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21: 1.11 Zeros of Polynomials

… ►where . … ► $f(z)=z^3-6z^2+6z-2$, $g(w)=w^3-6w-6$, $A=3\sqrt[3]{4}$, $B=3\sqrt[3]{2}$. … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. … ►Let … ►Then $f(z)$, with $a_n\ne 0$, is stable iff $a_0\ne 0$; $D_{2k}>0$, $k=1,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$; $\mathrm{sign}{D}_{2k+1}=\mathrm{sign}{a}_0$, $k=0,1,\mathrm{\dots },\lfloor \frac{1}{2}n-\frac{1}{2}\rfloor$.

22: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that …Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. For general $𝚪$, it is difficult to decide which root needs to be used. … ►($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) …

23: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ 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 $(1,1)$. … ►

Table 26.6.1: Delannoy numbers $D(m,n)$.

► ►►

$m$	$n$
	…

►

… ►

26.6.4 $$r(n)=D(n,n)-D(n+1,n-1),$$ $n\ge 1$.

ⓘ

Defines:

$r(n)$: Schröder number (locally)

Symbols:

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

Referenced by:

§26.6(i)

Permalink:

http://dlmf.nist.gov/26.6.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

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

…

24: 24.2 Definitions and Generating Functions

… ►

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

… ►

24.2.4 $$B_n=B_n\left(0\right),$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials and $n$: integer

A&S Ref:

23.1.2

Referenced by:

(25.11.7)

Permalink:

http://dlmf.nist.gov/24.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

… ►

${\tilde{B}}_n\left(x\right)=B_n\left(x\right)$ ,

… ►

${\tilde{B}}_n\left(x+1\right)={\tilde{B}}_n\left(x\right),$

… ►

Table 24.2.3: Bernoulli numbers $B_n=N/D$.

► ►►

$n$	$N$	$D$	$B_n$
…

►

…

25: 27.2 Functions

… ►($\nu \left(1\right)$ is defined to be 0.) …It can be expressed as a sum over all primes $p\le x$: … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

26: 28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►

28.8.4 $$U_m(\xi )\sim D_m\left(\xi \right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi \right)-4!\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)\right)+\frac{1}{2^{13}{h}^2}\left(D_{m+8}\left(\xi \right)-2^5(m+2)D_{m+4}\left(\xi \right)+4!{\mathrm{\hspace{0.17em}2}}^5(m-1)\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)+8!\left(\genfrac{}{}{0.0pt}{}{m}{8}\right)D_{m-8}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$U_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.17 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

… ►

28.8.6 $${\widehat{C}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1+\frac{2m+1}{8h}+\frac{m^4+2m^3+263m^2+262m+108}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2},$$

ⓘ

Defines:

${\widehat{C}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.19 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

►

28.8.7 $${\widehat{S}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1-\frac{2m+1}{8h}+\frac{m^4+2m^3-121m^2-122m-84}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2}.$$

ⓘ

Defines:

${\widehat{S}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.20 (in slightly different form)

Referenced by:

§28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

… ►It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. …

27: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

§19.27(iv) $R_D(x,y,z)$

… ►

19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

… ►These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice. …

28: 19.29 Reduction of General Elliptic Integrals

… ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the $8+8+12=28$ formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking $x^2$ as the variable of integration in 3. …where the arguments of the $R_D$ function are, in order, $(a-b)(u-c)$, $(b-c)(a-u)$, $(a-b)(b-c)$. … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting $x^2=t$ in some cases). … ►If each has real zeros, then (19.29.4) may be simpler than …where …

29: 30.3 Eigenvalues

… ►

30.3.2 $${\lambda }_n^m\left({\gamma }^2\right)=n(n+1)-\frac{1}{2}{\gamma }^2+O\left(n^{-2}\right),$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

… ►

30.3.11 $${\mathrm{\ell }}_8=2{(4m^2-1)}^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,$$

ⓘ

Symbols:

$m$: nonnegative integer, ${\mathrm{\ell }}_j$: coefficients, $A$ , $B$ , $C$ and $D$

Permalink:

http://dlmf.nist.gov/30.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(iv), §30.3 and Ch.30

… ►

$$B=\frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-7){(2n-5)}^2{(2n-3)}^3{(2n-1)}^4(2n+1)}-\frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n+m+3)(n+m+4)}{(2n+1){(2n+3)}^4{(2n+5)}^3{(2n+7)}^2(2n+9)},$$

… ►

$$D=\frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-3){(2n-1)}^4{(2n+1)}^2{(2n+3)}^4(2n+5)}.$$

►Further coefficients can be found with the Maple program SWF9; see §30.18(i).

30: 12.3 Graphics

… ►

►►

Figure 12.3.1: $U(a,x)$, $a$ = 0. …5, 5, 8. Magnify

►

►►

Figure 12.3.2: $V(a,x)$, $a$ = 0. …5, 5, 8. Magnify

… ►

►►

Figure 12.3.5: $U(-8,x)$, $\overline{U}(-8,x)$, $F(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

►

►►

Figure 12.3.6: $U^{\prime }(-8,x)$, ${\overline{U}}^{\prime }(-8,x)$, $G(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

…