BernoulliEuler

BernoulliEuler is a term used in mathematics to refer to objects that combine the families of Bernoulli numbers and Euler numbers or polynomials. It is most often encountered in the context of Bernoulli–Euler numbers or Bernoulli–Euler polynomials, which arise when the generating functions of Bernoulli and Euler sequences are multiplied or convolved.

Bernoulli numbers B_n are defined by the generating function t/(e^t − 1) = sum B_n t^n / n!. Euler numbers E_n are defined by 2 e^{t} /(e^{t} + 1) = sum E_n t^n / n!.

The Bernoulli–Euler polynomials B_n^{(E)}(x) are defined by the generating function sum B_n^{(E)}(x) t^n / n! = (t e^{x t})/(e^t − 1) * 2/(e^t + 1). The numbers B_n^{(E)} := B_n^{(E)}(0) are called Bernoulli–Euler numbers.

These objects encode combinations of the two classical families and can be expressed as finite sums involving ordinary Bernoulli and Euler numbers. They satisfy linear recurrences obtained from the generating function and have symmetry properties relating to the parity of n.

Applications include the evaluation of sums of powers, finite sums with alternating signs, and the use of asymptotic expansions in analytic number theory and combinatorics.

See also Bernoulli numbers, Euler numbers, Bernoulli polynomials, and Euler polynomials.