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21: 16.8 Differential Equations

… ►

D=\frac{\mathrm{d}}{\mathrm{d}z},

… ►

16.8.4

z^{q}D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w=0,

p\leq q

ⓘ

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $𝐷$ : differential operator, $w$ : solution, $\alpha_{j}$ : constant and $\beta_{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.5

z^{q}(1-z)D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{% 0}w=0,

p=q+1

ⓘ

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $𝐷$ : differential operator, $w$ : solution, $\alpha_{j}$ : constant and $\beta_{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.8

{{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=% \sum_{j=1}^{q+1}\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\frac{\Gamma\left(a_{k}-a_{j}\right)}{\Gamma\left(% a_{k}\right)}}{\prod\limits_{k=1}^{q}\frac{\Gamma\left(b_{k}-a_{j}\right)}{% \Gamma\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),

|\operatorname{ph}\left(-z\right)|\leq\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.9

\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a% _{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0% }-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*% \left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=% 1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a% _{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}% \right)\left(z-z_{0}\right)^{-n}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

…

22: 1.15 Summability Methods

… ►Here

u(x,y)=A(r,\theta)

is the Abel (or Poisson) sum of

f(\theta)

, and

v(x,y)

has the series representation … ►

1.15.51

D^{\alpha}f(x)=\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}I^{n-\alpha}f(x),

ⓘ

Defines:: $D^{\alpha}$ : fractional derivative (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $I^{\alpha}$ : fractional integral
Referenced by:: §1.15(vii)
Permalink:: http://dlmf.nist.gov/1.15.E51
Encodings:: pMML, png, TeX
See also:: Annotations for §1.15(vii), §1.15 and Ch.1

… ►

1.15.52

D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k},

k=1,2,\dots,n

ⓘ

Symbols:: $k$ : integer, $n$ : nonnegative integer, $I^{\alpha}$ : fractional integral and $D^{\alpha}$ : fractional derivative
Referenced by:: §1.15(vii)
Permalink:: http://dlmf.nist.gov/1.15.E52
Encodings:: pMML, png, TeX
See also:: Annotations for §1.15(vii), §1.15 and Ch.1

… ►

1.15.53

D^{\alpha}D^{\beta}=D^{\alpha+\beta}.

ⓘ

Symbols:: $D^{\alpha}$ : fractional derivative
Permalink:: http://dlmf.nist.gov/1.15.E53
Encodings:: pMML, png, TeX
See also:: Annotations for §1.15(vii), §1.15 and Ch.1

►Note that

D^{1/2}D\not=D^{3/2}

. …

23: 1.9 Calculus of a Complex Variable

… ►Any point whose neighborhoods always contain members and nonmembers of

D

is a boundary point of

D

. … ►A function

f(z)

is analytic in a domain

D

if it is analytic at each point of

D

. … ►at all points of

D

. … ►Suppose

f(z)

is analytic in a domain

D

and

C_{1},C_{2}

are two arcs in

D

passing through

z_{0}

. … ►Suppose the series

\sum^{\infty}_{n=0}f_{n}(z)

, where

f_{n}(z)

is continuous, converges uniformly on every compact set of a domain

D

, that is, every closed and bounded set in

D

. …

24: 18.38 Mathematical Applications

… ►

[K_{1},K_{2}]_{q}=B\,K_{1}+C_{0}\,K_{0}+D_{0},

►

[K_{2},K_{0}]_{q}=B\,K_{0}+C_{1}\,K_{1}+D_{1},

… ►

D_{0}=-q^{-3}\left(1-q\right)^{2}\left(1+q\right)\left(e_{4}+qe_{2}+q^{2}% \right),

►

D_{1}=-q^{-3}\left(1-q\right)^{2}\left(1+q\right)\left(e_{1}e_{4}+qe_{3}\right),

… ►The abstract associative algebra with generators

K_{0},K_{1},K_{2}

and relations (18.38.4), (18.38.6) and with the constants

B,C_{0},C_{1},D_{0},D_{1}

in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). …

25: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.9

\det[a_{jk}]^{2}\leq\left(\sum^{n}_{k=1}a^{2}_{1k}\right)\left(\sum^{n}_{k=1}a% ^{2}_{2k}\right)\dots\left(\sum^{n}_{k=1}a^{2}_{nk}\right).

