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21: Bibliography G

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Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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ISSN 0001-0782, Document

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§33.26(ii).

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Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

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ISSN 0001-0782, Document

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§7.25(iii).

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Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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FullText, MathReview (L. Carlitz), ZentralBlatt

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§26.1, §26.1, Notations S, Notations S.

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Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

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22: Bibliography H

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25D Table of the First One Hundred Values of $j_{0,s}$,$J_1(j_{0,s})$, $j_{1,s}$,$J_0(j_{1,s})=J_0(j_{0,s+1}^{\prime })$,$j_{1,s}^{\prime }$,$J_1(j_{1,s}^{\prime })$ . Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

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Ms. of six pages deposited in the UMT file

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8th item.

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

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Document

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§15.18.

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

Referenced by:

§10.21(xiv).

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

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

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

Referenced by:

§8.28(iv).

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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ZentralBlatt

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§24.6.

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23: Publications

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Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

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Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

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From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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24: Bibliography D

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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

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§18.32.

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Real zeros of ${}_2{}^{}F_1^{}$ hypergeometric polynomials. J. Comput. Appl. Math. 247, pp. 152–161.

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ISSN 0377-0427, Document, FullText, MathReview (Nasser Saad), ZentralBlatt

Referenced by:

§15.13, §15.13.

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Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

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ISSN 0001-0782, Document

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§8.28(iv).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

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ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

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§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

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25: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. … ►These are given by the following equations in which $a_1,a_2,\mathrm{\dots },a_n$ are nonnegative integers such that …$M_1$ is the multinominal coefficient (26.4.2): …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. … ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …

26: 3.2 Linear Algebra

… ►where $u_j=c_j$, $j=1,2,\mathrm{\dots },n-1$, $d_1=b_1$, and …Forward elimination for solving $𝐀𝐱=𝐟$ then becomes $y_1=f_1$, …and back substitution is $x_n=y_n/d_n$, followed by … ►Define the Lanczos vectors ${𝐯}_j$ and coefficients ${\alpha }_j$ and ${\beta }_j$ by ${𝐯}_0=\mathrm{𝟎}$, a normalized vector ${𝐯}_1$ (perhaps chosen randomly), ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$, and for $j=1,2,\mathrm{\dots },n-1$ by the recursive scheme … ►Start with ${𝐯}_0=\mathrm{𝟎}$, vector ${𝐯}_1$ such that ${𝐯}_1^{\mathrm{T}}𝐒{𝐯}_1=1$, ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$. …

27: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor $A_{jk}$ of $a_{jk}$ is … ►For real-valued $a_{jk}$, … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►If ${𝐷}_n[a_{j,k}]$ tends to a limit $L$ as $n\to \mathrm{\infty }$, then we say that the infinite determinant ${𝐷}_{\mathrm{\infty }}[a_{j,k}]$ converges and ${𝐷}_{\mathrm{\infty }}[a_{j,k}]=L$. … ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. …

28: 8.26 Tables

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Pearson (1965) tabulates the function $I(u,p)$ ($=P(p+1,u)$) for $p=-1(.05)0(.1)5(.2)50$, $u=0(.1)u_p$ to 7D, where $I(u,u_p)$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$, $u=0(.1)6$ to 5D.

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Pearson (1968) tabulates $I_x(a,b)$ for $x=0.01(.01)1$, $a,b=0.5(.5)11(1)50$, with $b\le a$, to 7D.

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Chiccoli et al. (1988) presents a short table of $E_p\left(x\right)$ for $p=-\frac{9}{2}(1)-\frac{1}{2}$, $0\le x\le 200$ to 14S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Stankiewicz (1968) tabulates $E_n\left(x\right)$ for $n=1(1)10$, $x=0.01(.01)5$ to 7D.

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29: Bibliography W

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

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9th item.

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

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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).

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Asymptotics for the ${}_4{}^{}F_3^{}$ polynomials. J. Approx. Theory 66 (1), pp. 58–71.

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ISSN 0021-9045, Document, MathReview (R. G. Buschman), ZentralBlatt

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§18.26(v).

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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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ISSN 1044-677X, FullText

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§28.4(vii).

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

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§10.46.

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30: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).