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21: 16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ is well-poised if … ►See Raynal (1979) for a statement in terms of $\mathit{3}j$ symbols (Chapter 34). … ► … ►Transformations for both balanced ${}_4{}^{}F_3^{}\left(1\right)$ and very well-poised ${}_7{}^{}F_6^{}\left(1\right)$ are included in Bailey (1964, pp. 56–63). A similar theory is available for very well-poised ${}_9{}^{}F_8^{}\left(1\right)$’s which are 2-balanced. …

22: 15.8 Transformations of Variable

… ►All functions in this subsection and §15.8(ii) assume their principal values. … ►When $b-a$ is an integer limits are taken in (15.8.2) and (15.8.3) as follows. … ►In (15.8.8) when $c-a-k-m$ is a nonpositive integer $\psi \left(c-a-k-m\right)/\mathrm{\Gamma }\left(c-a-k-m\right)$ is interpreted as ${(-1)}^{m+k+a-c+1}(m+k+a-c)!$. … ►Alternatively, if $b-a$ is a negative integer, then we interchange $a$ and $b$ in $𝐅(a,b;c;z)$. ►In a similar way, when $c-a-b$ is an integer limits are taken in (15.8.4) and (15.8.5) as follows. …

23: Bibliography M

… ►

Two-point quasi-fractional approximations to the Airy function $\mathrm{Ai}(x)$ . J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:

There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$.

External Links:

ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

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On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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External Links:

ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

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… ►

Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. Lond. Math. Soc. 9, pp. 6–13 (German).

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External Links:

ISSN 0024-6107; 1469-7750/e, ZentralBlatt

Referenced by:

§18.2(xii).

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An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

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External Links:

ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

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A connection formula for the $q$-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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External Links:

ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt

Referenced by:

§17.6(ii), §17.6(ii).

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…

24: Bibliography S

… ►

Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

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Perturbational results for diffraction of water-waves by nearly-vertical barriers. IMA J. Appl. Math. 34 (1), pp. 99–117.

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External Links:

ISSN 0272-4960, Document, MathReview, ZentralBlatt

Referenced by:

§11.12.

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… ►

A new Fortran program for the $9$-$j$ angular momentum coefficient. Comput. Phys. Comm. 56 (2), pp. 231–248.

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External Links:

Document, ZentralBlatt

Referenced by:

§34.13.

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… ►

An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

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…

25: 19.5 Maclaurin and Related Expansions

… ►where ${}_2{}^{}F_1^{}$ is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$ is an Appell function (§16.13). … ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when see Erdélyi et al. (1953b, §13.6(9)). …

26: 19.21 Connection Formulas

… ►The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►The complete case of $R_J$ can be expressed in terms of $R_F$ and $R_D$: … ► $R_D(x,y,z)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using …Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)). … ►Connection formulas for $R_{-a}(𝐛;𝐳)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). …

27: 34.4 Definition: $\mathit{6}j$ Symbol

§34.4 Definition: $\mathit{6}j$ Symbol

►The $\mathit{6}j$ symbol is defined by the following double sum of products of $\mathit{3}j$ symbols: …where the summation is taken over all admissible values of the $m$’s and $m^{\prime }$’s for each of the four $\mathit{3}j$ symbols; compare (34.2.2) and (34.2.3). … ►where ${}_4{}^{}F_3^{}$ is defined as in §16.2. ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

28: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

29: 18.13 Continued Fractions

… ►The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii). … ►

18.13.3 $$\frac{a_1}{x+}\frac{-\frac{1}{2}}{\frac{3}{2}x+}\frac{-\frac{2}{3}}{\frac{5}{3}x+}\frac{-\frac{3}{4}}{\frac{7}{4}x+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E3

Encodings:

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See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.4 $$\frac{a_1}{1-x+}\frac{-\frac{1}{2}}{\frac{1}{2}(3-x)+}\frac{-\frac{2}{3}}{\frac{1}{3}(5-x)+}\frac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E4

Encodings:

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See also:

Annotations for §18.13, §18.13 and Ch.18

… ►See also Cuyt et al. (2008, pp. 91–99).

30: 34.10 Zeros

§34.10 Zeros

►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to $\mathit{9}j$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).