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If y has period 2pi, y^' is L^2, and int_0^(2pi)ydx=0, (1) then int_0^(2pi)y^2dx<int_0^(2pi)y^('2)dx (2) unless y=Acosx+Bsinx (3) (Hardy et al. 1988). Another inequality ...

Let points A^', B^', and C^' be marked off some fixed distance x along each of the sides BC, CA, and AB. Then the lines AA^', BB^', and CC^' concur in a point U known as the ...

sum_(n=0)^(infty)[(q)_infty-(q)_n] = g(q)+(q)_inftysum_(k=1)^(infty)(q^k)/(1-q^k) (1) = g(q)+(q)_inftyL(q) (2) = g(q)+(q)_infty(psi_q(1)+ln(1-q))/(lnq) (3) = ...

The de Longchamps line is central line L_(32) with trilinear equation a^3alpha+b^3beta+c^3gamma=0 (Droussent 1953; Kimberling 1998, p. 150) which is the anticomplement of the ...

The h-statistic h_r is the unique symmetric unbiased estimator for a central moment of a distribution <h_r>=mu_r. (1) In addition, the variance var(h_r)=<(h_r-mu_r)^2> (2) is ...

The j-function is the modular function defined by j(tau)=1728J(tau), (1) where tau is the half-period ratio, I[tau]>0, ...

A k-subset is a subset of a set on n elements containing exactly k elements. The number of k-subsets on n elements is therefore given by the binomial coefficient (n; k). For ...

A q-analog of the beta function B(a,b) = int_0^1t^(a-1)(1-t)^(b-1)dt (1) = (Gamma(a)Gamma(b))/(Gamma(a+b)), (2) where Gamma(z) is a gamma function, is given by B_q(a,b) = ...

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 q-analog of Gauss's theorem due to Jacobi and Heine, _2phi_1(a,b;c;q,c/(ab))=((c/a;q)_infty(c/b;q)_infty)/((c;q)_infty(c/(ab);q)_infty) (1) for |c/(ab)|<1 (Gordon and ...