generalized functions

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31—40 of 376 matching pages

31: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of

\Gamma\left(a_{k}+s\right)

k=1,\dots,p

, from those of

\Gamma\left(-s\right)

. … ►Lastly, when

p>q+1

the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. … ►Laplace transforms and inverse Laplace transforms of generalized hypergeometric functions are given in Prudnikov et al. (1992a, §3.38) and Prudnikov et al. (1992b, §3.36). …

32: 16.18 Special Cases

§16.18 Special Cases

►The

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions introduced in Chapters 13 and 15, as well as the more general

{{}_{p}F_{q}}

functions introduced in the present chapter, are all special cases of the Meijer

G

-function. … ►

16.18.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).

ⓘ

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, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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/16.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.18 and Ch.16

… ►

33: 10.39 Relations to Other Functions

… ►

Generalized Hypergeometric Functions and Hypergeometric Function

►

10.39.9

I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{}% _{0}F_{1}}\left(-;\nu+1;\tfrac{1}{4}z^{2}\right),

ⓘ

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, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/10.39.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §10.39, §10.39 and Ch.10

…

34: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

… ►

16.10.1

{{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},\dots,c_{r}\atop b_{1},\dots,b% _{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty}\frac{{\left(\mathbf{% a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}(-z)^{k}}{{% \left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}k!}{{}_{p+2}F_{q+1}}% \left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_% {q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop% \alpha,\beta,d_{1},\dots,d_{s}};\zeta\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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : 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/16.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

… ►

16.10.2

{{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};z\zeta\right% )=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}}{k!}{{}_{p+1% }F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left% (\frac{z}{z-1}\right)^{k}.

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : 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/16.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

… ►Expansions of the form

\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)

are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

35: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

►For the confluent hypergeometric function

{{}_{1}F_{1}}

and the generalized hypergeometric function

{{}_{2}F_{0}}

, the Laguerre polynomial

L^{(\alpha)}_{n}

and the Whittaker function

W_{\kappa,\mu}

see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. ►

18.34.1

y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right)=n!\left(-\tfrac{1}{2}x\right)^{n}L^{(1-a-2n)}% _{n}\left(2x^{-1}\right)=\left(\tfrac{1}{2}x\right)^{1-\frac{1}{2}a}{\mathrm{e% }}^{1/x}W_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).

ⓘ

Defines:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial
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, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Proved:: Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$ (proved)
Referenced by:: (18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1)
Permalink:: http://dlmf.nist.gov/18.34.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.
See also:: Annotations for §18.34(i), §18.34 and Ch.18

… ►

18.34.7_1

\phi_{n}(x;\lambda)={\mathrm{e}}^{-\lambda\,{\mathrm{e}}^{-x}}\left(2\lambda\,% {\mathrm{e}}^{-x}\right)^{\lambda-\frac{1}{2}}\ifrac{y_{n}\left(\lambda^{-1}{% \mathrm{e}}^{x};2-2\lambda\right)}{n!}=(-1)^{n}{\mathrm{e}}^{-\lambda\,{% \mathrm{e}}^{-x}}\left(2\lambda\,{\mathrm{e}}^{-x}\right)^{\lambda-n-\frac{1}{% 2}}L^{(2\lambda-2n-1)}_{n}\left(2\lambda\,{\mathrm{e}}^{-x}\right)=(2\lambda)^% {-\frac{1}{2}}\,{\mathrm{e}}^{x/2}\,W_{\lambda,n+\frac{1}{2}-\lambda}\left(2% \lambda{\mathrm{e}}^{-x}\right)/n!\,,

n=0,1,\dots,N=\left\lceil\lambda-\frac{3}{2}\right\rceil

\lambda>\frac{1}{2}

ⓘ

Symbols:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\left\lceil\NVar{x}\right\rceil$ : ceiling of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Proof sketch:: The second and third equality follow from (18.34.1).
Referenced by:: (18.34.7_2), (18.34.7_3), §18.34(iii), (18.39.17), (18.39.18), §18.39(i), Erratum (V1.2.0) Section 18.34
Permalink:: http://dlmf.nist.gov/18.34.E7_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.34(iii), §18.34 and Ch.18

…

36: 10.16 Relations to Other Functions

… ►

Generalized Hypergeometric Functions

►

10.16.9

J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{% }_{0}F_{1}}\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right).

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : 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, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §10.16, §10.16 and Ch.10

…

37: 4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). ►For an application of the Lambert

W

-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …

38: 8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions

\operatorname{si}\left(a,z\right)

\operatorname{ci}\left(a,z\right)

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

is a multivalued function of

z

with branch point at

z=0

. Furthermore,

\operatorname{si}\left(a,z\right)

and

\operatorname{ci}\left(a,z\right)

are entire functions of

a

, and

\operatorname{Si}\left(a,z\right)

and

\operatorname{Ci}\left(a,z\right)

are meromorphic functions of

a

with simple poles at

a=-1,-3,-5,\dots

and

a=0,-2,-4,\dots

, respectively. … ►From here on it is assumed that unless indicated otherwise the functions

\operatorname{si}\left(a,z\right)

\operatorname{ci}\left(a,z\right)

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

have their principal values. … ►

Spherical-Bessel-Function Expansions

… ►

§8.21(vii) Auxiliary Functions

…

39: 18.20 Hahn Class: Explicit Representations

… ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

►For the definition of hypergeometric and generalized hypergeometric functions see §16.2. … ►

18.20.5

Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),

n=0,1,\dots,N

ⓘ

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, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.20(ii), §18.21(ii), §18.23, §18.26(ii), §18.27(ii), §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

… ►

18.20.8

C_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).

ⓘ

Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §13.6(v), §18.19, §18.20(ii), §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.20.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

►

18.20.9

p_{n}\left(x;a,b,\overline{a},\overline{b}\right)=\frac{{\mathrm{i}}^{n}{\left% (a+\overline{a}\right)_{n}}{\left(a+\overline{b}\right)_{n}}}{n!}\*{{}_{3}F_{2% }}\left({-n,n+2\Re\left(a+b\right)-1,a+\mathrm{i}x\atop a+\overline{a},a+% \overline{b}};1\right).

…

40: 34.13 Methods of Computation

… ►Methods of computation for

\mathit{3j}

and

\mathit{6j}

symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …

generalized functions

31—40 of 376 matching pages

31: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

32: 16.18 Special Cases

§16.18 Special Cases

33: 10.39 Relations to Other Functions

Generalized Hypergeometric Functions and Hypergeometric Function

34: 16.10 Expansions in Series of F q p Functions

§16.10 Expansions in Series of F q p Functions

35: 18.34 Bessel Polynomials

§18.34(i) Definitions and Recurrence Relation

36: 10.16 Relations to Other Functions

Generalized Hypergeometric Functions

37: 4.44 Other Applications

38: 8.21 Generalized Sine and Cosine Integrals

Spherical-Bessel-Function Expansions

§8.21(vii) Auxiliary Functions

39: 18.20 Hahn Class: Explicit Representations

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

40: 34.13 Methods of Computation

34: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions