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11: 14.26 Uniform Asymptotic Expansions

… ►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

12: 26.2 Basic Definitions

… ►Thus $231$ is the permutation $\sigma (1)=2$, $\sigma (2)=3$, $\sigma (3)=1$. … ►As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …See Table 26.2.1 for $n=0(1)50$. For the actual partitions ($\pi$) for $n=1(1)5$ see Table 26.4.1. … ►The example $\{1,1,1,2,4,4\}$ has six parts, three of which equal 1. …

13: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant … ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. ►Let the columns of matrix $𝐒$ be these eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$, then ${𝐒}^{-1}={𝐒}^{\mathrm{H}}$, and the similarity transformation (1.2.73) is now of the form ${𝐒}^{\mathrm{H}}𝐀𝐒={\lambda }_i{\delta }_{i,j}$. …

14: 34.11 Higher-Order $3nj$ Symbols

… ►For information on $12j,15j$,…, symbols, see Varshalovich et al. (1988, §10.12) and Yutsis et al. (1962, pp. 62–65 and 122–153).

15: 26.10 Integer Partitions: Other Restrictions

… ►Throughout this subsection it is assumed that . … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). … ►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …

16: 24.2 Definitions and Generating Functions

… ►

$B_{2n+1}=0$ ,

►

${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

… ►

$E_{2n+1}=0$ ,

►

${(-1)}^n{E}_{2n}>0$ .

… ►

${\tilde{E}}_n\left(x\right)=E_n\left(x\right)$ , ,

…

17: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. …

18: 34.1 Special Notation

… ►

►

►

►An often used alternative to the $\mathit{3}j$ symbol is the Clebsch–Gordan coefficient …see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). …

19: 26.9 Integer Partitions: Restricted Number and Part Size

… ►The conjugate to the example in Figure 26.9.1 is $6+5+4+2+1+1+1$. … ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. … ►

$p_3\left(n\right)=1+\lfloor {\displaystyle \frac{n^2+6n}{12}}\rfloor .$

… ►It is also assumed everywhere that . … ►Also, when …

20: 26.6 Other Lattice Path Numbers

… ► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ► $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ► $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

26.6.10 $$D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$$ $m,n\ge 1$,

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iii), §26.6 and Ch.26

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