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The Euler-Maclaurin integration and sums formulas can be derived from Darboux's formula by substituting the Bernoulli polynomial B_n(t) in for the function phi(t). ...

There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by B_n(x) (Abramowitz and Stegun 1972), and the archaic form of the ...

Some authors define a general Airy differential equation as y^('')+/-k^2xy=0. (1) This equation can be solved by series solution using the expansions y = ...

A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set. Graded modules arise naturally in ...

The representation of a number as a sum of powers of a base b, followed by expression of each of the exponents as a sum of powers of b, etc., until the process stops. For ...

A game in which the possible moves are the same for each player in any position. All positions in all impartial games form an additive Abelian group. For impartial games in ...

Denote the sum of two matrices A and B (of the same dimensions) by C=A+B. The sum is defined by adding entries with the same indices c_(ij)=a_(ij)+b_(ij) over all i and j. ...

The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. ...

Let s_b(n) be the sum of the base-b digits of n, and epsilon(n)=(-1)^(s_2(n)) the Thue-Morse sequence, then product_(n=0)^infty((2n+1)/(2n+2))^(epsilon(n))=1/2sqrt(2).

Let F be a finite field with q elements, and let F_s be a field containing F such that [F_s:F]=s. Let chi be a nontrivial multiplicative character of F and chi^'=chi ...