View source for Bernoulli polynomials

The polynomials

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156502.png" /> are the [[Bernoulli numbers|Bernoulli numbers]]. Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156503.png" />

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Bernoulli polynomials may be computed by the recurrence formula

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For a natural argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156507.png" /> Bernoulli polynomials were first studied by Jacob Bernoulli in 1713, in connection with the computation of the sum

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L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitrary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156509.png" />. 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

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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|Appell polynomials]], i.e. they satisfy the condition

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and are closely connected with the [[Euler polynomials|Euler polynomials]]

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The generating function of the Bernoulli polynomials is

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Bernoulli polynomials are expandable into [[Fourier series|Fourier series]]: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565014.png" />

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and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565016.png" />

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Bernoulli polynomials satisfy the relationships

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(the multiplication theorem);

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(the complement theorem);

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(the theorem on addition of arguments).

Bernoulli polynomials are employed to express the residual term of the [[Euler–MacLaurin formula|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

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

Various generalizations of the Bernoulli polynomials have been proposed. N.E. Nörlund introduced generalized Bernoulli polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565023.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565024.png" />:

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

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and

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then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565028.png" /> is successively determined as the polynomial solution of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565029.png" /> of the finite-difference equation

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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565031.png" /> with initial conditions

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565033.png" /> (a generalized Bernoulli number) is found from the recurrence relations

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,   "Institutiones calculi differentialis"  G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; opera mat.'' , '''10''' , Teubner  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Nörlund,   "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Likhin,   ''Istor Mat. Issled.''  (1959)  pp. 59–134</TD></TR></table>