triangle conditions

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1: 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. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

2: 34.2 Definition: $\mathit{3}j$ Symbol

… ►They therefore satisfy the triangle conditions …

3: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►Then assuming the triangle conditions are satisfied … ►Again it is assumed that in (34.3.7) the triangle conditions are satisfied. … ►In the following three equations it is assumed that the triangle conditions are satisfied by each $\mathit{3}j$ symbol. …

4: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►In the following equation it is assumed that the triangle conditions are satisfied. …

5: 18.37 Classical OP’s in Two or More Variables

… ►The following three conditions, taken together, determine $R_{m,n}^{(\alpha )}\left(z\right)$ uniquely: … ►

§18.37(ii) OP’s on the Triangle

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Definition in Terms of Jacobi Polynomials

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18.37.7 $$P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)=P_{m-n}^{(\alpha ,\beta +\gamma +2n+1)}\left(2x-1\right)x^n{P}_n^{(\beta ,\gamma )}\left(2x^{-1}y-1\right),$$ $m\ge n\ge 0$, $\alpha ,\beta ,\gamma >-1$.

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Defines:

$P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.37(ii)

Permalink:

http://dlmf.nist.gov/18.37.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

… ►

18.37.8 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

…

6: 28.29 Definitions and Basic Properties

… ►Given $\lambda$ together with the condition (28.29.6), the solutions $\pm \nu$ of (28.29.9) are the characteristic exponents of (28.29.1). … ►For a given $\nu$, the characteristic equation $△(\lambda )-2\cos \left(\pi \nu \right)=0$ has infinitely many roots $\lambda$. Conversely, for a given $\lambda$, the value of $△(\lambda )$ is needed for the computation of $\nu$. … ►

28.29.16 $${\lambda }_n,n=0,1,2,\mathrm{\dots },\text{ with }△({\lambda }_n)=2,$$

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Defines:

${\lambda }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(iii), §28.29 and Ch.28

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28.29.17 $${\mu }_n,n=1,2,3,\mathrm{\dots },\text{ with }△({\mu }_n)=-2.$$

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Defines:

${\mu }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(iii), §28.29 and Ch.28

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7: 10.22 Integrals

… ►(Thus if $a,b,c$ are the sides of a triangle, then $A^{\frac{1}{2}}$ is the area of the triangle.) … ►Sufficient conditions for the validity of (10.22.77) are that when $\nu \ge -\frac{1}{2}$, or that and when ; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). … ►A sufficient condition for the validity is . …Sufficient conditions for the validity of (10.22.79) are that when , or that and when ; see Titchmarsh (1962a, pp. 88–90). …

8: Bibliography G

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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

… ►

Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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External Links:

ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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