Gauss transformations

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1: 19.15 Advantages of Symmetry

… ►Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … …

2: 19.22 Quadratic Transformations

… ►If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when (implying ), and descending Gauss transformations when (implying ). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …

3: 19.8 Quadratic Transformations

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§19.8(iii) Gauss Transformation

►We consider only the descending Gauss transformation because its (ascending) inverse moves $F(\varphi ,k)$ closer to the singularity at $k=\sin \varphi =1$. …

4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Kummer Transformation

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Thomae Transformation

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5: 19.36 Methods of Computation

… ►The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). … ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function $\mathrm{el2}(x,k_c,a,b)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). …

6: 16.6 Transformations of Variable

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16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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).

7: Bibliography F

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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8: Bibliography V

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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9: 16.4 Argument Unity

… ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ►Transformations for both balanced ${}_4{}^{}F_3^{}\left(1\right)$ and very well-poised ${}_7{}^{}F_6^{}\left(1\right)$ are included in Bailey (1964, pp. 56–63). …

10: 16.5 Integral Representations and Integrals

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16.5.2 $${}_{p+1}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_0,\mathrm{\dots },b_q};z)=\frac{\mathrm{\Gamma }\left(b_0\right)}{\mathrm{\Gamma }\left(a_0\right)\mathrm{\Gamma }\left(b_0-a_0\right)}{\int }_0^1{t}^{a_0-1}{(1-t)}^{b_0-a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};zt)dt,$$ $\mathrm{\Re }{b}_0>\mathrm{\Re }{a}_0>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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

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16.5.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{\mathrm{\Gamma }\left(a_0\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\cdots },b_q};zt)dt,$$ , $\mathrm{\Re }{a}_0>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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

►

16.5.4 $${}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_0,\mathrm{\dots },b_q};z)=\frac{\mathrm{\Gamma }\left(b_0\right)}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^t{t}^{-b_0}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\frac{z}{t})dt,$$ $c>0$, $\mathrm{\Re }{b}_0>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Mellin transform

Referenced by:

§16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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