ⓘ

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►for every distinct pair of

j, k

, or when one of the factors

\sum^{n}_{k=1}a^{2}_{jk}

vanishes. … ►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 ►

1.3.18

D_{n}[a_{j,k}]=\begin{vmatrix}a_{-n,-n}&a_{-n,-n+1}&\dots&a_{-n,n}\\ a_{-n+1,-n}&a_{-n+1,-n+1}&\dots&a_{-n+1,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n,-n}&a_{n,-n+1}&\dots&a_{n,n}\end{vmatrix}.

ⓘ

Defines:: $D_{\NVar{n}}[\NVar{a}_{\NVar{j},\NVar{k}}]$ : $(2\NVar{n}+1)\times(2\NVar{n}+1)$ determinant (locally)
Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iii), §1.3 and Ch.1

►If

D_{n}[a_{j,k}]

tends to a limit

L

n\to\infty

, then we say that the infinite determinant

D_{\infty}[a_{j,k}]

converges and

D_{\infty}[a_{j,k}]=L

. …

26: 1.6 Vectors and Vector-Valued Functions

… ►In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over). … ►with

(u,v)\in D

, an open set in the plane. … ►If

\boldsymbol{{\Phi}}_{1}

and

\boldsymbol{{\Phi}}_{2}

are both orientation preserving or both orientation reversing parametrizations of

S

defined on open sets

D_{1}

and

D_{2}

respectively, then ►

1.6.56

\iint_{\boldsymbol{{\Phi}}_{1}(D_{1})}\mathbf{F}\cdot\,\mathrm{d}\mathbf{S}=% \iint_{\boldsymbol{{\Phi}}_{2}(D_{2})}\mathbf{F}\cdot\,\mathrm{d}\mathbf{S};

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\boldsymbol{{\Phi}}(x,y,z)$ : parameterization and $D$ : open set in the plane
Permalink:: http://dlmf.nist.gov/1.6.E56
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(v), §1.6 and Ch.1

…

27: 3.3 Interpolation

… ►If

f

is analytic in a simply-connected domain

D

(§1.13(i)), then for

z\in D

, ►

3.3.6

R_{n}(z)=\frac{\omega_{n+1}(z)}{2\pi\mathrm{i}}\int_{C}\frac{f(\zeta)}{(\zeta-% z)\omega_{n+1}(\zeta)}\,\mathrm{d}\zeta,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $f(z)$ : function, $\omega_{n+1}(z)$ : nodal polynomial, $R_{n}(x)$ : remainder and $C$ : simple closed contour in $D$
A&S Ref:: 25.2.5
Permalink:: http://dlmf.nist.gov/3.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §3.3(i), §3.3 and Ch.3

►where

C

is a simple closed contour in

D

described in the positive rotational sense and enclosing the points

z,z_{1},z_{2},\dots,z_{n}

. … ►If

f

is analytic in a simply-connected domain

D

, then for

z\in{D}

, …where

\omega_{n+1}(\zeta)

is given by (3.3.3), and

C

is a simple closed contour in

{D}

described in the positive rotational sense and enclosing

z_{0},z_{1},\dots,z_{n}

. …

28: 19.15 Advantages of Symmetry

… ►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s

F_{D}

(Carlson (1961b)). The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …

29: 3.10 Continued Fractions

… ►can be converted into a continued fraction

C

of type (3.10.1), and with the property that the

n

th convergent

C_{n}=A_{n}/B_{n}

C

is equal to the

n

th partial sum of the series in (3.10.3), that is, … ►The

n

th partial sum

t_{0}+t_{1}+\dots+t_{n-1}

equals the

n

th convergent of (3.10.13),

n=1,2,3,\dots

. … ►

D_{1}=1/b_{1},

►

\nabla C_{1}=a_{1}D_{1},

… ►

D_{n}=\frac{1}{D_{n-1}a_{n}+b_{n}},

…

30: 30.3 Eigenvalues

… ►

30.3.8

\lambda^{m}_{n}\left(\gamma^{2}\right)=\sum_{k=0}^{\infty}\ell_{2k}\gamma^{2k},

|\gamma^{2}|<r_{n}^{m}

ⓘ

Symbols:: $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $\ell_{j}$ : coefficients
Referenced by:: §30.16(i), item
Permalink:: http://dlmf.nist.gov/30.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §30.3(iv), §30.3 and Ch.30

… ►

30.3.11

\ell_{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,

ⓘ

Symbols:: $m$ : nonnegative integer, $\ell_{j}$ : 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

… ►

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)}.

…