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1: 24.1 Special Notation

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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. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … … ►Let $A$ and $B$ be the $n\times n$ matrices with $(j,k)$th elements $s(j,k)$, and $S(j,k)$, respectively. … ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

3: 26.1 Special Notation

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$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
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$s(n,k)$	Stirling numbers of the first kind.
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… ►Other notations for $s(n,k)$, the Stirling numbers of the first kind, include $S_n^{(k)}$ (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), $S_n^k$ (Jordan (1939), Moser and Wyman (1958a)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

4: 24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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§24.15(ii) Tangent Numbers

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§24.15(iii) Stirling Numbers

►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). … ►

§24.15(iv) Fibonacci and Lucas Numbers

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5: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. ►Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.

6: 26.13 Permutations: Cycle Notation

… ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►The Stirling cycle numbers of the first kind, denoted by $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$, count the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. They are related to Stirling numbers of the first kind by … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: …

7: Bibliography O

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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External Links:

ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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Formulae and Tables, The Modified Quotients of Cylinder Functions. Technical report Technical Report UDC 517.564.3:518.25, Vol. 4, Report of the Institute of Industrial Science, University of Tokyo, Institute of Industrial Science, Chiba City, Japan.

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Referenced by:

§10.29(i), §10.6(i).

Encodings:

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8: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

Encodings:

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Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

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An Introduction to the Theory of Numbers. 5th edition, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-62546-9, MathReview, ZentralBlatt

Referenced by:

Ch.27.

Encodings:

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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

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9: 24.16 Generalizations

§24.16 Generalizations

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Polynomials and Numbers of Integer Order

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Bernoulli Numbers of the Second Kind

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Degenerate Bernoulli Numbers

… ►Here $s(n,m)$ again denotes the Stirling number of the first kind. …

10: Bibliography Y

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

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