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1: 26.14 Permutations: Order Notation

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26.14.1

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Symbols:

$j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer and $\sigma$: permutation

Permalink:

http://dlmf.nist.gov/26.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(i), §26.14 and Ch.26

►Equivalently, this is the sum over of the number of integers less than $\sigma (j)$ that lie in positions to the right of the $j$th position: $inv(35247816)=2+3+1+1+2+2+0=11.$ … ►

26.14.2

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Symbols:

$j$: nonnegative integer, $n$: nonnegative integer, $\sigma$: permutation and $maj(\sigma )$: major index

Permalink:

http://dlmf.nist.gov/26.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(i), §26.14 and Ch.26

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26.14.3 $$\sum _{\sigma \in {𝔖}_n}{q}^{inv(\sigma )}=\sum _{\sigma \in {𝔖}_n}{q}^{maj(\sigma )}=\prod _{j=1}^n\frac{1-q^j}{1-q}.$$

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Symbols:

$\in$: element of, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $j$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(ii), §26.14 and Ch.26

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26.14.6 $n\ge 1$,

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Symbols:

: Eulerian number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

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2: 33.3 Graphics

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Figure 33.3.4: $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$ with $\mathrm{\ell }=0$, $\eta =10$. The turning point is at ${\rho }_{\mathrm{tp}}(10,0)=20$. Magnify

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33.3.1 $$M_{\mathrm{\ell }}(\eta ,\rho )={(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))}^{1/2}=\left|H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\right|.$$

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Defines:

$M_{\mathrm{\ell }}(\eta ,\rho )$: envelope of Coulomb functions

Symbols:

$G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.3(i), §33.3 and Ch.33

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3: 26.10 Integer Partitions: Other Restrictions

§26.10 Integer Partitions: Other Restrictions

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§26.10(i) Definitions

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§26.10(ii) Generating Functions

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§26.10(iii) Recurrence Relations

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4: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►Rational solutions of ${\text{P}}_{\text{II}}$ exist for $\alpha =n(\in \mathbb{Z})$ and are generated using the seed solution $w(z;0)=0$ and the Bäcklund transformations (32.7.1) and (32.7.2). … ►with $n\in \mathbb{Z}$. … ►with $m,n\in \mathbb{Z}$. The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv). …

5: 36.4 Bifurcation Sets

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36.4.6 $$27x^2=-8y^3.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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$x=\frac{9}{20}{z}^2.$

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$x=-\frac{3}{20}{z}^2,$

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36.4.11 $$x+\mathrm{i}y=-z^2\exp \left(\frac{2}{3}\mathrm{i}\pi m\right),$$ $m=0,1,2$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $n$: integer and $x$: real parameter

Referenced by:

§36.7(iii)

Permalink:

http://dlmf.nist.gov/36.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.13 $$x=y=-\frac{1}{4}{z}^2.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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6: 28.8 Asymptotic Expansions for Large $q$

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28.8.1 $$\begin{array}{c}{a}_m\left(h^2\right)\\ b_{m+1}\left(h^2\right)\end{array}\}\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter

A&S Ref:

20.2.30 (in different form)

Referenced by:

§28.8(i), 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.E1

Encodings:

pMML, png, TeX

See also:

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

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7: 30.9 Asymptotic Approximations and Expansions

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30.9.1 $${\lambda }_n^m\left({\gamma }^2\right)\sim -{\gamma }^2+\gamma q+{\beta }_0+{\beta }_1{\gamma }^{-1}+{\beta }_2{\gamma }^{-2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and ${\beta }_n$: coeffients

A&S Ref:

21.7.6

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(i), §30.9 and Ch.30

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$$2^{20}{\beta }_5=-527q^7-61529q^5-\mathrm{10\hspace{0.33em}43961}{q}^3-\mathrm{22\hspace{0.33em}41599}q+32m^2(5739q^5+\mathrm{1\hspace{0.33em}27550}{q}^3+\mathrm{2\hspace{0.33em}98951}q)-2048m^4(355q^3+1505q)+65536m^{6}q.$$

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

30.9.4 $${\lambda }_n^m\left({\gamma }^2\right)\sim 2q|\gamma |+c_0+c_1{|\gamma |}^{-1}+c_2{|\gamma |}^{-2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and $c_n$: coeffients

A&S Ref:

21.8.2

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(ii), §30.9 and Ch.30

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

8: 25.11 Hurwitz Zeta Function

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25.11.3 $$\zeta (s,a)=\zeta (s,a+1)+a^{-s},$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

recurrence

Proof sketch:

Derivable from (25.11.1) by comparing its $n=0$ term with its sum over $n\ge 1$.

Permalink:

http://dlmf.nist.gov/25.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(i), §25.11 and Ch.25

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►►

Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

… ►where $h,k$ are integers with $1\le h\le k$ and $n=1,2,3,\mathrm{\dots }$. … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

9: 11.6 Asymptotic Expansions

… ►For fixed $\lambda (>1)$ ►

11.6.6 $${𝐊}_{\nu }\left(\lambda \nu \right)\sim \frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

A&S Ref:

12.1.34

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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10: 12.11 Zeros

… ►If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$. … ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

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12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

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Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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