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11: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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12: 22.9 Cyclic Identities

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22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

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Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.11 $${\left(d_{1,2}^{(2)}\right)}^2{d}_{2,2}^{(2)}\pm {\left(d_{2,2}^{(2)}\right)}^2{d}_{1,2}^{(2)}=k^{\prime }\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$$

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Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.17 $$d_{1,4}^{(2)}{d}_{2,4}^{(2)}{d}_{3,4}^{(2)}\pm d_{2,4}^{(2)}{d}_{3,4}^{(2)}{d}_{4,4}^{(2)}+d_{3,4}^{(2)}{d}_{4,4}^{(2)}{d}_{1,4}^{(2)}\pm d_{4,4}^{(2)}{d}_{1,4}^{(2)}{d}_{2,4}^{(2)}=k^{\prime }\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_{4,4}^{(2)}\right),$$

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Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.18 $${\left(d_{1,4}^{(2)}\right)}^2{d}_{3,4}^{(2)}\pm {\left(d_{2,4}^{(2)}\right)}^2{d}_{4,4}^{(2)}+{\left(d_{3,4}^{(2)}\right)}^2{d}_{1,4}^{(2)}\pm {\left(d_{4,4}^{(2)}\right)}^2{d}_{2,4}^{(2)}=k^{\prime }\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,4}^{(2)}\pm d_{4,4}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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13: 12.15 Generalized Parabolic Cylinder Functions

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12.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\nu +{\lambda }^{-1}-{\lambda }^{-2}{z}^{\lambda }\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex parameter

Referenced by:

§12.15

Permalink:

http://dlmf.nist.gov/12.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.15 and Ch.12

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14: 17.15 Generalizations

… ►For higher-dimensional basic hypergometric functions, see Milne (1985a, b, c, d, 1988, 1994, 1997) and Gustafson (1987).

15: 7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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${\displaystyle \frac{d\mathrm{f}\left(z\right)}{dz}}=-\pi z\mathrm{g}\left(z\right),$

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${\displaystyle \frac{d\mathrm{g}\left(z\right)}{dz}}=\pi z\mathrm{f}\left(z\right)-1.$

16: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
$d|n$	$d$ divides $n$.
…
$\left(d_1,\mathrm{\dots },d_n\right)$	greatest common divisor of $d_1,\mathrm{\dots },d_n$.
${\sum }_{d|n}$, ${\prod }_{d|n}$	sum, product taken over divisors of $n$.
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17: 32.2 Differential Equations

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32.2.7 $$\frac{d^{2}w}{{dz}^2}=F(z,w,\frac{dw}{dz}),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: real

Permalink:

http://dlmf.nist.gov/32.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(i), §32.2 and Ch.32

►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. … ►then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{III}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$. … ►then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{V}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$.

18: 17.13 Integrals

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17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

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Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

►

17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

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19: 1.13 Differential Equations

… ►where $z\in D$, a simply-connected domain, and $f(z)$, $g(z)$ are analytic in $D$, has an infinite number of analytic solutions in $D$. A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►with $f(z)$, $g(z)$, and $r(z)$ analytic in $D$ has infinitely many analytic solutions in $D$. … ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. … ►where $\ddot{w}$ now denotes $\frac{d^{2}w}{{dt}^2}$, via the transformation …

20: 30.16 Methods of Computation

… ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements …and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. … ►

30.16.2 $${\alpha }_{j,d+1}\le {\alpha }_{j,d},$$

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Symbols:

$d$: dimension and ${\alpha }_{j,k}$: real eigenvalues

Permalink:

http://dlmf.nist.gov/30.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(i), §30.16 and Ch.30

… ►Let $𝐀$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to …