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11: 26.5 Lattice Paths: Catalan Numbers

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§26.5(iv) Limiting Forms

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

12: 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 …Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums. … ►It may even fail altogether by not being limit-preserving. …

13: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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18.21.3 $$\underset{t\to \mathrm{\infty }}{lim}{Q}_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E3

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Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.4 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(x;\beta -1,N(c^{-1}-1),N)=M_n(x;\beta ,c).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E4

Encodings:

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Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.6 $$\underset{N\to \mathrm{\infty }}{lim}{K}_n(x;N^{-1}a,N)=C_n(x;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E6

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Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►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. …

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

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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

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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

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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

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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:

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See also:

Annotations for §22.12 and Ch.22

15: 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 $\rho$

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§33.5(iii) Small $|\eta |$

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§33.5(iv) Large $\mathrm{\ell }$

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16: 10.30 Limiting Forms

§10.30 Limiting Forms

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§10.30(i) $z\to 0$

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17: 1.4 Calculus of One Variable

… ►When this limit exists $f$ is differentiable at $x$. … ►when the last limit exists. … ►If the limit exists then $f$ is called Riemann integrable. … ►when this limit exists. … ►when this limit exists. …

18: 1.9 Calculus of a Complex Variable

… ►Also, the union of $S$ and its limit points is the closure of $S$. … ►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►or its limiting form, and is invariant under bilinear transformations. … ►

§1.9(vii) Inversion of Limits

… ►Then both repeated limits equal $z$. …

19: 20.5 Infinite Products and Related Results

… ►The left-hand sides of (20.5.10) and (20.5.11) are replaced by their limiting values when $\cot z$ or $\tan z$ are undefined. … ►

20.5.15 $${\theta }_2\left(z\right|\tau )={\theta }_2\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=1-M}^M\left(1+\frac{z}{(m-\frac{1}{2}+n\tau )\pi }\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E15

Encodings:

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See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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20.5.16 $${\theta }_3\left(z\right|\tau )={\theta }_3\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=1-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=1-M}^M\left(1+\frac{z}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )\pi }\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E16

Encodings:

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See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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20.5.17 $${\theta }_4\left(z\right|\tau )={\theta }_4\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=1-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=-M}^M\left(1+\frac{z}{(m+(n-\frac{1}{2})\tau )\pi }\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.5.E17

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Annotations for §20.5(iii), §20.5 and Ch.20

►These double products are not absolutely convergent; hence the order of the limits is important. …

20: 22.5 Special Values

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§22.5(ii) Limiting Values of $k$

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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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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
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