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Bernoulli Polynomial

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BernoulliPolynomials

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 Bernoulli polynomial by B_n^*(x) (or sometimes phi_n(x)). When evaluated at zero, these definitions correspond to the Bernoulli numbers,

B_n=B_n(0)
(1)
B_n^*=B_n^*(0).
(2)

The Bernoulli polynomials are an Appell sequence with

g(t)=(e^t-1)/t
(3)

(Roman 1984, p. 31), giving the generating function

(te^(tx))/(e^t-1)=sum_(n=0)^inftyB_n(x)(t^n)/(n!)
(4)

(Abramowitz and Stegun 1972, p. 804), first obtained by Euler (1738). The first few Bernoulli polynomials are

B_0(x)=1
(5)
B_1(x)=x-1/2
(6)
B_2(x)=x^2-x+1/6
(7)
B_3(x)=x^3-3/2x^2+1/2x
(8)
B_4(x)=x^4-2x^3+x^2-1/(30)
(9)
B_5(x)=x^5-5/2x^4+5/3x^3-1/6x
(10)
B_6(x)=x^6-3x^5+5/2x^4-1/2x^2+1/(42).
(11)

Whittaker and Watson (1990, p. 126) define an older type of "Bernoulli polynomial" by writing

t(e^(zt)-1)/(e^t-1)=sum_(n=1)^infty(phi_n(z)t^n)/(n!)
(12)

instead of (12). This gives the polynomials

phi_n(x)=B_n(x)-B_n,
(13)

where B_n is a Bernoulli number, the first few of which are

phi_1(x)=x
(14)
phi_2(x)=x^2-x
(15)
phi_3(x)=x^3-3/2x^2+1/2x
(16)
phi_4(x)=x^4-2x^3+x^2
(17)
phi_5(x)=x^5-5/2x^4+5/3x^3-1/6x.
(18)

The Bernoulli polynomials also satisfy

B_n(1)=(-1)^nB_n(0)
(19)

and

B_n(1-x)=(-1)^nB_n(x)
(20)

(Lehmer 1988). For n!=1,

B_n(1)=B_n,
(21)

so

B_n(1)=B_n=0
(22)

for odd n>1.

They also satisfy the relation

B_n(x+1)-B_n(x)=nx^(n-1)
(23)

(Whittaker and Watson 1990, p. 127).

For rational values of x, B_n(x) can be expressed for positive integers n in terms of Bernoulli and Euler numbers, for example

B_n(1)=(-1)^nB_n
(24)
B_n(1/2)=(2^(1-n)-1)B_n
(25)
B_n(1/4)=-2^(-n)(1-2^(1-n))B_n-4^(-n)nE_(n-1)
(26)
B_(2n)(1/3)=-1/2(1-3^(1-2n))B_(2n)
(27)
B_(2n)(1/6)=1/2(1-2^(1-2n))(1-3^(1-2n))B_(2n).
(28)

Bernoulli (1713) defined the polynomials in terms of sums of the powers of consecutive integers,

sum_(k=0)^(m-1)k^(n-1)=1/n[B_n(m)-B_n(0)].
(29)

The Bernoulli polynomials satisfy the recurrence relation

(dB_n)/(dx)=nB_(n-1)(x)
(30)

(Appell 1882), and obey the identity

B_n(x)=(B+x)^n,
(31)

where B^k is interpreted as the Bernoulli number B_k=B_k(0). Another related identity is

B_n=(B-x)^n,
(32)

where B^k is interpreted as the Bernoulli polynomial B_k(x).

Hurwitz gave the Fourier series

B_n(x)=-(n!)/((2pii)^n)sum^'_(k=-infty)^inftyk^(-n)e^(2piikx),
(33)

for 0<x<1, where the prime in the summation indicates that the term k=0 is omitted. Performing the sum gives

B_n(x)=-(n!)/((2pii)^n)[(-1)^nLi_n(e^(-2piix))+Li_n(e^(2piix))],
(34)

where Li_n(x) is the polylogarithm function. Raabe (1851) found

1/msum_(k=0)^(m-1)B_n(x+k/m)=m^(-n)B_n(mx).
(35)

A sum identity involving the Bernoulli polynomials is

sum_(k=0)^m(m; k)B_k(alpha)B_(m-k)(beta) =-(m-1)B_m(alpha+beta)+m(alpha+beta-1)B_(m-1)(alpha+beta)
(36)

for m an integer. A sum identity due to S. M. Ruiz is

sum_(k=0)^n(-1)^(k+n)(n; k)B_n(k)=n!,
(37)

where (n; k) is a binomial coefficient. The Bernoulli polynomials are also given by the formula

B_n(x)=B_n(0)+sum_(k=1)^nn/kS(n-1,k-1)(x)_k,
(38)

where S(n,m) is a Stirling number of the second kind and (x)_k is a falling factorial (Roman 1984, p. 94). A general identity is given by

(n)_mx^(n-m)=sum_(k=m)^n((n)_k)/((k-m+1)!)B_(n-k)(x),
(39)

which simplifies to

nx^(n-1)=sum_(k=1)^n(n; k)B_(n-k)(x)
(40)

(Roman 1984, p. 97). Gosper gave the identity

sum_(j=0)^i([2(i-j)-1]3^(2j)(2^((2j+1))+1)B_(2(i-j))B_(2j+1)(1/3))/([2(i-j)]!(2j+1)!) =(2i·3^(2(i-1))(2^(2i-1)+1)B_(2i-1)(1/3)-(i-1/2)B_(2i))/((2i)!).
(41)

A generalization B_n^((alpha))(x) of the Bernoulli polynomials with an additional free parameter can be defined such that B_n(x)=B_n^((1))(x) (Roman 1984, p. 93). These polynomials have the generating function

e^(zt)(t/(e^t-1))^alpha=sum_(n=0)^inftyB_n^((alpha))(z)(t^n)/(n!),
(42)

and are implemented in the Wolfram Language as NorlundB[n, alpha, z].

See also

Bernoulli Number, Bernoulli Polynomial of the Second Kind, Euler-Maclaurin Integration Formulas, Euler Polynomial, Norlund Polynomial

Related Wolfram sites

http://functions.wolfram.com/Polynomials/BernoulliB2/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972.Appell, P. E. "Sur une classe de polynomes." Annales d'École Normal Superieur, Ser. 2 9, 119-144, 1882.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 330, 1985.Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. Published posthumously.Euler, L. "Methodus generalis summandi progressiones." Comment. Acad. Sci. Petropol. 6, 68-97, 1738.Lehmer, D. H. "A New Approach to Bernoulli Polynomials." Amer. Math. Monthly. 95, 905-911, 1988.Lucas, E. Ch. 14 in Théorie des Nombres. Paris, 1891.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function zeta(s,x), Bernoulli Polynomials B_n(x), Euler Polynomials E_n(x), and Polylogarithms Li_nu(x)." §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23-24, 1990.Raabe, J. L. "Zurückführung einiger Summen und bestimmten Integrale auf die Jakob Bernoullische Function." J. reine angew. Math. 42, 348-376, 1851.Roman, S. "The Bernoulli Polynomials." §4.2.2 in The Umbral Calculus. New York: Academic Press, pp. 93-100, 1984.Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial B_n(x)." Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167-173, 1987.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Bernoulli Polynomial

Cite this as:

Weisstein, Eric W. "Bernoulli Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoulliPolynomial.html

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