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1: 24.1 Special Notation

… ►

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: Diego Dominici

… ► 1972 in Buenos Aires, Argentina, d. 2023) studied Pure Mathematics at the Universidad de Buenos Aires, Argentina, and did his Ph. …

… ► 1954 in Wabern-Harle, Germany) is Professor in the Department of Mathematics and Statistics at Dalhousie University in Halifax, Nova Scotia, Canada. … ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

4: Bibliography F

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

►

H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:: Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp(i\theta)$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.
External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii), §19.37(ii).
Encodings:: BibTeX

… ►

S. Fillebrown (1992) Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

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External Links:: ISSN 0196-6774, MathReview (P. K. Subramanian), ZentralBlatt
Referenced by:: §24.19(i).
Encodings:: BibTeX

… ►

A. S. Fokas, A. R. Its, and X. Zhou (1992) Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:: Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990
External Links:: MathReview (F. Pempinelli), ZentralBlatt
Referenced by:: §32.15.
Encodings:: BibTeX

… ►

Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

5: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ►

s\left(n,k\right)

denotes the Stirling number of the first kind:

(-1)^{n-k}

times the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. … … ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. … ►For asymptotic approximations for

s\left(n+1,k+1\right)

and

S\left(n,k\right)

that apply uniformly for

1\leq k\leq n

n\to\infty

see Temme (1993) and Temme (2015, Chapter 34). …

6: Sidebar 9.SB1: Supernumerary Rainbows

… ►Photograph by Dr. Roy Bishop, Physics Department, Acadia University, Nova Scotia, Canada. See Bishop (1981). ©R. L. Bishop.

7: 24 Bernoulli and Euler Polynomials

…

8: Stephen M. Watt

… ► 1959 in Montreal, Canada) is Professor of Computer Science in the David R. …

9: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…
$s\left(n,k\right)$	Stirling numbers of the first kind.
…

… ►Other notations for

s\left(n,k\right)

, 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)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

10: Bibliography C

… ►

L. Carlitz (1953) 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

►

L. Carlitz (1954a)

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

►

L. Carlitz (1954b) 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

… ►

R. Cazenave (1969) Intégrales et Fonctions Elliptiques en Vue des Applications. Préface de Henri Villat. Publications Scientifiques et Techniques du Ministère de l’Air, No. 452, Centre de Documentation de l’Armement, Paris.

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External Links:: MathReview (B. Ć. Carlson)
Referenced by:: §19.11(i), §19.12, §19.14(i), §19.36(ii), §19.4(ii), §19.8(ii), §19.8(iii), Ch.19.
Encodings:: BibTeX

… ►

Combinatorial Object Server (website) Department of Computer Science, University of Victoria, Canada.

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Notes:: Permutations, set and integer partitions, sequences, graphs, and trees.
External Links:: Combinatorial Object Server
Referenced by:: 1st item.
Encodings:: BibTeX

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