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1: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

2: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►

$D_{-1}^{\pm }=0,$

►

$D_0^{\pm }=1,$

… ►

28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

►The upper signs correspond to ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when …

3: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. …

4: 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

…

5: 1.11 Zeros of Polynomials

… ►Set $z=w-\frac{1}{3}a$ to reduce $f(z)=z^3+a{z}^2+bz+c$ to $g(w)=w^3+pw+q$, with $p=(3b-a^2)/3$, $q=(2a^3-9ab+27c)/27$. … ►Addition of $-\frac{1}{3}a$ to each of these roots gives the roots of $f(z)=0$. … ►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 $a_0\ne 0$; $D_{2k}>0$, $k=1,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$; $\mathrm{sign}{D}_{2k+1}=\mathrm{sign}{a}_0$, $k=0,1,\mathrm{\dots },\lfloor \frac{1}{2}n-\frac{1}{2}\rfloor$.

6: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that ►

21.5.1

ⓘ

Defines:

Matrix $𝚪$ (locally)

Permalink:

http://dlmf.nist.gov/21.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.4 $$\theta \left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\xi (𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, Matrix $𝚪$ and $\xi (𝚪)$: eighth root of unity

Referenced by:

§21.5(i)

Permalink:

http://dlmf.nist.gov/21.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

►Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. … ►

21.5.9 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝐃𝜶-𝐂𝜷+\frac{1}{2}\mathrm{diag}[𝐂{𝐃}^{\mathrm{T}}]}{-𝐁𝜶+𝐀𝜷+\frac{1}{2}\mathrm{diag}[𝐀{𝐁}^{\mathrm{T}}]}\right]\left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\kappa (𝜶,𝜷,𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀),$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(ii), §21.5 and Ch.21

…

7: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►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

►

28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$V_m(\xi )$: 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:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Suggested 2017-02-02 by Daniel Karlsson

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

…

8: 27.2 Functions

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►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 $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
…

►

9: 19.27 Asymptotic Approximations and Expansions

… ►

§19.27(ii) $R_F(x,y,z)$

… ►

§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

►

19.27.8 $$R_D(x,y,z)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}{R}_G(x,y,0)\left(1+O\left(\frac{z}{g}\right)\right),$$ $z/g\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $g$

Referenced by:

§19.20(iv), §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.27.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

19.27.10 $$R_D(x,y,z)=R_D(0,y,z)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt{\frac{x}{h}}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $h$

Permalink:

http://dlmf.nist.gov/19.27.E10

Encodings:

pMML, png, TeX

See also:

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

…

10: 19.29 Reduction of General Elliptic Integrals

… ►and $\alpha ,\beta ,\gamma ,\delta$ is any permutation of the numbers $1,2,3,4$, then … ►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). … ►where …