tables of coefficients

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11: 29.21 Tables

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

12: 18.39 Applications in the Physical Sciences

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. …Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. …

13: 3.11 Approximation Techniques

… ►with coefficients given in Table 3.11.1. ►

Table 3.11.1: Coefficients $p_j$, $q_j$ for the minimax rational approximation $R_{3,3}(x)$.

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$j$	$p_j$	$q_j$
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… ►Any five approximants arranged in the Padé table as …

14: 34.14 Tables

… ►Biedenharn and Louck (1981) give tables of algebraic expressions for Clebsch–Gordan coefficients and $\mathit{6}j$ symbols, together with a bibliography of tables produced prior to 1975. …

15: Bibliography N

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Tables of Lagrangian Interpolation Coefficients. Columbia University Press, New York.

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Notes:

Technical Director: Arnold N. Lowan

External Links:

MathReview (W. Feller), ZentralBlatt

Referenced by:

§3.3(ii).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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16: Bibliography G

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Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

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External Links:

MathReview (M. S. Cheema), ZentralBlatt

Referenced by:

§26.21.

Encodings:

BibTeX

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17: 10.75 Tables

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Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including $\zeta$ and ${(4\zeta /(1-x^2))}^{\frac{1}{4}}$ as functions of $x$ $(=z)$ and the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, $D_k(\zeta )$ as functions of $\zeta$. These enable $J_{\nu }\left(\nu x\right)$, $Y_{\nu }\left(\nu x\right)$, $J_{\nu }^{\prime }\left(\nu x\right)$, $Y_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 15$, except in the neighborhoods of zeros.

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Olver (1960) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(\frac{1}{2})20⁤\frac{1}{2}$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$; see §10.21(viii), and more fully Olver (1954).

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Olver (1962) provides tables for the uniform asymptotic expansions given in §10.41(ii), including $\eta$ and the coefficients $U_k(p)$, $V_k(p)$ as functions of $p={(1+x^2)}^{-\frac{1}{2}}$. These enable $I_{\nu }\left(\nu x\right)$, $K_{\nu }\left(\nu x\right)$, $I_{\nu }^{\prime }\left(\nu x\right)$, $K_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 16$.

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Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

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18: 18.22 Hahn Class: Recurrence Relations and Differences

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18.22.2 $$-x{p}_n(x)=A_n{p}_{n+1}(x)-\left(A_n+C_n\right)p_n(x)+C_n{p}_{n-1}(x),$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $A_n$: coefficient and $C_n$: coefficient

Referenced by:

§16.4(iii), §18.21(ii), §18.22(i), Table 18.22.1

Permalink:

http://dlmf.nist.gov/18.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

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18.22.12 $$A(x)p_n(x+1)-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-1)+{\lambda }_n{p}_n(x)=0.$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $C(x)$: coefficient, ${\lambda }_n$: coefficient and $A(x)$: coefficient

Referenced by:

§18.22(ii), Table 18.22.2

Permalink:

http://dlmf.nist.gov/18.22.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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19: 3.5 Quadrature

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Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions $p_n(x)$ and orthonormal versions $q_n(x)$ of the classical orthogonal polynomials.

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$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
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20: Bibliography S

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Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§28.1, 7th item, Notations S, Notations S.

Encodings:

BibTeX

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Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0262190028, 0262692910, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

1st item, Ch.30.

Encodings:

BibTeX

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