homogeneous equations

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21: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function

F_{D}

(1963), he defined the

R

-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

22: 20.9 Relations to Other Functions

…

23: 19.16 Definitions

… ►

19.16.3Moved to (19.23.6_5).

ⓘ

Referenced by:: §19.16(i), (19.23.6_5), §19.23, Erratum (V1.1.0) for Rearrangement
Permalink:: http://dlmf.nist.gov/19.16.E3
Rearrangement (effective with 1.1.0):: This equation has been moved to (19.23.6_5).
See also:: Annotations for §19.16(i), §19.16 and Ch.19

… ►which is homogeneous and of degree

-a

in the

z

’s, and unchanged when the same permutation is applied to both sets of subscripts

1,\dots,n

. … ►

19.16.9

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}\,% \mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a% -1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\,\mathrm{d}t,

b_{1}+\cdots+b_{n}>a>0

b_{j}\in\mathbb{R}

z_{j}\in\mathbb{C}\setminus(-\infty,0]

ⓘ

Defines:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function
Symbols:: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\mathbb{R}$ : real line, $\setminus$ : set subtraction and $n$ : nonnegative integer
Referenced by:: §19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)
Permalink:: http://dlmf.nist.gov/19.16.E9
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.18):: The constraint $a,a^{\prime}>0$ , was replaced with $b_{1}+\cdots+b_{n}>a>0$ , $b_{j}\in\mathbb{R}$ . It therefore follows from (19.16.10) that $a^{\prime}>0$ .
Suggested 2017-07-25 by Bastien Roucariès
See also:: Annotations for §19.16(ii), §19.16 and Ch.19

… ►

19.16.10

a^{\prime}=-a+\sum_{j=1}^{n}b_{j}.

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Symbols:: $n$ : nonnegative integer
Referenced by:: (19.16.9), Erratum (V1.0.18) for Equation (19.16.9)
Permalink:: http://dlmf.nist.gov/19.16.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.16(ii), §19.16 and Ch.19

…

24: 19.26 Addition Theorems

… ►with corresponding equations for

y+\mu

and

z+\mu

obtained by permuting

x, y, z

. … ►The equations inverse to

z+\lambda=(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})

and the two other equations obtained by permuting

x, y, z

(see (19.26.19)) are …and two similar equations obtained by exchanging

z

with

x

(and

\zeta

with

\xi

), or

z

with

y

(and

\zeta

with

\eta

). …

25: Bibliography G

… ►

W. Gautschi (1997b) The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:: ISBN 90-5699-521-9, MathReview, ZentralBlatt
Referenced by:: §3.6(iii), §3.6(vii).
Encodings:: BibTeX

… ►

A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:: ISSN 0036-1429, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(v).
Encodings:: BibTeX

… ►

S. G. Gindikin (1964) Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89
External Links:: Document, MathReview (A. Korányi), ZentralBlatt
Referenced by:: §35.3(i).
Encodings:: BibTeX

… ►

J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:: ISBN 0-8176-3837-7, MathReview, ZentralBlatt
Referenced by:: §15.17(v).
Encodings:: BibTeX

… ►

V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 18(3), pp. 317–326.
External Links:: ISSN 0374-0641, MathReview, ZentralBlatt
Referenced by:: §32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).
Encodings:: BibTeX

…

26: 19.25 Relations to Other Functions

… ►Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of

R_{D}

; see (19.21.7). … ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … ►

19.25.35

z+2\omega=\pm R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp% \left(z\right)-e_{3}\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots and $z$ : complex number
Proof sketch:: Use (23.6.36) with $z=\wp\left(\omega\right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\wp\left(\omega\right)$ and compare with (19.16.1). The undetermined $\pm$ is a consequence of the multivaluedness of the square-roots in (19.16.4).
Referenced by:: (19.25.38), (19.25.40), §19.25(v), §19.25(vi), §19.25(vi), §20.9(i), Erratum (V1.1.0) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E35
Encodings:: pMML, png, TeX
Correction (effective with 1.1.0):: This equation has been corrected, namely the left-hand side which was originally $z$ , has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$ .
See also:: Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.37

\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)=\pm 2R_{G}\left(\wp% \left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function and $z$ : complex number
Referenced by:: §19.25(vi), Erratum (V1.0.18) for Equation (19.25.37), Erratum (V1.1.0) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E37
Encodings:: pMML, png, TeX
Correction (effective with 1.1.0):: This equation has been corrected, namely the left-hand side which was originally $\zeta\left(z\right)+z\wp\left(z\right)$ , has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$ , and the right-hand side has been multiplied by $\pm 1$ .
Clarification (effective with 1.0.18):: The Weierstrass zeta function in this equation incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the functions appeared correct.
See also:: Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.40

z+2\omega=\pm\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $z$ : complex number and $\sigma_{j}(z)$ : function
Proof sketch:: Combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1).
Referenced by:: §19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E40
Encodings:: pMML, png, TeX
Correction (effective with 1.1.0):: This equation has been corrected, namely the left-hand side which was originally $z$ , has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$ .
See also:: Annotations for §19.25(vi), §19.25 and Ch.19

…