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1: 34.6 Definition: 9 ⁒ j Symbol
§34.6 Definition: 9 ⁒ j Symbol
β–ΊThe 9 ⁒ j symbol may be defined either in terms of 3 ⁒ j symbols or equivalently in terms of 6 ⁒ j symbols: β–Ί
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  ⁒ m r ⁒ s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ⁒ ( j 21 j 22 j 23 m 21 m 22 m 23 ) ⁒ ( j 31 j 32 j 33 m 31 m 32 m 33 ) ⁒ ( j 11 j 21 j 31 m 11 m 21 m 31 ) ⁒ ( j 12 j 22 j 32 m 12 m 22 m 32 ) ⁒ ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
β–Ί
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 ⁒ j ⁒ ( 2 ⁒ j + 1 ) ⁒ { j 11 j 21 j 31 j 32 j 33 j } ⁒ { j 12 j 22 j 32 j 21 j j 23 } ⁒ { j 13 j 23 j 33 j j 11 j 12 } .
β–ΊThe 9 ⁒ j symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
2: Customize DLMF
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3: Ian J. Thompson
Profile
Ian J. Thompson
β–ΊIan J. Thompson (b. …
4: 27.20 Methods of Computation: Other Number-Theoretic Functions
β–ΊThe recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ⁑ ( n ) for n < N . See Calkin et al. (2007), and Lehmer (1941, pp. 5–83). … β–ΊA recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function Ο„ ⁑ ( n ) , and the values can be checked by the congruence (27.14.20). …
5: Preface
β–ΊAbramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authoritative status of the existing Handbook, and its orientation toward applications in science, statistics, engineering and computation, will be preserved. β–ΊThus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. The term digital library has gained acceptance for this kind of information resource, and our choice of project title reflects our hope that the NIST DLMF will be a vehicle for revolutionizing the way applicable mathematics in general is practiced and delivered.
6: Viewing DLMF Interactive 3D Graphics
β–ΊBelow we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer. We have strived to provide the most current information available, but if any details appear to be out of date or confusing, please notify us. … β–ΊUntil these issues are resolved we cannot guarantee that the DLMF WebGL visualizations can be viewed in Internet Explorer. … β–ΊBoth are expected to be superseded by WebGL. Any installed VRML or X3D browser should be removed before installing a new one. …
7: 29.20 Methods of Computation
β–ΊThe normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … β–ΊInitial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … β–ΊThese matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that n has to be chosen sufficiently large. … β–ΊAlternatively, the zeros can be found by locating the maximum of function g in (29.12.11).
8: 17.19 Software
β–ΊCitations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. …
9: 31.18 Methods of Computation
β–ΊIndependent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see LaΔ­ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …
10: 21.11 Software
β–ΊCitations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. …