Bernoulli polynomial

Bernoulli polynomial

[ber‚nü·lē ‚päl·ə′nō·mē·əl]
(mathematics)
The n th such polynomial is where (nk) is a binomial coefficient, and Bk is a Bernoulli number.
References in periodicals archive ?
for k [member of] [Z.sup.+], n= 1, 2, ..., [2.sup.k-1], and m = 0, 1, 2, ..., M - 1 is the order of the Bernoulli polynomial and M is a fixed positive integer.
[B.sub.n], [B.sub.n](x) are the Bernoulli number and Bernoulli polynomial, respectively.
Let [B.sub.k] (x) be the Bernoulli polynomial of degree k [11] and let us set
and [t.sub.0] [member of] (0, 1/2) is a unique zero point (in the interval (0,1/2)) of the Bernoulli polynomial [B.sub.k](*), k = 2r.
where k is a nonnegative integer, [B.sub.2](x) = [x.sup.2] - x + 1/6 is the second Bernoulli polynomial, and {t} denotes the fractional part of t.
is the usual n-th Bernoulli polynomial. Balanzario and Sanchez [5] derive the following generating function for [B.sub.n](x) defined in (5):
It is well known that [B.sub.n] = [B.sub.n](0), where [B.sub.n](x) is the classical Bernoulli polynomial (see, e.g., [4,7-10]).
where [B.sub.1](x) = x ?1/2 is the first Bernoulli polynomial.
with [B.sub.2k](t) the Bernoulli polynomial and {t} = t - [t] the fractional part of t.
Yalijinbas, "Bernoulli polynomial approach to high-order linear differential-difference equations," AIP Conference Proceedings, vol.
Bernardini, "A generalization of the Bernoulli polynomials," Journal of Applied Mathematics, no.
Dolgy, "On q-analogs of degenerate Bernoulli polynomials," Adv.