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21: Bibliography W

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

Encodings:

BibTeX

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

BibTeX

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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22: 18.35 Pollaczek Polynomials

… ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) …the recurrence relation of form (18.2.11_5) becomes … ►As in the coefficients of the above recurrence relations $n$ and $c$ only occur in the form $n+c$, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30. … ►we have the explicit representations … ►For type 3 orthogonality (18.35.5) generalizes to …

23: 1.9 Calculus of a Complex Variable

… ►($z_0$ may or may not belong to $S$.) … ►When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open. … ►A function analytic at every point of $ℂ$ is said to be entire. … ►(principal value), where …(principal value), where $\nu \in ℂ$, …

24: Bibliography T

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Painlevé-Calogero correspondence revisited. J. Math. Phys. 42 (3), pp. 1443–1473.

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External Links:

ISSN 0022-2488, Document, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.2(iii).

Encodings:

BibTeX

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Modified Bessel functions $I_{\nu }(z)$ and $K_{\nu }(z)$ of real order and complex argument, to selected accuracy. Comput. Phys. Comm. 47 (2-3), pp. 245–257.

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Notes:

For erratum see same journal 159 (2004), no. 3, p. 243. Double-precision Fortran, accuracy 24S, but can be increased.

External Links:

ISSN 0010-4655, Document, Code, MathReview (John P. Coleman)

Referenced by:

3rd item, 4th item.

Encodings:

BibTeX

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

Encodings:

BibTeX

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Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.

External Links:

ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt

Referenced by:

§34.3(vii).

Encodings:

BibTeX

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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25: 8.21 Generalized Sine and Cosine Integrals

… ►Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

§8.21(v) Special Values

… ►For ${𝗃}_n\left(z\right)$ see §10.47(ii). … ►When $z\to \mathrm{\infty }$ with $|\mathrm{ph}z|\le \pi -\delta$ (), …

26: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

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A new identity for some six-$j$ symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

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External Links:

ISSN 0022-2488, Document, MathReview

Referenced by:

§34.5(vi).

Encodings:

BibTeX

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Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

BibTeX

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27: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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Toward symbolic integration of elliptic integrals. J. Symbolic Comput. 28 (6), pp. 739–753.

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Notes:

Code written by J. FitzSimons to generate reduction tables for elliptic integrals symbolically in the Derive computer algebra system is available.

External Links:

ISSN 0747-7171, Document, Code, MathReview (Axel Riese), ZentralBlatt

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii), §19.29(ii).

Encodings:

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Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§10.21(x), §10.21(x).

Encodings:

BibTeX

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

Encodings:

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An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

Encodings:

BibTeX

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28: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. ►

Ultraspherical

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Laguerre

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Hermite

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29: Bibliography K

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Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:

ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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External Links:

ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

Encodings:

BibTeX

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Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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External Links:

FullText, MathReview (T. Oda), ZentralBlatt

Referenced by:

§21.6(ii).

Encodings:

BibTeX

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The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:

ISSN 0377-0427, Document, Link, MathReview Entry

Referenced by:

(18.9.18_5).

Encodings:

BibTeX

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An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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External Links:

ISSN 0747-7171, MathReview, ZentralBlatt

Referenced by:

§31.14(ii).

Encodings:

BibTeX

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30: 14.17 Integrals

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14.17.6 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^m\left(x\right)dx=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}{\delta }_{l,n},$$

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.11 8.14.13

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.7 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^{-m}\left(x\right)dx=\frac{{(-1)}^m}{l+\frac{1}{2}}{\delta }_{l,n},$$

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^m$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.8 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^m\left(x\right)}{1-x^2}dx=\frac{(n+m)!}{(n-m)!m}{\delta }_{l,m},$$ $m>0$,

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.12 8.14.14

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.9 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^{-m}\left(x\right)}{1-x^2}dx=\frac{{(-1)}^l}{l}{\delta }_{l,m},$$ $l>0$.

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^l$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). …