24 Bernoulli and Euler PolynomialsNotation24 Bernoulli and Euler Polynomials24.2 Definitions and Generating Functions

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

$j,k,\mathrm{\ell},m,n$ | integers, nonnegative unless stated otherwise. |
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$t,x$ | real or complex variables. |

$p$ | prime. |

$p|m$ | $p$ divides $m$. |

$\left(k,m\right)$ | greatest common divisor of $m,n$. |

$\left(k,m\right)=1$ | $k$ 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 $n$.

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

${B}_{1}$ | $=\frac{1}{6}$, | ||

${B}_{2}$ | $=\frac{1}{30}$, | ||

${B}_{3}$ | $=\frac{1}{42}$, | ||

${B}_{4}$ | $=\frac{1}{30},\mathrm{\dots}$. |

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