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1: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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2: 18.25 Wilson Class: Definitions

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§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.

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$p_n(x)$	$k_n$
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3: 18.19 Hahn Class: Definitions

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Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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4: 18.15 Asymptotic Approximations

… ►The leading coefficients are given by … ►The leading coefficients are given by $A_0(\xi )=1$ and … ►The leading coefficients are given by $E_0(\zeta )=1$ 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

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28.2.18 $$w(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2n}{\mathrm{e}}^{\mathrm{i}(\nu +2n)z}$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution and $c_{2n}$: coefficients

A&S Ref:

20.3.8

Permalink:

http://dlmf.nist.gov/28.2.E18

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See also:

Annotations for §28.2(iv), §28.2 and Ch.28

… ►The coefficients $c_{2n}$ satisfy ►

28.2.19 $$q{c}_{2n+2}-\left(a-{(\nu +2n)}^2\right)c_{2n}+q{c}_{2n-2}=0,$$ $n\in \mathbb{Z}$.

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Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $q=h^2$: parameter, $n$: integer, $\nu$: complex parameter, $a$: parameter and $c_{2n}$: coefficients

A&S Ref:

20.3.12 20.3.13

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E19

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Annotations for §28.2(iv), §28.2 and Ch.28

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28.2.20 $$\underset{n\to \pm \mathrm{\infty }}{lim}{|c_{2n}|}^{1/|n|}=0$$

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Symbols:

$n$: integer and $c_{2n}$: coefficients

Permalink:

http://dlmf.nist.gov/28.2.E20

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Annotations for §28.2(iv), §28.2 and Ch.28

►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 $$g_k=\sqrt{2}{\left(\frac{1}{2}\right)}_k{a}_{2k},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $g_k$: coefficients and $a_k$: coefficient

Referenced by:

§5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E5

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Annotations for §5.11(i), §5.11 and Ch.5

►where $a_0=\frac{1}{2}\sqrt{2}$ 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 $$G_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{k}\right)B_k^{(a-b+1)}\left(a\right),$$

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Defines:

$G_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.11.E17

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Annotations for §5.11(iii), §5.11 and Ch.5

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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 $n\to \mathrm{\infty }$. … ►

§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 $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula …Logarithmic differentiation of the generating function $1/f\left(x\right)$ leads to another recursion: … ►where ►

27.14.10 $$A_k(n)=\sum _{\begin{array}{c}\hfill h=1\hfill \\ \hfill \left(h,k\right)=1\hfill \end{array}}^k\exp \left(\pi \mathrm{i}s(h,k)-2\pi \mathrm{i}n\frac{h}{k}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $k$: positive integer, $n$: positive integer, $A_k(n)$: coefficients and $s(h,k)$: Dedekind sum

A&S Ref:

24.2.1 I.C

Permalink:

http://dlmf.nist.gov/27.14.E10

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Annotations for §27.14(iii), §27.14 and Ch.27

… ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: …