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Ball point picking is the selection of points randomly placed inside a ball. n random points can be picked in a unit ball in the Wolfram Language using the function ...

The spherical harmonics form a complete orthogonal system, so an arbitrary real function f(theta,phi) can be expanded in terms of complex spherical harmonics by ...

A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. ...

The Gram series is an approximation to the prime counting function given by G(x)=1+sum_(k=1)^infty((lnx)^k)/(kk!zeta(k+1)), (1) where zeta(z) is the Riemann zeta function ...

Given an n-ball B^n of radius R, find the distribution of the lengths s of the lines determined by two points chosen at random within the ball. The probability distribution ...

erf(z) is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by ...

The polynomials G_n(x;a,b) given by the associated Sheffer sequence with f(t)=e^(at)(e^(bt)-1), (1) where b!=0. The inverse function (and therefore generating function) ...

A q-analog of the Chu-Vandermonde identity given by where _2phi_1(a,b;c;q,z) is the q-hypergeometric function. The identity can also be written as ...

A q-analog of the Saalschütz theorem due to Jackson is given by where _3phi_2 is the q-hypergeometric function (Koepf 1998, p. 40; Schilling and Warnaar 1999).

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 ...