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A generalization of the complete beta function defined by B(z;a,b)=int_0^zu^(a-1)(1-u)^(b-1)du, (1) sometimes also denoted B_z(a,b). The so-called Chebyshev integral is given ...

Kummer's first formula is (1) where _2F_1(a,b;c;z) is the hypergeometric function with m!=-1/2, -1, -3/2, ..., and Gamma(z) is the gamma function. The identity can be written ...

A generalization of the factorial and double factorial, n! = n(n-1)(n-2)...2·1 (1) n!! = n(n-2)(n-4)... (2) n!!! = n(n-3)(n-6)..., (3) etc., where the products run through ...

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 ...

A partition whose conjugate partition is equivalent to itself. The Ferrers diagrams corresponding to the self-conjugate partitions for 3<=n<=10 are illustrated above. The ...

Let V be a real vector space (e.g., the real continuous functions C(I) on a closed interval I, two-dimensional Euclidean space R^2, the twice differentiable real functions ...

A two-regular graph is a regular graph for which all local degrees are 2. A two-regular graph consists of one or more (disconnected) cycles. The numbers a_n of two-regular ...

There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field K can be transformed, given ...

The q-analog of the binomial theorem (1-z)^n=1-nz+(n(n-1))/(1·2)z^2-(n(n-1)(n-2))/(1·2·3)z^3+... (1) is given by (1-z/(q^n))(1-z/(q^(n-1)))...(1-z/q) ...

A triangle line lalpha+mbeta+ngamma=0 defined relative to a reference triangle is called a central line iff l:m:n is a triangle center (Kimberling 1998, p. 127). If l:m:n is ...