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1: 24 Bernoulli and Euler Polynomials

Chapter 24 Bernoulli and Euler Polynomials

…

2: Bibliography X

… ►

G. L. Xu and J. K. Li (1994) Variable precision computation of elementary functions. J. Numer. Methods Comput. Appl. 15 (3), pp. 161–171 (Chinese).

ⓘ

Notes:: English translation in Chinese J. Numer. Math. Appl. 17 (1995), no. 1, pp. 13–24
External Links:: ISSN 1000-3266, MathReview, ZentralBlatt
Referenced by:: §4.48(iii).
Encodings:: BibTeX

3: Adri B. Olde Daalhuis

… ►On August 24, 2012, he was appointed Mathematics Editor for the DLMF Project. …

4: Karl Dilcher

… ►

24 Bernoulli and Euler Polynomials

…

5: David M. Bressoud

… ►His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …

6: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

34.7.2

\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(iv), §34.7 and Ch.34

… ►

34.7.3

\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.9
Permalink:: http://dlmf.nist.gov/34.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

…

7: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

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\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. …

8: 26.1 Special Notation

… ►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).

9: Bibliography Q

… ►

H. Qin and Y. Lu (2008) A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

ⓘ

External Links:: ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt
Referenced by:: §32.11(i).
Encodings:: BibTeX

…

10: 27.2 Functions

… ►

Table 27.2.2: Functions related to division.

► ►►►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
2	1	2	3	15	8	4	24	28	12	6	56	41	40	2	42
…
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
9	6	3	13	22	10	4	36	35	24	4	48	48	16	10	124
…
13	12	2	14	26	12	4	42	39	24	4	56	52	24	6	98

►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
2	1	2	3	15	8	4	24	28	12	6	56	41	40	2	42
…
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
9	6	3	13	22	10	4	36	35	24	4	48	48	16	10	124
…
13	12	2	14	26	12	4	42	39	24	4	56	52	24	6	98

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
2	1	2	3	15	8	4	24	28	12	6	56	41	40	2	42
…
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
9	6	3	13	22	10	4	36	35	24	4	48	48	16	10	124
…
13	12	2	14	26	12	4	42	39	24	4	56	52	24	6	98