects of continuous space and time. 1. Dichotomy paradox: Before an object can travel a given ...

Let Xi be the xi-function defined by Xi(iz)=1/2(z^2-1/4)pi^(-z/2-1/4)Gamma(1/2z+1/4)zeta(z+1/2). (1) Xi(z/2)/8 can be viewed as the Fourier transform of the signal ...

A q-analog of the beta function B(a,b) = int_0^1t^(a-1)(1-t)^(b-1)dt (1) = (Gamma(a)Gamma(b))/(Gamma(a+b)), (2) where Gamma(z) is a gamma function, is given by B_q(a,b) = ...

There are several q-analogs of the cosine function. The two natural definitions of the q-cosine defined by Koekoek and Swarttouw (1998) are given by cos_q(z) = ...