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1: 35.12 Software

… ►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

2: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

3: Bibliography B

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:

ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

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Prolate angular spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 187–192.

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Document, Code

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1st item.

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Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.

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External Links:

Document, Code

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1st item.

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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

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4: Bibliography Z

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Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry

Referenced by:

3rd item, 6th item, §7.25(iii), §7.25(vi).

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom I, Izdat. Akad. Nauk SSSR, Moscow, 1960.

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

4th item.

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom II, Izdat. Akad. Nauk SSSR, Moscow, 1962.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

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Tablitsy vyrozhdennoi gipergeometricheskoi funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

Translated title: Tables of the confluent hypergeometric function

External Links:

MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

1st item.

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5: Bibliography L

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Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.

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Notes:

For a Maple implementation of the algorithm given in this paper for Jack and zonal polynomials see Zeilberger (website).

External Links:

ISSN 0010-3616, Link, MathReview (Kazuhiro Hikami), ZentralBlatt

Referenced by:

§35.12.

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Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds. Comput. Phys. Comm. 60 (3), pp. 319–327.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§19.36(iii), §19.5.

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Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

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Notes:

Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.

External Links:

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

Referenced by:

§8.28(iv).

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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.

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External Links:

ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§8.13(i).

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Miniaturized tables of Bessel functions. Math. Comp. 25 (114), pp. 323–330.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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6: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Version 1.1.6 (June 30, 2022)

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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There has been disagreement about the identification of the Chebyshev polynomials of the third and fourth kinds, denoted $V_n\left(x\right)$ and $W_n\left(x\right)$, in published references. Originally, DLMF used the definitions given in (Andrews et al., 1999, Remark 2.5.3). However, those definitions were the reverse of those used by Mason and Handscomb (2003), Gautschi (2004) following Mason (1993) and Gautschi (1992), as was noted in several warnings added in Version 1.0.10 (August 7, 2015) of the DLMF. Since the latter definitions are more widely established, the DLMF is now adopting the definitions of Mason and Handscomb (2003). Essentially, what we previously denoted $V_n(x)$ is now written as $W_n(x)$, and vice-versa.

This notational interchange necessitated changes in Tables 18.3.1, 18.5.1, and 18.6.1, and in Equations (18.5.3), (18.5.4), (18.7.5), (18.7.6), (18.7.17), (18.7.18), (18.9.11), and (18.9.12).

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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7: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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8: 14.33 Tables

§14.33 Tables

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Zhang and Jin (1996, Chapter 4) tabulates ${𝖯}_n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${𝖯}_n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖰}_n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${𝖰}_n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖯}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${𝖰}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${𝖯}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P_n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${𝖯}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

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Žurina and Karmazina (1964, 1965) tabulate the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

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Žurina and Karmazina (1963) tabulates the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

►For tables prior to 1961 see Fletcher et al. (1962) and Lebedev and Fedorova (1960).

9: 24.2 Definitions and Generating Functions

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§24.2(iv) Tables

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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Table 24.2.2: Bernoulli and Euler polynomials.

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$n$	$B_n\left(x\right)$	$E_n\left(x\right)$
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10: 24.1 Special Notation

… ►The present notation, as defined in §24.2(i), was used in Lucas (1891) and Nörlund (1924), and has become the prevailing notation; see Table 24.2.1. … ►

$B_2=\frac{1}{30}$ ,

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$B_4=\frac{1}{30},\mathrm{\dots }$ .

… ►Other historical remarks on notations can be found in Cajori (1929, pp. 42–44). …