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Consider the decimal expansion of the reciprocal of the number seven, 1/7=0.142857142857...=0.142857^_, (1) which is a repeating decimal. Now take overlapping pairs of these ...

If (1-z)^(alpha+beta-gamma-1/2)_2F_1(2alpha,2beta;2gamma;z)=sum_(n=0)^inftya_nz^n, (1) where _2F_1(a,b;c;z) is a hypergeometric function, then (2) where (a)_n is a Pochhammer ...

The golden ratio phi can be written in terms of a nested radical in the beautiful form phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...)))), (1) which can be written recursively as ...

The Rogers mod 14 identities are a set of three Rogers-Ramanujan-like identities given by A(q) = sum_(n=0)^(infty)(q^(n^2))/((q;q)_n(q;q^2)_n) (1) = ...

Let the sum of squares function r_k(n) denote the number of representations of n by k squares, then the summatory function of r_2(k)/k has the asymptotic expansion ...

A transformation that preserves angles and changes all distances in the same ratio, called the ratio of magnification. A similarity can also be defined as a transformation ...

The expansion of the two sides of a sum equality in terms of polynomials in x^m and y^k, followed by closed form summation in terms of x and y. For an example of the ...

A formula also known as the Legendre addition theorem which is derived by finding Green's functions for the spherical harmonic expansion and equating them to the generating ...

An extended form of Bürmann's theorem. Let f(z) be a function of z analytic in a ring-shaped region A, bounded by another curve C and an inner curve c. Let theta(z) be a ...

A transformation consisting of a constant offset with no rotation or distortion. In n-dimensional Euclidean space, a translation may be specified simply as a vector giving ...