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11: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Krawtchouk

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Hahn $\to$ Meixner

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Meixner $\to$ Charlier

… ►A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1. …

12: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

§26.5(iv) Limiting Forms

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

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See also:

Annotations for §26.5(iv), §26.5 and Ch.26

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26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

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Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E7

Encodings:

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See also:

Annotations for §26.5(iv), §26.5 and Ch.26

13: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence $\{s_n\}$ with limit $\sigma$ into a sequence $\{t_n\}$ is called limit-preserving if $\{t_n\}$ converges to the same limit $\sigma$. ►The transformation is accelerating if it is limit-preserving and if … ►It may even fail altogether by not being limit-preserving. … ►, sequences $s_n$ converging to $\sigma$ such that …

14: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

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Annotations for §22.12 and Ch.22

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22.12.4 $$2\mathrm{i}K\mathrm{dn}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-(n+\frac{1}{2})\tau }\right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E4

Encodings:

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See also:

Annotations for §22.12 and Ch.22

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22.12.7 $$2\mathrm{i}K{k}^{\prime }\mathrm{nd}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\underset{M\to \mathrm{\infty }}{lim}\left(\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E7

Encodings:

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See also:

Annotations for §22.12 and Ch.22

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22.12.10 $$-2K{k}^{\prime }\mathrm{sc}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E10

Encodings:

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See also:

Annotations for §22.12 and Ch.22

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

15: 10.30 Limiting Forms

§10.30 Limiting Forms

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§10.30(i) $z\to 0$

►When $\nu$ is fixed and $z\to 0$, … ►

§10.30(ii) $z\to \mathrm{\infty }$

►When $\nu$ is fixed and $z\to \mathrm{\infty }$, …

16: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

►As $\rho \to 0$ with $\eta$ fixed, … ►

§33.5(iii) Small $|\eta |$

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§33.5(iv) Large $\mathrm{\ell }$

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17: 22.5 Special Values

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§22.5(ii) Limiting Values of $k$

►If $k\to 0+$, then $K\to \pi /2$ and $K^{\prime }\to \mathrm{\infty }$; if $k\to 1-$, then $K\to \mathrm{\infty }$ and $K^{\prime }\to \pi /2$. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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… ►Expansions for $K,K^{\prime }$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. … ►

18: Notices

… ►Pursuant to Title 17 USC 105, the National Institute of Standards and Technology (NIST), United States Department of Commerce, is authorized to receive and hold copyrights transferred to it by assignment or otherwise. … ►Limited copying and internal distribution of the content of these pages is permitted for research and teaching. …Questions regarding this copyright policy should be directed to dlmf-feedback. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …Thus, we seek to provide DLMF users with links to sources of such software. …

19: 33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: ►

When $r\to \pm \mathrm{\infty }$ with $ϵ>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

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When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

… ►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

20: 1.4 Calculus of One Variable

… ►when the last limit exists. … ►Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist. … ►When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. … ►when this limit exists. … ►when this limit exists. …