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31: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
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1	1	3	5	7	9	11	13	15	17	19	21
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5	1	11	61	231	681	1683	3653	7183	13073	22363	36365
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Table 26.6.2: Motzkin numbers $M(n)$.

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$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
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3	4	7	127	11	5798	15	3 10572	19	181 99284

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Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
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6	0	1	15	50	50	15	1
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Table 26.6.4: Schröder numbers $r(n)$.

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$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
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$3$	$22$	$7$	$8558$	$11$	$\mathrm{52\hspace{0.33em}93446}$	$15$	$\mathrm{39376\hspace{0.33em}03038}$	$19$	$\mathrm{323\hspace{0.33em}67243\hspace{0.33em}17174}$

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32: Bibliography F

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

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Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

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External Links:

ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

Encodings:

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

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Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

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Notes:

Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.

External Links:

ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(ii).

Encodings:

BibTeX

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33: 34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

34: Richard B. Paris

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11 Struve and Related Functions

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35: 9.18 Tables

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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36: Bibliography W

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

Encodings:

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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:

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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:

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The generalized Bessel function of order greater than one. Quart. J. Math., Oxford Ser. 11, pp. 36–48.

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External Links:

Document, MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§10.46.

Encodings:

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37: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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38: 24.20 Tables

… ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

39: 30.7 Graphics

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Figure 30.7.4: Eigenvalues ${\lambda }_n^{10}\left({\gamma }^2\right)$, $n=10,11,12,13$, $-50\le {\gamma }^2\le 150$. Magnify

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Figure 30.7.9: ${\mathrm{𝖯𝗌}}_2^0(x,{\gamma }^2)$, $-1\le x\le 1$, $-50\le {\gamma }^2\le 50$. Magnify 3D Help

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Figure 30.7.10: ${\mathrm{𝖯𝗌}}_3^1(x,{\gamma }^2)$, $-1\le x\le 1$, $-50\le {\gamma }^2\le 50$. Magnify 3D Help

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40: Bibliography J

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Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

Encodings:

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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