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1: 18.3 Definitions
β–Ί
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
β–Ί β–Ίβ–Ί
Name p n ⁑ ( x ) ( a , b ) w ⁑ ( x ) h n k n k ~ n / k n Constraints
β–Ί
β–ΊFor explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
2: 29.20 Methods of Computation
β–ΊSubsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … β–ΊA third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as n . … β–Ί
§29.20(ii) Lamé Polynomials
β–ΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
3: 15.17 Mathematical Applications
β–ΊIn combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. … β–ΊThe three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …
4: 5.11 Asymptotic Expansions
β–Ίwhere … β–Ί
5.11.5 g k = 2 ⁒ ( 1 2 ) k ⁒ a 2 ⁒ k ,
β–Ίwhere a 0 = 1 2 ⁒ 2 and …Wrench (1968) gives exact values of g k up to g 20 . … β–ΊThe expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). …
5: 18.25 Wilson Class: Definitions
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§18.25(iv) Leading Coefficients
β–ΊTable 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. β–Ί
Table 18.25.2: Wilson class OP’s: leading coefficients.
β–Ί β–Ίβ–Ί
p n ⁑ ( x ) k n
β–Ί
6: 29.7 Asymptotic Expansions
β–Ί
29.7.1 a Ξ½ m ⁑ ( k 2 ) p ⁒ ΞΊ Ο„ 0 Ο„ 1 ⁒ ΞΊ 1 Ο„ 2 ⁒ ΞΊ 2 β‹― ,
β–Ί
29.7.3 Ο„ 0 = 1 2 3 ⁒ ( 1 + k 2 ) ⁒ ( 1 + p 2 ) ,
β–Ί
29.7.4 Ο„ 1 = p 2 6 ⁒ ( ( 1 + k 2 ) 2 ⁒ ( p 2 + 3 ) 4 ⁒ k 2 ⁒ ( p 2 + 5 ) ) .
β–Ί
29.7.6 Ο„ 2 = 1 2 10 ⁒ ( 1 + k 2 ) ⁒ ( 1 k 2 ) 2 ⁒ ( 5 ⁒ p 4 + 34 ⁒ p 2 + 9 ) ,
β–ΊIn Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions 𝐸𝑐 Ξ½ m ⁑ ( z , k 2 ) and 𝐸𝑠 Ξ½ m ⁑ ( z , k 2 ) . …
7: 18.19 Hahn Class: Definitions
β–Ί
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
β–Ί β–Ίβ–Ί
p n ⁑ ( x ) k n
β–Ί
8: 16.23 Mathematical Applications
β–ΊA variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. … β–ΊMany combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. …
9: 26.3 Lattice Paths: Binomial Coefficients
§26.3 Lattice Paths: Binomial Coefficients
β–Ί
§26.3(i) Definitions
β–Ί
§26.3(ii) Generating Functions
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§26.3(iii) Recurrence Relations
β–Ί
§26.3(iv) Identities
10: Ronald F. Boisvert
β–Ί 1951 in Manchester, New Hampshire) leads the Applied and Computational Mathematics Division of the NIST Information Technology Laboratory. …