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1: 1.13 Differential Equations

… ►

§1.13(vii) Closed-Form Solutions

… ►

§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes

\frac{\mathrm{d}}{\mathrm{d}x}

. Assuming that

u(x)

satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

…

2: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►

$m$	$n$
$m$	…
3	1	4	10	20	35	56	84	120	165
…

► … ►

26.3.4

\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}x^{m}=\frac{1}{(1-x)^{n+1}},

|x|<1

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.1 (in slightly different form)
Permalink:: http://dlmf.nist.gov/26.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(ii), §26.3 and Ch.26

… ►

26.3.11

\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{2^{n}(2n-1)(2n-3)\cdots 3\cdot 1}{n!}.

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer
A&S Ref:: 24.1.1
Permalink:: http://dlmf.nist.gov/26.3.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(iv), §26.3 and Ch.26

… ►

§26.3(v) Limiting Form

►

26.3.12

\genfrac{(}{)}{0.0pt}{}{2n}{n}\sim\frac{4^{n}}{\sqrt{\pi n}},

n\to\infty

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Symbols:: $\sim$ : asymptotic equality, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter and $n$ : nonnegative integer
Referenced by:: §26.5(iv)
Permalink:: http://dlmf.nist.gov/26.3.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(v), §26.3 and Ch.26

3: 26.5 Lattice Paths: Catalan Numbers

… ►

26.5.1

C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}% \genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}% {0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n% -1}{n+1}.

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Defines:: $C\left(\NVar{n}\right)$ : Catalan number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $n$ : nonnegative integer
Referenced by:: §26.5(iv)
Permalink:: http://dlmf.nist.gov/26.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(i), §26.5 and Ch.26

… ►

26.5.4

C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),

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Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.5

C\left(n+1\right)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\genfrac{(}{)}{0.0% pt}{}{n}{2k}2^{n-2k}C\left(k\right).

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

►

§26.5(iv) Limiting Forms

… ►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

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Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

4: 20 Theta Functions

Chapter 20 Theta Functions

…

5: 33.18 Limiting Forms for Large $\ell$

§33.18 Limiting Forms for Large $\ell$

… ►

f\left(\epsilon,\ell;r\right)\sim\frac{(2r)^{\ell+1}}{(2\ell+1)!},

►

h\left(\epsilon,\ell;r\right)\sim\frac{(2\ell)!}{\pi(2r)^{\ell}}.

6: 26.9 Integer Partitions: Restricted Number and Part Size

… ►The conjugate to the example in Figure 26.9.1 is

6+5+4+2+1+1+1

. … ►

Figure 26.9.2: The partition

5+5+3+2

represented as a lattice path.

►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer

k

. … ►

p_{2}\left(n\right)=1+\left\lfloor n/2\right\rfloor,

… ►

§26.9(iv) Limiting Form

…

7: 26.10 Integer Partitions: Other Restrictions

… ►The set

\{2,3,4,\ldots\}

is denoted by

T

. If more than one restriction applies, then the restrictions are separated by commas, for example,

p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

m

for which

n-\tfrac{1}{2}km^{2}-m+\tfrac{1}{2}km\geq 0

. … ►

§26.10(v) Limiting Form

…

8: 26.12 Plane Partitions

… ►The number of self-complementary plane partitions in

B(2r,2s,2t)

is …in

B(2r+1,2s,2t)

it is …in

B(2r+1,2s+1,2t)

it is … ►in

B(2r+1,2r+1,2t)

it is … ►

§26.12(iv) Limiting Form

…

9: Bibliography D

… ►

C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction

\zeta(s)

de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

►

C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

►

T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

ⓘ

Notes:: Errata: In eq. (2.8) replace $(x^{2}/4)^{2}$ by $(x^{2}/4)^{s}$ . In eq. (4.2) $\phi(\zeta)$ should be replaced by $\psi(\zeta)$ . In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$ . In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.
External Links:: ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.24, §10.45, §2.8(iv).
Encodings:: BibTeX

…

10: 6.16 Mathematical Applications

… ►These limits are not approached uniformly, however. The first maximum of

\frac{1}{2}\operatorname{Si}\left(x\right)

for positive

x

occurs at

x=\pi

and equals

(1.1789\dots)\times\frac{1}{4}\pi

; compare Figure 6.3.2. Hence if

x=\pi/(2n)

and

n\to\infty

, then the limiting value of

S_{n}(x)

overshoots

\frac{1}{4}\pi

by approximately 18%. Similarly if

x=\pi/n

, then the limiting value of

S_{n}(x)

undershoots

\frac{1}{4}\pi

by approximately 10%, and so on. … ►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then …

limit%20forms%20as%20%E2%84%91%CF%84%E2%86%920%2B

1—10 of 837 matching pages

1: 1.13 Differential Equations

§1.13(vii) Closed-Form Solutions

§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

Transformation to Liouville normal Form

2: 26.3 Lattice Paths: Binomial Coefficients

§26.3(v) Limiting Form

3: 26.5 Lattice Paths: Catalan Numbers

§26.5(iv) Limiting Forms

4: 20 Theta Functions

Chapter 20 Theta Functions

5: 33.18 Limiting Forms for Large ℓ

§33.18 Limiting Forms for Large ℓ

6: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9(iv) Limiting Form

7: 26.10 Integer Partitions: Other Restrictions

§26.10(v) Limiting Form

8: 26.12 Plane Partitions

§26.12(iv) Limiting Form

9: Bibliography D

10: 6.16 Mathematical Applications

5: 33.18 Limiting Forms for Large $\ell$

§33.18 Limiting Forms for Large $\ell$