24 Bernoulli and Euler Polynomials Notation 24 Bernoulli and Euler Polynomials 24.2 Definitions and Generating Functions

§24.1 Special Notation

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Referenced by:: Erratum (V1.1.3) for Section 24.1
Permalink:: http://dlmf.nist.gov/24.1
Errata (effective with 1.1.3):: The definition “greatest common divisor of $m, n$ ” was replaced with “greatest common divisor of $k, m$ ”.
See also:: Annotations for Ch.24

(For other notation see Notation for the Special Functions.)

$j,k,\ell,m,n$	integers, nonnegative unless stated otherwise.
$t, x$	real or complex variables.
$p$	prime.
$p\mathbin{\|}m$	$p$ divides $m$ .
$(k,m)$	greatest common divisor of $k, m$ .
$(k,m)=1$	$k$ and $m$ relatively prime.

Unless otherwise noted, the formulas in this chapter hold for all values of the variables $x$ and $t$ , and for all nonnegative integers $n$ .

Bernoulli Numbers and Polynomials

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Keywords:: Bernoulli numbers, Bernoulli polynomials, notation
See also:: Annotations for §24.1 and Ch.24

The origin of the notation $B_{n}$ , $B_{n}\left(x\right)$ , is not clear. The present notation, as defined in §24.2(i), was used in Lucas (1891) and Nörlund (1924), and has become the prevailing notation; see Table 24.2.1. Among various older notations, the most common one is

	$\displaystyle B_{1}$	$\displaystyle=\tfrac{1}{6}$ ,
	$\displaystyle B_{2}$	$\displaystyle=\tfrac{1}{30}$ ,
	$\displaystyle B_{3}$	$\displaystyle=\tfrac{1}{42}$ ,
	$\displaystyle B_{4}$	$\displaystyle=\tfrac{1}{30},\ldots$ .

It was used in Saalschütz (1893), Nielsen (1923), Schwatt (1962), and Whittaker and Watson (1927).

Euler Numbers and Polynomials

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Keywords:: Euler numbers, Euler polynomials, notation
See also:: Annotations for §24.1 and Ch.24

The secant series (4.19.5) first occurs in the work of Gregory in 1671. Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_{n}$ , $E_{n}\left(x\right)$ , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924).

Other historical remarks on notations can be found in Cajori (1929, pp. 42–44). Various systems of notation are summarized in Adrian (1959) and D’Ocagne (1904).

24 Bernoulli and Euler Polynomials 24.2 Definitions and Generating Functions

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