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21: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …

22: 34.8 Approximations for Large Parameters

… ►

34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

… ►

34.8.2

\cos\theta=\frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_% {1}+1)j_{2}(j_{2}+1)}},

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

…

23: 17.4 Basic Hypergeometric Functions

… ►

§17.4(i) ${{}_{r}\phi_{s}}$ Functions

… ►Here and elsewhere it is assumed that the

b_{j}

do not take any of the values

q^{-n}

. … ►

§17.4(ii) ${{}_{r}\psi_{s}}$ Functions

… ►Here and elsewhere the

b_{j}

must not take any of the values

q^{-n}

, and the

a_{j}

must not take any of the values

q^{n+1}

. … ►For the function

{{}_{r}H_{r}}

see §16.4(v). …

24: 16.11 Asymptotic Expansions

… ►For subsequent use we define two formal infinite series,

E_{p,q}(z)

and

H_{p,q}(z)

, as follows: …and

b_{q+1}=1

. Explicit representations for the coefficients

c_{k}

are given in Volkmer (2023). … ►In this subsection we assume that none of

a_{1},a_{2},\dots,a_{p}

is a nonpositive integer. … ►Explicit representations for the coefficients

c_{k}

are given in Volkmer and Wood (2014). …

26: 16.19 Identities

… ►

16.19.1

{G^{m,n}_{p,q}}\left(\frac{1}{z};{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{n,m}_{q,p}}\left(z;{1-b_{1},\dots,1-b_{q}\atop 1-a_{1},\dots,1-a_{% p}}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : 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:: §16.21
Permalink:: http://dlmf.nist.gov/16.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.2

z^{\mu}{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right% )={G^{m,n}_{p,q}}\left(z;{a_{1}+\mu,\dots,a_{p}+\mu\atop b_{1}+\mu,\dots,b_{q}% +\mu}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : 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
Permalink:: http://dlmf.nist.gov/16.19.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.3

{G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q},a_{0}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : 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
Permalink:: http://dlmf.nist.gov/16.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

… ►

16.19.5

\vartheta{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{% q}}\right)+(a_{1}-1){G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q}}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : 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 $\vartheta$ : differential operator
Permalink:: http://dlmf.nist.gov/16.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.6

\int_{0}^{1}t^{-a_{0}}(1-t)^{a_{0}-b_{q+1}-1}{G^{m,n}_{p,q}}\left(zt;{a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}}\right)\,\mathrm{d}t=\Gamma\left(a_{0}-b_{q% +1}\right){G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{% q+1}}\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p$ : nonnegative integer, $q$ : 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:: §16.19
Permalink:: http://dlmf.nist.gov/16.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

…

27: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

§17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

… ►

17.5.1

{{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
Permalink:: http://dlmf.nist.gov/17.5.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.11):: The constraint $|z|<1$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.2

{{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},

|z|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3

{{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.4

{{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},

|z|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii)
Permalink:: http://dlmf.nist.gov/17.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

…

28: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

… ►

•

Achenbach (1986) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0(.1)8$ , 19D or 19–21S.

… ►

•

Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_{n}\left(z\right)$ , for $n=2(1)10$ , 29S.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

…

29: 16.8 Differential Equations

… ►is a value

z_{0}

z

at which all the coefficients

f_{j}(z)

j=0,1,\dots,n-1

, are analytic. If

z_{0}

is not an ordinary point but

(z-z_{0})^{n-j}f_{j}(z)

j=0,1,\dots,n-1

, are analytic at

z=z_{0}

, then

z_{0}

is a regular singularity. … ►where

\alpha_{j}

and

\beta_{j}

are constants. … ►where

*

indicates that the entry

1+b_{j}-b_{j}

is omitted. … ►where

*

indicates that the entry

1-a_{j}+a_{j}

is omitted. …

30: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: … ►The coefficients of the power series of

a_{2n}\left(q\right)

b_{2n}\left(q\right)

and also

a_{2n+1}\left(q\right)

b_{2n+1}\left(q\right)

are the same until the terms in

q^{2n-2}

and

q^{2n}

, respectively. … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. Here

j=1

for

a_{2n}\left(q\right)

j=2

for

b_{2n+2}\left(q\right)

, and

j=3

for

a_{2n+1}\left(q\right)

and

b_{2n+1}\left(q\right)

. … ►

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

…

.世界杯瑞士预选赛战绩__『wn4.com_』_fifa18更新世界杯_w6n2c9o_2022年11月29日9时24分48秒_w6ecaw2qa.com

21—30 of 784 matching pages

21: 3.6 Linear Difference Equations

22: 34.8 Approximations for Large Parameters

23: 17.4 Basic Hypergeometric Functions

§17.4(i) ϕ s r Functions

§17.4(ii) ψ s r Functions

24: 16.11 Asymptotic Expansions

25: 5.10 Continued Fractions

26: 16.19 Identities

27: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions

§17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions