(0.001 seconds)

## 1—10 of 21 matching pages

βΊ
##### 2: 18.25 Wilson Class: Definitions
βΊ
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
βΊ
##### 4: 18.15 Asymptotic Approximations
βΊThe leading coefficients are given by … βΊThe leading coefficients are given by $A_{0}(\xi)=1$ and … βΊThe leading coefficients are given by $E_{0}(\zeta)=1$ and …
##### 5: 2.6 Distributional Methods
βΊThis leads to integrals of the form … βΊThe replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … βΊIn the sense of distributions, they can be written … βΊAlso, …Multiplication of these expansions leads to …
##### 6: 28.2 Definitions and Basic Properties
βΊ
28.2.18 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$
βΊThe coefficients $c_{2n}$ satisfy βΊ βΊ
28.2.20 $\lim_{n\to\pm\infty}|c_{2n}|^{1/|n|}=0$
βΊleads to a Floquet solution. …
##### 7: 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. …
##### 8: 5.11 Asymptotic Expansions
βΊwhere … βΊ βΊwhere $a_{0}=\tfrac{1}{2}\sqrt{2}$ and … 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)). … βΊ
##### 9: 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 $n\to\infty$. … βΊ
###### §29.20(ii) Lamé Polynomials
βΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
##### 10: 27.14 Unrestricted Partitions
βΊMultiplying the power series for $\mathit{f}\left(x\right)$ with that for $1/\mathit{f}\left(x\right)$ and equating coefficients, we obtain the recursion formula …Logarithmic differentiation of the generating function $1/\mathit{f}\left(x\right)$ leads to another recursion: … βΊwhere βΊ
27.14.10 $A_{k}(n)=\sum_{\begin{subarray}{c}h=1\\ \left(h,k\right)=1\end{subarray}}^{k}\exp\left(\pi\mathrm{i}s(h,k)-2\pi\mathrm% {i}n\frac{h}{k}\right),$
βΊThe 24th power of $\eta\left(\tau\right)$ in (27.14.12) with $e^{2\pi\mathrm{i}\tau}=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau\left(n\right)$: …