Bernoulli polynomials

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The polynomials

where are the Bernoulli numbers. Thus, for

Bernoulli polynomials may be computed by the recurrence formula

For a natural argument Bernoulli polynomials were first studied by Jacob Bernoulli in 1713, in connection with the computation of the sum

L. Euler [1] was the first to study Bernoulli polynomials for arbitrary values of . The term "Bernoulli polynomials" was introduced by J.L. Raabe in 1851. The fundamental property of such polynomials is that they satisfy the finite-difference equation

and therefore play the same role in finite-difference calculus as do power functions in differential calculus.

Bernoulli polynomials belong to the class of Appell polynomials, i.e. they satisfy the condition

and are closely connected with the Euler polynomials

The generating function of the Bernoulli polynomials is

Bernoulli polynomials are expandable into Fourier series: For

and for

Bernoulli polynomials satisfy the relationships

(the multiplication theorem);

(the complement theorem);

(the theorem on addition of arguments).

Bernoulli polynomials are employed to express the residual term of the Euler–MacLaurin formula, and for the expansion of functions into series. Many important properties of Bernoulli numbers are a consequence of the properties of Bernoulli polynomials. Bernoulli polynomials are employed in the integral representation of differentiable periodic functions

and play an important part in the theory of approximation of such functions by trigonometric polynomials and other aggregates. Cf. Favard problem.

Various generalizations of the Bernoulli polynomials have been proposed. N.E. Nörlund introduced generalized Bernoulli polynomials of order and degree :

(certain special cases of such polynomials had been previously considered by V.G. Immenetskii, N.Ya. Sonin and D.M. Sintsov). Let


then is successively determined as the polynomial solution of degree of the finite-difference equation

with initial conditions

where (a generalized Bernoulli number) is found from the recurrence relations


[1] L. Euler, "Institutiones calculi differentialis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 10 , Teubner (1980)
[2] N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)
[3] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[4] V.V. Likhin, Istor Mat. Issled. (1959) pp. 59–134
How to Cite This Entry:
Bernoulli polynomials. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article