F. H. Jackson transformations

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21: 33.3 Graphics

… ►

§33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$

►

See accompanying text — Figure 33.3.1: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=-2$ . Magnify

►

… ►

33.3.1

M_{\ell}\left(\eta,\rho\right)=({F_{\ell}}^{2}\left(\eta,\rho\right)+{G_{\ell}% }^{2}\left(\eta,\rho\right))^{1/2}=\left|{H^{\pm}_{\ell}}\left(\eta,\rho\right% )\right|.

ⓘ

Defines:: $M_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : envelope of Coulomb functions
Symbols:: $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.3(i), §33.3 and Ch.33

… ►

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_{0}\left(\eta,\rho\right)$ and $G_{0}\left(\eta,\rho\right)$

…

22: 15.6 Integral Representations

… ►The function

\mathbf{F}\left(a,b;c;z\right)

(not

F\left(a,b;c;z\right)

) has the following integral representations: ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Proof sketch:: Follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.9

\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^% {a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}% \left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\lambda\in\mathbb{C}

\Re c>\Re d>0

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Notes:: With $\lambda=a+b$ , this equation specializes to (15.6.8).
Referenced by:: (15.6.8), §15.6, §15.6, Erratum (V1.0.18) for Equation (15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E9
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
Addition (effective with 1.0.18):: The constraint $\lambda\in\mathbb{C}$ was added.
See also:: Annotations for §15.6 and Ch.15

…

23: 33.23 Methods of Computation

… ►§33.8 supplies continued fractions for

F_{\ell}'/F_{\ell}

and

{H^{\pm}_{\ell}}'/{H^{\pm}_{\ell}}

. Combined with the Wronskians (33.2.12), the values of

F_{\ell}

G_{\ell}

, and their derivatives can be extracted. … ►Bardin et al. (1972) describes ten different methods for the calculation of

F_{\ell}

and

G_{\ell}

, valid in different regions of the (

\eta,\rho

)-plane. … ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for

F_{0}

and

G_{0}

in the region inside the turning point:

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

24: 16.5 Integral Representations and Integrals

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as

z\to 0

in the sector

|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2

, where

\delta

is an arbitrary small positive constant. … ►

16.5.2

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

\Re b_{0}>\Re a_{0}>0

ⓘ

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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $p$ : 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
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

ⓘ

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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : 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
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

… ►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). …

25: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.1

F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

►

33.6.2

F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

… ►

33.6.4

A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}% }\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

►

33.6.5

{H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),

ⓘ

Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a$ : variable
A&S Ref:: 14.1.14 (equivalent)
Referenced by:: §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)
Permalink:: http://dlmf.nist.gov/33.6.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.0.19):: On the right-hand side the factor in the denominator was incorrectly written as $\Gamma\left(-\ell+\mathrm{i}\eta\right)$ . This has been corrected to $\Gamma\left(-\ell\pm\mathrm{i}\eta\right)$ .
Reported 2018-05-15 by Ian Thompson
See also:: Annotations for §33.6 and Ch.33

… ►Corresponding expansions for

{H^{\pm}_{\ell}}'\left(\eta,\rho\right)

can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

26: 32.10 Special Function Solutions

… ►

32.10.1

(w^{\prime})^{n}+\sum_{j=0}^{n-1}F_{j}(w,z)(w^{\prime})^{j}=0,

ⓘ

Symbols:: $n$ : integer and $z$ : real
Permalink:: http://dlmf.nist.gov/32.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(i), §32.10 and Ch.32

►where

F_{j}(w,z)

is polynomial in

w

with coefficients that are rational functions of

z

. … ►Solutions for other values of

\alpha

are derived from

w(z;\pm\tfrac{1}{2})

by application of the Bäcklund transformations (32.7.1) and (32.7.2). … ►

32.10.20

w(z;-m,-2(m-1)^{2})=-\frac{H_{m-1}'\left(z\right)}{H_{m-1}\left(z\right)},

m=1,2,3,\dots

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $m$ : integer and $z$ : real
Permalink:: http://dlmf.nist.gov/32.10.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iv), §32.10 and Ch.32

… ►

32.10.31

\phi(\zeta)=C_{1}F\left(b,-a;b+c;\zeta\right)+C_{2}\zeta^{-b+1-c}\*F\left(-a-b% -c+1,-c+1;2-b-c;\zeta\right),

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $a$ , $b$ , $c$ , $\phi(z)$ : function and $C$ : constant
Permalink:: http://dlmf.nist.gov/32.10.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(vi), §32.10 and Ch.32

…

27: 13.6 Relations to Other Functions

… ►

13.6.1

M\left(a,a,z\right)=e^{z},

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 13.6.12
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(i), §13.6 and Ch.13

… ►

13.6.16

M\left(-n,\tfrac{1}{2},z^{2}\right)=(-1)^{n}\frac{n!}{(2n)!}H_{2n}\left(z% \right),

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 13.6.17
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

►

13.6.17

M\left(-n,\tfrac{3}{2},z^{2}\right)=(-1)^{n}\frac{n!}{(2n+1)!2z}H_{2n+1}\left(% z\right),

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 13.6.18 (as corrected in later editions)
Permalink:: http://dlmf.nist.gov/13.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

… ►

13.6.21

U\left(a,b,z\right)=z^{-a}{{}_{2}F_{0}}\left(a,a-b+1;-;-z^{-1}\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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.6(vi)
Permalink:: http://dlmf.nist.gov/13.6.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(vi), §13.6 and Ch.13

►For the definition of

{{}_{2}F_{0}}\left(a,a-b+1;-;-z^{-1}\right)

when neither

a

nor

a-b+1

is a nonpositive integer see §16.5. …

28: 33.20 Expansions for Small $|\epsilon|$

… ►

33.20.3

f\left(\epsilon,\ell;r\right)=\sum_{k=0}^{\infty}\epsilon^{k}{\sf F}_{k}(\ell;% r),

ⓘ

Symbols:: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\mathsf{F}_{k}(\ell;r)$ : coefficient
Referenced by:: §33.20(ii)
Permalink:: http://dlmf.nist.gov/33.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

►where ►

33.20.4

{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}J_{2\ell+1+p}\left(% \sqrt{8r}\right),

r>0

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

►

33.20.5

{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2|r|)^{(p+1)/2}C_{k,p}I_{2% \ell+1+p}\left(\sqrt{8|r|}\right),

r<0

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

… ►where

A(\epsilon,\ell)

is given by (33.14.11), (33.14.12), and …

29: 1.13 Differential Equations

… ►

Transformation of the Point at Infinity

►The substitution

\xi=1/z

in (1.13.1) gives … ►

Liouville Transformation

… ►

Transformation to Liouville normal Form

►Equation (1.13.26) with

x\in[a,b]

may be transformed to the Liouville normal form …

30: 18.5 Explicit Representations

… ►In (18.5.4_5) see §26.11 for the Fibonacci numbers

F_{n}

. … ►In this equation

w(x)

is as in Table 18.3.1, (reproduced in Table 18.5.1), and

F(x)

\kappa_{n}

are as in Table 18.5.1. … ►For the definitions of

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

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

, and

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

see §16.2. … ►

18.5.13

H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^% {\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1% }{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{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, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.10
Referenced by:: §18.17(v), §18.17(vii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

H_{0}\left(x\right)=1,

…