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1: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
2: 18.25 Wilson Class: Definitions
§18.25(iv) Leading Coefficients
Table 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials.
Table 18.25.2: Wilson class OP’s: leading coefficients.
p n ( x ) k n
3: 18.19 Hahn Class: Definitions
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
p n ( x ) k n
4: 18.15 Asymptotic Approximations
The leading coefficients are given by … The leading coefficients are given by A 0 ( ξ ) = 1 and … The leading coefficients are given by E 0 ( ζ ) = 1 and …
5: 2.6 Distributional Methods
This leads to integrals of the form … The replacement of f ( t ) by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … In the sense of distributions, they can be written … Also, …Multiplication of these expansions leads to …
6: 28.2 Definitions and Basic Properties
28.2.18 w ( z ) = n = - c 2 n e i ( ν + 2 n ) z
The coefficients c 2 n satisfy
28.2.19 q c 2 n + 2 - ( a - ( ν + 2 n ) 2 ) c 2 n + q c 2 n - 2 = 0 , n .
28.2.20 lim n ± | c 2 n | 1 / | n | = 0
leads to a Floquet solution. …
7: 15.17 Mathematical Applications
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. … The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …
8: 5.11 Asymptotic Expansions
where …
5.11.5 g k = 2 ( 1 2 ) k a 2 k ,
where a 0 = 1 2 2 and … The 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)). …
5.11.17 G k ( a , b ) = ( a - b k ) B k ( a - b + 1 ) ( a ) ,
9: 29.20 Methods of Computation
Subsequently, 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. … A 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 . …
§29.20(ii) Lamé Polynomials
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
10: 27.14 Unrestricted Partitions
Multiplying the power series for f ( x ) with that for 1 / f ( x ) and equating coefficients, we obtain the recursion formula …Logarithmic differentiation of the generating function 1 / f ( x ) leads to another recursion: … where
27.14.10 A k ( n ) = h = 1 ( h , k ) = 1 k exp ( π i s ( h , k ) - 2 π i n h k ) ,
The 24th power of η ( τ ) in (27.14.12) with e 2 π i τ = x is an infinite product that generates a power series in x with integer coefficients called Ramanujan’s tau function τ ( n ) : …