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The Jacobi polynomials, also known as hypergeometric polynomials, occur in the study of rotation groups and in the solution to the equations of motion of the symmetric top. ...

Let where (alpha)_j is a Pochhammer symbol, and let alpha be a negative integer. Then S(alpha,beta,m;z)=(Gamma(beta+1-m))/(Gamma(alpha+beta+1-m)), where Gamma(z) is the gamma ...

Let there be N_i observations of the ith phenomenon, where i=1, ..., p and N = sumN_i (1) y^__i = 1/(N_i)sum_(alpha)y_(ialpha) (2) y^_ = 1/Nsum_(i)sum_(alpha)y_(ialpha). (3) ...

For a power function f(x)=x^k with k>=0 on the interval [0,2L] and periodic with period 2L, the coefficients of the Fourier series are given by a_0 = (2^(k+1)L^k)/(k+1) (1) ...

A group of linear fractional transformations which transform the arguments of Kummer solutions to the hypergeometric differential equation into each other. Define A(z) = 1-z ...

Laplace's integral is one of the following integral representations of the Legendre polynomial P_n(x), P_n(x) = 1/piint_0^pi(du)/((x+sqrt(x^2-1)cosu)^(n+1))du (1) = ...

The integral transform (Kf)(x)=int_0^infty((x-t)_+^(c-1))/(Gamma(c))_2F_1(a,b;c;1-t/x)f(t)dt, where Gamma(x) is the gamma function, _2F_1(a,b;c;z) is a hypergeometric ...

A function of more than one variable.

A generalization of Student's t-distribution known as the noncentral Student's t-distribution is given by (1) where Gamma(z) is the gamma function and _1F_1(a;b;z) is a ...

where R[mu+nu-lambda+1]>0, R[lambda]>-1, 0<a<b, J_nu(x) is a Bessel function of the first kind, Gamma(x) is the gamma function, and _2F_1(a,b;c;x) is a hypergeometric ...