Gauss%20multiplication%20formula

SearchAdvancedHelp

(0.001 seconds)

21—30 of 413 matching pages

21: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

22: Bibliography G

… ►

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

… ►

Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

… ►

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

… ►

Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

… ►

Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

ⓘ

External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

…

23: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(iii) ${}_3{}^{}F_2^{}$ Case

►

Kummer Transformation

… ►

Pfaff–Saalschütz Formula

… ►

Thomae Transformation

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${}_p{}^{}F_q^{}$ and ${}_{p+1}{}^{}F_p^{}$ of matrix argument. …

24: 15.8 Transformations of Variable

… ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). … ►

15.8.13 $$F(\genfrac{}{}{0pt}{}{a,b}{2b};z)={\left(1-\frac{1}{2}z\right)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{b+\frac{1}{2}};{\left(\frac{z}{2-z}\right)}^2),$$ ,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.8(iv)

Permalink:

http://dlmf.nist.gov/15.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.14 $$F(\genfrac{}{}{0pt}{}{a,b}{2b};z)={\left(1-z\right)}^{-a/2}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,b-\frac{1}{2}a}{b+\frac{1}{2}};\frac{z^2}{4z-4}),$$ .

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►

15.8.15 $$F(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)={(1+z)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{a-b+1};\frac{4z}{{(1+z)}^2}),$$ ,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.4(i), §15.8(iv)

Permalink:

http://dlmf.nist.gov/15.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.16 $$F(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)={(1-z)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}}{a-b+1};\frac{-4z}{{(1-z)}^2}),$$ .

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

…

25: 16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ is well-poised if … ►The function ${}_{q+1}{}^{}F_q^{}$ with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For generalizations involving ${}_{r+3}{}^{}F_{r+2}^{}$ functions see Kim et al. (2013). … ►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). …

26: 3.5 Quadrature

… ►

§3.5(v) Gauss Quadrature

… ►

Gauss–Legendre Formula

… ►

Gauss–Chebyshev Formula

… ►

Gauss–Jacobi Formula

… ►

Gauss–Laguerre Formula

…

27: 8.17 Incomplete Beta Functions

… ►

8.17.7 $${\mathrm{B}}_x(a,b)=\frac{x^a}{a}F(a,1-b;a+1;x),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.8

Permalink:

http://dlmf.nist.gov/8.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

►

8.17.8 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^b}{a}F(a+b,1;a+1;x),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

►

8.17.9 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^{b-1}}{a}F(\genfrac{}{}{0pt}{}{1,1-b}{a+1};\frac{x}{x-1}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

…

28: Bibliography I

… ►

The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

… ►

The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

ⓘ

Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

… ►

Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

BibTeX

… ►

From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

ⓘ

External Links:

ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§32.2(vi), §32.7(vii).

Encodings:

BibTeX

29: 15.9 Relations to Other Functions

… ►

15.9.2 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

►

15.9.3 $$C_n^{(\lambda )}\left(x\right)={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,\frac{1}{2}(1-n)}{1-\lambda -n};\frac{1}{x^2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►

15.9.5 $$T_n\left(x\right)=F(\genfrac{}{}{0pt}{}{-n,n}{\frac{1}{2}};\frac{1-x}{2}).$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►

15.9.7 $$P_n\left(x\right)=F(\genfrac{}{}{0pt}{}{-n,n+1}{1};\frac{1-x}{2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n\left(x\right)$: 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 following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …

30: 31.7 Relations to Other Functions

… ►

§31.7(i) Reductions to the Gauss Hypergeometric Function

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. … ►

31.7.2 $$H\mathrm{\ell }(2,\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta -2\gamma +1;z)={}_2{}^{}F_1^{}(\frac{1}{2}\alpha ,\frac{1}{2}\beta ;\gamma ;1-{(1-z)}^2),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

►

31.7.3 $$H\mathrm{\ell }(4,\alpha \beta ;\alpha ,\beta ,\frac{1}{2},\frac{2}{3}(\alpha +\beta );z)={}_2{}^{}F_1^{}(\frac{1}{3}\alpha ,\frac{1}{3}\beta ;\frac{1}{2};1-{(1-z)}^2(1-\frac{1}{4}z)),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

… ►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta =K$, $K+\mathrm{i}{K}^{\prime }$, and $\mathrm{i}{K}^{\prime }$, where $K$ and $K^{\prime }$ are related to $k$ as in §19.2(ii).