There are two definitions of Bernoulli polynomials in use. The th Bernoulli polynomial is denoted here by (Abramowitz and Stegun 1972), and the archaic form of the Bernoulli polynomial by (or sometimes ). When evaluated at zero, these definitions correspond to the Bernoulli numbers,
(1)
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(2)
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The Bernoulli polynomials are an Appell sequence with
(3)
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(Roman 1984, p. 31), giving the generating function
(4)
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(Abramowitz and Stegun 1972, p. 804), first obtained by Euler (1738). The first few Bernoulli polynomials are
(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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Whittaker and Watson (1990, p. 126) define an older type of "Bernoulli polynomial" by writing
(12)
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instead of (12). This gives the polynomials
(13)
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where is a Bernoulli number, the first few of which are
(14)
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(15)
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(16)
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(17)
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(18)
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The Bernoulli polynomials also satisfy
(19)
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and
(20)
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(Lehmer 1988). For ,
(21)
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so
(22)
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for odd .
They also satisfy the relation
(23)
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(Whittaker and Watson 1990, p. 127).
For rational values of , can be expressed for positive integers in terms of Bernoulli and Euler numbers, for example
(24)
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(25)
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(26)
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(27)
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(28)
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Bernoulli (1713) defined the polynomials in terms of sums of the powers of consecutive integers,
(29)
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The Bernoulli polynomials satisfy the recurrence relation
(30)
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(Appell 1882), and obey the identity
(31)
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where is interpreted as the Bernoulli number . Another related identity is
(32)
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where is interpreted as the Bernoulli polynomial .
Hurwitz gave the Fourier series
(33)
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for , where the prime in the summation indicates that the term is omitted. Performing the sum gives
(34)
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where is the polylogarithm function. Raabe (1851) found
(35)
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A sum identity involving the Bernoulli polynomials is
(36)
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for an integer. A sum identity due to S. M. Ruiz is
(37)
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where is a binomial coefficient. The Bernoulli polynomials are also given by the formula
(38)
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where is a Stirling number of the second kind and is a falling factorial (Roman 1984, p. 94). A general identity is given by
(39)
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which simplifies to
(40)
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(Roman 1984, p. 97). Gosper gave the identity
(41)
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A generalization of the Bernoulli polynomials with an additional free parameter can be defined such that (Roman 1984, p. 93). These polynomials have the generating function
(42)
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and are implemented in the Wolfram Language as NorlundB[n, alpha, z].