Gauss transformations

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21: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

22: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by …If both

m, n

are positive, then

G(m,n)

allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …

23: 16.16 Transformations of Variables

§16.16 Transformations of Variables

►

§16.16(i) Reduction Formulas

… ►

§16.16(ii) Other Transformations

… ►For quadratic transformations of Appell functions see Carlson (1976).

24: Bibliography G

… ►

M. J. Gander and A. H. Karp (2001) Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

ⓘ

Referenced by:: §18.39(iii).
Encodings:: BibTeX

… ►

C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).

ⓘ

Notes:: Reprinted by Georg Olms Verlag, Hildesheim, 1973.
External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §27.2(i).
Encodings:: BibTeX

… ►

W. Gautschi (1968) Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

ⓘ

External Links:: ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)
Referenced by:: §18.39(iii).
Encodings:: BibTeX

… ►

W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.29(iii), §15.19(iii).
Encodings:: BibTeX

… ►

G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.

ⓘ

Notes:: Loose microfiche suppl. A1–A10
External Links:: FullText, MathReview (F. Stenger), ZentralBlatt
Referenced by:: §18.39(iii), §3.5(vi).
Encodings:: BibTeX

…

25: Bibliography M

… ►

J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

ⓘ

External Links:: ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt
Referenced by:: §2.6(ii).
Encodings:: BibTeX

… ►

N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.

ⓘ

External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §15.19(i), §15.19(i).
Encodings:: BibTeX

… ►

A. E. Milne, P. A. Clarkson, and A. P. Bassom (1997) Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

ⓘ

External Links:: ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.10(iii), §32.7(iii), §32.8(iii), §32.9(i).
Encodings:: BibTeX

… ►

S. C. Milne (1988) A

q

-analog of the Gauss summation theorem for hypergeometric series in

U(n)

. Adv. in Math. 72 (1), pp. 59–131.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

►

S. C. Milne (1994) A

q

-analog of a Whipple’s transformation for hypergeometric series in

U(n)

. Adv. Math. 108 (1), pp. 1–76.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

26: 15.10 Hypergeometric Differential Equation

… ►

f_{1}(z)=F\left({a,b\atop c};z\right),

… ►

f_{1}(z)=F\left({a,b\atop a+b+1-c};1-z\right),

… ►(b) If

c

equals

n=1,2,3,\dots

, and

a\neq 1,2,\dots,n-1

, then fundamental solutions in the neighborhood of

z=0

are given by

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

and … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. ►

15.10.11

w_{1}(z)=F\left({a,b\atop c};z\right)=(1-z)^{c-a-b}F\left({c-a,c-b\atop c};z% \right)=(1-z)^{-a}F\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}F\left(% {c-a,b\atop c};\frac{z}{z-1}\right).

ⓘ

Defines:: $w_{1}(z)$ : Kummer’s solution $w_{1}$ (locally)
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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.10.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

…

27: 15.9 Relations to Other Functions

… ►

15.9.7

P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{1-x}{2}\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, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $x$ : real variable and $n$ : integer
Permalink:: http://dlmf.nist.gov/15.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►The Jacobi transform is defined as …with inverse … … ►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. …

28: 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

►

15.6.2_5

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{\infty}\frac{t^{b-1}\left(t+1\right)^{a-c}}{\left(t-zt+1% \right)^{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:: §15.6, Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/15.6.E2_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: This equation was added.
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

…

29: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing

\operatorname{Ai}\left(z\right)

in the complex plane, once values of this function can be generated on the positive real axis. … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …

30: 18.38 Mathematical Applications

… ►It has elegant structures, including

N

-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►For the generalized hypergeometric function

{{}_{3}F_{2}}

see (16.2.1). … ►

Radon Transform

… ►The

3j

symbol (34.2.6), with an alternative expression as a terminating

{{}_{3}F_{2}}

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. …