scaled gamma function

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21: 15.1 Special Notation

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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22: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►However, this may be unstable; also overflow and underflow may occur when evaluating $A_n$ and $B_n$ (making it necessary to re-scale from time to time). … ►In contrast to the preceding algorithms in this subsection no scaling problems arise and no a priori information is needed. ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). … ►Again, no scaling problems arise and no a priori information is needed. …

23: 13.4 Integral Representations

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13.4.12 $$𝐌(a,c,z)=\frac{\mathrm{\Gamma }\left(b\right)}{2\pi \mathrm{i}}{z}^{1-b}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;1/t)dt,$$ $b\ne 0,-1,-2,\mathrm{\dots }$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

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13.4.15 $$\frac{U(a,b,z)}{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt}{t}^{-c}{}_2{}^{}𝐅_1^{}(a,c;a+c-b+1;1-\frac{1}{t})dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

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24: 13.10 Integrals

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13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;a+b-c+1;1-\frac{1}{z}),$$ $\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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25: 10.39 Relations to Other Functions

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Elementary Functions

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Airy Functions

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Parabolic Cylinder Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

… ►For the functions ${}_0{}^{}F_1^{}$ and $𝐅$ see (16.2.1) and §15.2(i).

26: 18.12 Generating Functions

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n,$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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27: 13.23 Integrals

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13.23.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{W}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right){}_2{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}-\mu +\nu ,\frac{1}{2}+\mu +\nu }{\nu -\kappa +1};\frac{1}{2}-z),$$ $\mathrm{\Re }\left(\nu +\frac{1}{2}\right)>|\mathrm{\Re }\mu |$, $\mathrm{\Re }z>-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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28: 10.16 Relations to Other Functions

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Elementary Functions

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Airy Functions

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Parabolic Cylinder Functions

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Generalized Hypergeometric Functions

… ►With $𝐅$ as in §15.2(i), and with $z$ and $\nu$ fixed, …

29: 32.13 Reductions of Partial Differential Equations

… ►has the scaling reduction … ►has the scaling reduction … ►has the scaling reduction …where $v(z)$ satisfies (32.2.10) with $\alpha =\frac{1}{2}$ and $\gamma =0$. In consequence if $w=\exp \left(-\mathrm{i}v\right)$, then $w(z)$ satisfies ${\text{P}}_{\text{III}}$ with $\alpha =-\beta =\frac{1}{2}$ and $\gamma =\delta =0$. …

30: 36.6 Scaling Relations

§36.6 Scaling Relations

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Diffraction Catastrophe Scaling

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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Indices for $k$-Scaling of Coordinates $x_m$

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Indices for $k$-Scaling of $𝐱$ Hypervolume

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