parameter constraints

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21: Bibliography C

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On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

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External Links:

ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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Constraints on the phase of scattering amplitudes due to positivity. Nuclear Phys. B 49, pp. 413–440.

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External Links:

Document

Referenced by:

§18.39(v).

Encodings:

BibTeX

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Constraints on the phases of helicity amplitudes due to positivity. Nuclear Phys. B 77, pp. 141–162.

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External Links:

Document, MathReview

Referenced by:

§18.39(v).

Encodings:

BibTeX

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$S$-matrix and $R$-matrix determination of the low-energy ${}_{}{}^5\mathrm{He}$ and ${}_{}{}^5\mathrm{Li}$ resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

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External Links:

Document

Referenced by:

2nd item.

Encodings:

BibTeX

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22: 18.35 Pollaczek Polynomials

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18.35.4 $$P_n^{(\lambda )}(\cos \theta ;a,b)=\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_n}{n!}{\mathrm{e}}^{\mathrm{i}n\theta }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{-n-\lambda +1+\mathrm{i}{\tau }_{a,b}(\theta )};{\mathrm{e}}^{-2\mathrm{i}\theta })=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta },$$

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Defines:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.10)); In the cited formula, in the lower parameter of the hypergeometric function, $-\lambda$ should be replaced by $-\lambda +1$.(proved)

Referenced by:

(18.35.4_5)

Permalink:

http://dlmf.nist.gov/18.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(i), §18.35 and Ch.18

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18.35.5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b)P_m^{(\lambda )}(x;a,b)w^{(\lambda )}(x;a,b)dx=\frac{\mathrm{\Gamma }\left(2\lambda +n\right)}{n!(\lambda +a+n)}{\delta }_{n,m},$$ $a\ge b\ge -a$, $\lambda >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, Theorem 5.4.2)(proved)

Referenced by:

(18.35.6), §18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation which was previously only valid for $n\ne m$, was updated to give the full normalization and the constraints which were originally given in (18.35.6) were moved to this equation. The constraint $\lambda >-\frac{1}{2}$ has been updated to be $\lambda >0$.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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18.35.6 $$w^{(\lambda )}(\cos \theta ;a,b)={\pi }^{-1}{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}{\left(2\sin \theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$

Source:

Ismail (2009, (5.4.13))

Referenced by:

(18.35.5), §18.35(ii), §18.39(iv), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

The constraints in this equation have been moved to (18.35.5) except for .

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

… ►with two possible constraints: $a>b>-a$, $2\lambda +c>0$, $c\ge 0$, or $a>b>-a$, $2\lambda +c\ge 1$, $c>-1$. …