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21: Diego Dominici

… ►He was elected as Program Director for the period 2011–2016 and served as OPSF-Talk moderator from 2010–2022 with Bonita Saunders, and co-editor for OPSF-Net from 2006–2015 with Martin Muldoon. … ►

9 Airy and Related Functions

…

22: 1.12 Continued Fractions

… ►

C_{n}

is called the

n

th approximant or convergent to $C$ .

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►Define … ►Conversely,

C

is called an extension of

C^{\prime}

. … ►Then the convergents

C_{n}

satisfy …

23: 30.3 Eigenvalues

… ►

30.3.11

\ell_{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,

ⓘ

Symbols:: $m$ : nonnegative integer, $\ell_{j}$ : coefficients, $A$ , $B$ , $C$ and $D$
Permalink:: http://dlmf.nist.gov/30.3.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §30.3(iv), §30.3 and Ch.30

►

A=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)^{2}(2n-3)(2n-1)^{7}(2n+1)(2n+3)^{2}}-% \frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)^{2}(2n+1)(2n+3)^{7}(2n+5)(2n+7)^{2}},

►

B=\frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-7)(2n-5)^{2}% (2n-3)^{3}(2n-1)^{4}(2n+1)}-\frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n% +m+3)(n+m+4)}{(2n+1)(2n+3)^{4}(2n+5)^{3}(2n+7)^{2}(2n+9)},

►

C=\frac{(n-m+1)^{2}(n-m+2)^{2}(n+m+1)^{2}(n+m+2)^{2}}{(2n+1)^{2}(2n+3)^{7}(2n+% 5)^{2}}-\frac{(n-m-1)^{2}(n-m)^{2}(n+m-1)^{2}(n+m)^{2}}{(2n-3)^{2}(2n-1)^{7}(2% n+1)^{2}},

►

D=\frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-3)(2n-1)^{4}% (2n+1)^{2}(2n+3)^{4}(2n+5)}.

…

24: Bibliography M

… ►

L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

… ►

R. C. McCann (1977) Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

ⓘ

External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

►

S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

25: 24.2 Definitions and Generating Functions

… ►

E_{2n+1}=0

… ►

24.2.9

E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.2
Permalink:: http://dlmf.nist.gov/24.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(ii), §24.2 and Ch.24

… ►

\widetilde{E}_{n}\left(x\right)=E_{n}\left(x\right)

0\leq x<1

… ►

Table 24.2.3: Bernoulli numbers

B_{n}=N/D

► ►►

$n$	$N$	$D$	$B_{n}$
…

► ►

Table 24.2.4: Euler numbers

E_{n}

► ►►►

$n$	$E_{n}$
…
$8$	$1385$
…

► …

26: Bibliography S

… ►

F. W. Schäfke and D. Schmidt (1966) Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

ⓘ

Notes:: Includes Algol program
External Links:: ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §28.36(ii).
Encodings:: BibTeX

… ►

J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: Erratum (V1.2.0) Section 18.39.
Encodings:: BibTeX

… ►

R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

ⓘ

External Links:: ISSN 0020-7225, Document, MathReview, ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

ⓘ

External Links:: FullText, MathReview (M. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

… ►

R. Sips (1967) Répartition du courant alternatif dans un conducteur cylindrique de section elliptique. Acad. Roy. Belg. Bull. Cl. Sci. (5) 53 (8), pp. 861–878.

ⓘ

Referenced by:: 6th item.
Encodings:: BibTeX

…

27: Bibliography L

… ►

D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

ⓘ

External Links:: FullText, MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

… ►

J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

ⓘ

External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

H. Levine and J. Schwinger (1948) On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

ⓘ

External Links:: Document, MathReview (C. J. Bouwkamp), ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

J. T. Lewis and M. E. Muldoon (1977) Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 171–178.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

N. A. Lukaševič (1968) Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 4(8), pp. 732–735
External Links:: MathReview (C. E. Langenhop), ZentralBlatt
Referenced by:: §32.10(v), §32.8(v).
Encodings:: BibTeX

…

28: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►and the coefficients

\widetilde{\cal{A}}_{s}(t)

and

\widetilde{\cal{B}}_{s}(t)

are given by … ►and the coefficients

\mathsf{A}_{s}(\tau)

are the product of

\tau^{s}

and a polynomial in

\tau

of degree

2s

. …starting with

\mathsf{A}_{0}(\tau)=1

. … ►The coefficients

A_{s}(\zeta)

and

B_{s}(\zeta)

are given by …The coefficients

C_{s}(\zeta)

and

D_{s}(\zeta)

in (12.10.36) and (12.10.38) are given by …

29: 3.9 Acceleration of Convergence

… ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). … ►

Table 3.9.1: Shanks’ transformation for

s_{n}=\sum_{j=1}^{n}(-1)^{j+1}j^{-2}

► ►►►►►

$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
4	0.82221 76684 88	0.82246 28314 41	0.82246 69467 93	0.82246 70314 36	0.82246 70333 75
…
8	0.82243 73137 33	0.82246 67719 32	0.82246 70301 49	0.82246 70333 73	0.82246 70334 23
9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
…

► …

30: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the

\mathit{3j}

and

\mathit{6j}

symbols in which all parameters are

\leq 8

are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols on pp. … ►Some selected

\mathit{9j}

symbols are also given. … 16-17; for

\mathit{9j}

symbols on p. … ► 310–332, and for the

\mathit{9j}

symbols on pp. …