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1: 24.1 Special Notation

… ►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

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. …

2: 3.6 Linear Difference Equations

… ►If $d_n=0$, $\forall n$, then the difference equation is homogeneous; otherwise it is inhomogeneous. … ►with $a_n\ne 0$, $\forall n$, can be computed recursively for $n=2,3,\mathrm{\dots }$. … ►Similar principles apply to equation (3.6.1) when $a_n{c}_n\ne 0$, $\forall n$, and $d_n\ne 0$ for some, or all, values of $n$. … ►More precisely, assume that $f_0\ne 0$, $g_n\ne 0$ for all sufficiently large $n$, and as $n\to \mathrm{\infty }$ … ►For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).

3: Bibliography C

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Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.

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External Links:

FullText, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(i).

Encodings:

BibTeX

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$q$-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.

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External Links:

FullText, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§17.3(iii), §24.16(iii).

Encodings:

BibTeX

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A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

BibTeX

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

Encodings:

BibTeX

… ►

A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

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External Links:

ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt

Referenced by:

§3.6(vii).

Encodings:

BibTeX

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4: DLMF Project News

error generating summary

5: 27.21 Tables

… ►Lehmer (1914) lists all primes up to 100 06721. …Table III lists all solutions $n\le {10}^4$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma (n)=m$ for all $m\le {10}^4$. …8 gives examples of primitive roots of all primes $\le 9973$; Table 24. 9 lists all primes that are less than 1 00000. … ►Lehmer (1941) also has a section that supplies errata and corrections to all tables cited. …

6: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). ►The Sieve of Eratosthenes (Crandall and Pomerance (2005, §3.2)) generates a list of all primes below a given bound. … ►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of $n-1,n+1$, or both; see Crandall and Pomerance (2005, §§4.1–4.2). … …

7: 27 Functions of Number Theory

Chapter 27 Functions of Number Theory

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8: 24.10 Arithmetic Properties

… ►where the summation is over all $p$ such that $p-1$ divides $2n$. The denominator of $B_{2n}$ is the product of all these primes $p$. … ►valid for fixed integers $\mathrm{\ell }(\ge 0)$, and for all $n(\ge 0)$ and $w(\ge 0)$ such that $2^{\mathrm{\ell }}|w$. … ►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. …valid for fixed integers $\mathrm{\ell }(\ge 1)$ and for all $n(\ge 1)$ such that $(p-1)p^{\mathrm{\ell }-1}|2n$.

9: Mathematical Introduction

… ►Also, valuable initial advice on all aspects of the project was provided by ten external associate editors. … ►All chapters went through several drafts (nine in some cases) before the authors, validators, and editors were fully satisfied. … ►The Notations section includes all the notations for the special functions adopted in this Handbook. … ►All of the special function chapters include sections devoted to mathematical, physical, and sometimes other applications of the main functions in the chapter. … ►All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions. …

10: 21.1 Special Notation

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$g,h$	positive integers.
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${\mathbb{Z}}^{g\times h}$	set of all $g\times h$ matrices with integer elements.
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$𝜶,𝜷$	$g$-dimensional vectors, with all elements in $[0,1)$, unless stated otherwise.
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$S_1{S}_2$	set of all elements of the form “$\text{element of }{S}_1\times \text{element of }{S}_2$”.
$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
…

►Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. …