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Given a homogeneous linear second-order ordinary differential equation, y^('')+P(x)y^'+Q(x)y=0, (1) call the two linearly independent solutions y_1(x) and y_2(x). Then ...

Consider the process of taking a number, adding its digits, then adding the digits of the number derived from it, etc., until the remaining number has only one digit. The ...

The Andrews-Schur identity states sum_(k=0)^nq^(k^2+ak)[2n-k+a; k]_q =sum_(k=-infty)^inftyq^(10k^2+(4a-1)k)[2n+2a+2; n-5k]_q([10k+2a+2]_q)/([2n+2a+2]_q) (1) where [n; m]_q is ...

The arithmetic-geometric matrix A_(AG) of a simple graph is a weighted adjacency matrix with weight f(d_i,d_j)=sqrt(d_i^2+d_j^2), (1) where d_i are the vertex degrees of the ...

One of the Zermelo-Fraenkel axioms which asserts the existence for any set a of the power set x consisting of all the subsets of a. The axiom may be stated symbolically as ...

The vector triple product identity Ax(BxC)=B(A·C)-C(A·B). This identity can be generalized to n dimensions,

A class of curve defined at integer values which hops from one value to another. Their name derives from the Greek word betaalphataurhoalphachiiotaomicronnu batrachion, which ...

Let T(m) denote the set of the phi(m) numbers less than and relatively prime to m, where phi(n) is the totient function. Define f_m(x)=product_(t in T(m))(x-t). (1) Then a ...

Let m>=3 be an integer and let f(x)=sum_(k=0)^na_kx^(n-k) be an integer polynomial that has at least one real root. Then f(x) has infinitely many prime divisors that are not ...

The bei_nu(z) function is defined through the equation J_nu(ze^(3pii/4))=ber_nu(z)+ibei_nu(z), (1) where J_nu(z) is a Bessel function of the first kind, so ...