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21: 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). …

22: 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

…

23: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor

A_{jk}

a_{jk}

is … ►For real-valued

a_{jk}

, … ►where

\omega_{1},\omega_{2},\dots,\omega_{n}

are the

n

th roots of unity (1.11.21). … ►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

. … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. …

24: 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

…

25: 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. …

26: 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}

. …

27: 16.2 Definition and Analytic Properties

… ►Throughout this chapter it is assumed that none of the bottom parameters

b_{1}

b_{2}

\dots

b_{q}

is a nonpositive integer, unless stated otherwise. Then formally …Equivalently, the function is denoted by

{{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

, and sometimes, for brevity, by

{{}_{p}F_{q}}\left(z\right)

. … ►Suppose first one or more of the top parameters

a_{j}

is a nonpositive integer. … ►See §16.5 for the definition of

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

as a contour integral when

p>q+1

and none of the

a_{k}

is a nonpositive integer. … ►When

p\leq q+1

and

z

is fixed and not a branch point, any branch of

{{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)

is an entire function of each of the parameters

a_{1},\dots,a_{p},b_{1},\dots,b_{q}

28: 16.6 Transformations of Variable

… ►

16.6.1

{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{-a}{{}_{3}F_{2}}% \left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z% )^{2}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $z$ : 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:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

… ►

16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $z$ : 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:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of

{{}_{2}F_{2}}

functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

29: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$

… ►

§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

►

Transformations of ${{}_{3}\phi_{2}}$ -Series

… ►

§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

►

Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

…

30: 10.63 Recurrence Relations and Derivatives

… ►

§10.63(i) $\operatorname{ber}_{\nu}x$ , $\operatorname{bei}_{\nu}x$ , $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$

►Let

f_{\nu}(x)

g_{\nu}(x)

denote any one of the ordered pairs: ►

\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x;

►

\operatorname{bei}_{\nu}x,-\operatorname{ber}_{\nu}x;

►

\operatorname{ker}_{\nu}x,\operatorname{kei}_{\nu}x;

…

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21—30 of 783 matching pages

21: 16.11 Asymptotic Expansions

22: 16.19 Identities

23: 1.3 Determinants, Linear Operators, and Spectral Expansions

24: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions

§17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions

25: 16.8 Differential Equations

26: 3.6 Linear Difference Equations

27: 16.2 Definition and Analytic Properties

28: 16.6 Transformations of Variable

29: 17.9 Further Transformations of ϕ r r + 1 Functions

§17.9(i) ϕ 1 2 → ϕ 2 2 , ϕ 1 3 , or ϕ 2 3

§17.9(ii) ϕ 2 3 → ϕ 2 3

Transformations of ϕ 2 3 -Series

§17.9(iii) Further ϕ s r Functions

Sears’ Balanced ϕ 3 4 Transformations