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11: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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On the computation of modified Bessel function ratios. Math. Comp. 32 (143), pp. 865–875.

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External Links:

ISSN 0025-5718, FullText, MathReview (James M. Horner), ZentralBlatt

Referenced by:

§10.74(v).

Encodings:

BibTeX

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Computing the real parabolic cylinder functions $U(a,x)$, $V(a,x)$ . ACM Trans. Math. Software 32 (1), pp. 70–101.

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External Links:

ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt

Referenced by:

§12.18.

Encodings:

BibTeX

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A new identity for some six-$j$ symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

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External Links:

ISSN 0022-2488, Document, MathReview

Referenced by:

§34.5(vi).

Encodings:

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A Treatise on Bessel Functions and their Applications to Physics. 2nd edition, Macmillan and Co., London.

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Notes:

Table errata: Math. Comp. v. 24 (1970), p. 240, v. 31 (1978), pp. 806 and 1046, and v. 32 (1978), pp. 318. Reprinted by Dover Publications Inc., New York, 1966.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.21(x), §10.73(i), §10.73(i).

Encodings:

BibTeX

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12: 28.6 Expansions for Small $q$

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28.6.10 $$a_5\left(q\right)=25+\frac{1}{48}{q}^2+\frac{11}{\mathrm{7\hspace{0.33em}74144}}{q}^4+\frac{1}{\mathrm{1\hspace{0.33em}47456}}{q}^5+\frac{37}{\mathrm{8918\hspace{0.33em}13888}}{q}^6+\mathrm{\cdots },$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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28.6.11 $$b_5\left(q\right)=25+\frac{1}{48}{q}^2+\frac{11}{\mathrm{7\hspace{0.33em}74144}}{q}^4-\frac{1}{\mathrm{1\hspace{0.33em}47456}}{q}^5+\frac{37}{\mathrm{8918\hspace{0.33em}13888}}{q}^6+\mathrm{\cdots },$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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28.6.14 $$\begin{array}{c}{a}_m\left(q\right)\\ b_m\left(q\right)\end{array}\}=m^2+\frac{1}{2(m^2-1)}{q}^2+\frac{5m^2+7}{32{(m^2-1)}^3(m^2-4)}{q}^4+\frac{9m^4+58m^2+29}{64{(m^2-1)}^5(m^2-4)(m^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer and $q=h^2$: parameter

A&S Ref:

20.2.26

Referenced by:

§28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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28.6.21 $$2^{1/2}{\mathrm{ce}}_0(z,q)=1-\frac{1}{2}q\cos 2z+\frac{1}{32}{q}^2\left(\cos 4z-2\right)-\frac{1}{128}{q}^3\left(\frac{1}{9}\cos 6z-11\cos 2z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable

A&S Ref:

20.2.27

Referenced by:

§28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.26 $${\mathrm{ce}}_m(z,q)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}\cos (m+2)z-\frac{1}{m-1}\cos (m-2)z\right)+\frac{q^2}{32}\left(\frac{1}{(m+1)(m+2)}\cos (m+4)z+\frac{1}{(m-1)(m-2)}\cos (m-4)z-\frac{2(m^2+1)}{{(m^2-1)}^2}\cos mz\right)+\mathrm{\cdots }.$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter and $z$: complex variable

A&S Ref:

20.2.28 (in slightly different form)

Referenced by:

§28.6(ii), §28.6(ii), §28.6(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.6.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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13: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$32$	$-\mathrm{770\hspace{0.33em}93210\hspace{0.33em}41217}$	$510$	$-\mathrm{1.51163\hspace{0.33em}1577}$	$\mathrm{\times 10¹⁰}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$32$	$\mathrm{17751\hspace{0.33em}93915\hspace{0.33em}79539\hspace{0.33em}28943\hspace{0.33em}66647\hspace{0.33em}89665}$
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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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$11$	$0$	$\frac{5}{6}$	$0$	$-\frac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\frac{55}{6}$	$-\frac{11}{2}$	$1$
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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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14: Vadim B. Kuznetsov

… ►He was also editor of The Kowalevski Property, CRM Proceedings and Lecture Notes 32, published by the American Mathematical Society in 2002. …

15: Bibliography

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Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:

MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§30.9(iii).

Encodings:

BibTeX

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

Encodings:

BibTeX

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Ly$\alpha$ line formation in starbursting galaxies. I. Moderately thick, dustless, and static H i media. Astrophysical J. 554, pp. 604–614.

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External Links:

FullText

Referenced by:

§7.21.

Encodings:

BibTeX

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Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:

Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.

External Links:

Document, MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

BibTeX

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Matching higher conserved charges for strings and spins. J. High Energy Phys. 2004 (3).

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Notes:

(electronic only, article id. JHEP03(2004)004, 32 pp.).

External Links:

ISSN 1126-6708, Document, MathReview

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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16: 34.1 Special Notation

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17: Bibliography W

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The irregular primes to $125000$ . Math. Comp. 32 (142), pp. 583–591.

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External Links:

FullText, Document, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.20.

Encodings:

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11 (4), pp. 690–701.

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External Links:

ISSN 0036-1410, Document, MathReview (R. G. Buschman), ZentralBlatt

Referenced by:

§18.25(i), §18.25(i), §18.25(ii), §18.25(iii), §18.26(i), §18.26(ii), §18.38(iii).

Encodings:

BibTeX

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:

ISSN 1044-677X, FullText

Referenced by:

§28.4(vii).

Encodings:

BibTeX

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18: Bibliography H

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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External Links:

Document

Referenced by:

§15.18.

Encodings:

BibTeX

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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

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Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

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Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

ISSN 0001-0782, Document, Remark 1, Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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19: 26.10 Integer Partitions: Other Restrictions

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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$11$	$12$	$7$	$4$	$5$
$12$	$15$	$9$	$6$	$6$
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$15$	$27$	$14$	$9$	$9$
$16$	$32$	$17$	$11$	$10$
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… ►It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

20: 10 Bessel Functions

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