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Given the left factorial function Sigma(n)=sum_(k=1)^nk!, SK(p) for p prime is the smallest integer n such that p|1+Sigma(n-1). The first few known values of SK(p) are 2, 4, ...

The confluent hypergeometric function of the first kind _1F_1(a;b;z) is a degenerate form of the hypergeometric function _2F_1(a,b;c;z) which arises as a solution the ...

A function f(x) is completely convex in an open interval (a,b) if it has derivatives of all orders there and if (-1)^kf^((2k))(x)>=0 for k=0, 1, 2, ... in that interval ...

A space-filling function which maps a one-dimensional interval into a two-dimensional area. Plane-filling functions were thought to be impossible until Hilbert discovered the ...

The Fourier transform of the Heaviside step function H(x) is given by F_x[H(x)](k) = int_(-infty)^inftye^(-2piikx)H(x)dx (1) = 1/2[delta(k)-i/(pik)], (2) where delta(k) is ...

Given a sequence {a_n}_(n=1)^infty, a formal power series f(s) = sum_(n=1)^(infty)(a_n)/(n^s) (1) = a_1+(a_2)/(2^s)+(a_3)/(3^s)+... (2) is called the Dirichlet generating ...

A real-valued function g defined on a convex subset C subset R^n is said to be quasi-convex if for all real alpha in R, the set {x in C:g(x)<alpha} is convex. This is ...

Lambda_0(phi|m)=(F(phi|1-m))/(K(1-m))+2/piK(m)Z(phi|1-m), where phi is the Jacobi amplitude, m is the parameter, Z is the Jacobi zeta function, and F(phi|m^') and K(m) are ...

Let p run over all distinct primitive ordered periodic geodesics, and let tau(p) denote the positive length of p, then the Selberg zeta function is defined as ...

The modular group Gamma is the set of all transformations w of the form w(t)=(at+b)/(ct+d), where a, b, c, and d are integers and ad-bc=1. A Gamma-modular function is then ...