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

A hypergeometric function in which one parameter changes by +1 or -1 is said to be contiguous. There are 26 functions contiguous to _2F_1(a,b;c;x) taking one pair at a time. ...

A transformation of a hypergeometric function,

An identity which relates hypergeometric functions,

The word quadrature has (at least) three incompatible meanings. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known ...

Seeks to obtain the best numerical estimate of an integral by picking optimal abscissas x_i at which to evaluate the function f(x). The fundamental theorem of Gaussian ...

Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist ...

where _2F_1(a,b;c;z) is a hypergeometric function and _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function.

The identity _2F_1(x,-x;x+n+1;-1)=(Gamma(x+n+1)Gamma(1/2n+1))/(Gamma(x+1/2n+1)Gamma(n+1)), or equivalently ...

When |x|<1/2, (1-x)^(-a)_2F_1(a,b;c;-x/(1-x))=_2F_1(a,c-b;c;x).