separable Gauss sum

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1: 27.10 Periodic Number-Theoretic Functions

… ►

G\left(n,\chi\right)

is separable for some

n

if … ►For any Dirichlet character

\chi\pmod{k}

G\left(n,\chi\right)

is separable for

n

\left(n,k\right)=1

, and is separable for every

n

if and only if

G\left(n,\chi\right)=0

whenever

\left(n,k\right)>1

. For a primitive character

\chi\pmod{k}

G\left(n,\chi\right)

is separable for every

n

, and … ►Conversely, if

G\left(n,\chi\right)

is separable for every

n

, then

\chi

is primitive (mod

k

). …

2: Errata

… ►

Equation (18.38.3)

18.38.3

\sum_{m=0}^{n}P^{(\alpha,0)}_{m}\left(x\right)=\frac{{\left(\alpha+2\right)_{n% }}}{n!}{{}_{3}F_{2}}\left({-n,n+\alpha+2,\frac{1}{2}(\alpha+1)\atop\alpha+1,% \frac{1}{2}(\alpha+3)};\tfrac{1}{2}(1-x)\right)\geq 0,

x\geq-1

\alpha\geq-2

n=0,1,\dots

This equation was updated to include the value of the sum in terms of the ${{}_{3}F_{2}}$ function. Also the constraint was previously $-1\leq x\leq 1$ , $\alpha>-1$ .

… ►

Subsection 17.7(iii)

The title of the paragraph which was previously “Andrews’ Terminating $q$ -Analog of (17.7.8)” has been changed to “Andrews’ $q$ -Analog of the Terminating Version of Watson’s ${{}_{3}F_{2}}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$ -Analog” has been changed to “Andrews’ $q$ -Analog of the Terminating Version of Whipple’s ${{}_{3}F_{2}}$ Sum (16.4.7)”.

… ►

Notation

The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

… ►

Paragraph Mellin–Barnes Integrals (in §8.6(ii))

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$ . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma\left(a,z\right)$ . In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$ .

Reported 2017-07-10 by Kurt Fischer.

… ►

Subsection 15.19(v)

A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

…

3: Bibliography M

… ►

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.

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

… ►

W. Miller (1974) Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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External Links:: Document, MathReview (G. R. Allcock), ZentralBlatt
Referenced by:: §12.13(ii), §12.17.
Encodings:: BibTeX

►

W. Miller (1977) Symmetry and Separation of Variables. Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam.

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Notes:: Encyclopedia of Mathematics and its Applications, Vol. 4
External Links:: ISBN 0-201-13503-5, MathReview (Ernest G. Kalnins), ZentralBlatt
Referenced by:: §31.17(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.

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

… ►

S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

4: 16.17 Definition

… ►where the integration path

L

separates the poles of the factors

\Gamma\left(b_{\ell}-s\right)

from those of the factors

\Gamma\left(1-a_{\ell}+s\right)

. … ►

16.17.2

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\sum_% {k=1}^{m}A_{p,q,k}^{m,n}(z){{}_{p}F_{q-1}}\left({1+b_{k}-a_{1},\dots,1+b_{k}-a% _{p}\atop 1+b_{k}-b_{1},\ldots*\dots,1+b_{k}-b_{q}};(-1)^{p-m-n}z\right),

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $A_{p,q}^{m,n}(z)$ : coefficient
Permalink:: http://dlmf.nist.gov/16.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.17 and Ch.16

…

5: 18.2 General Orthogonal Polynomials

… ►

§18.2(v) Christoffel–Darboux Formula

… ►The zeros of

p_{n}(x)

and

p_{n+1}(x)

separate each other, and if

m<n

then between any two zeros of

p_{m}(x)

there is at least one zero of

p_{n}(x)

. … ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►A system

\{p_{n}(x)\}

of OP’s satisfying (18.2.1) and (18.2.5) is complete if each

f(x)

in the Hilbert space

L_{w}^{2}((a,b))

can be approximated in Hilbert norm by finite sums

\sum_{n}\lambda_{n}p_{n}(x)

. … ►If these

x_{k}

satisfy

\sum_{k}(|x_{k}|-1)^{\ifrac{1}{2}}<\infty

then Szegő type asymptotics outside

[-1,1]

can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). …

6: Bibliography K

… ►

S. L. Kalla (1992) On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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External Links:: ISSN 0861-1459, MathReview, ZentralBlatt
Referenced by:: §15.15, §15.19(i).
Encodings:: BibTeX

… ►

E. G. Kalnins, W. Miller, and P. Winternitz (1976) The group

{\rm{O}}(4)

, separation of variables and the hydrogen atom. SIAM J. Appl. Math. 30 (4), pp. 630–664.

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External Links:: Document, MathReview (Anthony J. Bracken), ZentralBlatt
Referenced by:: §31.17(i).
Encodings:: BibTeX

… ►

E. G. Kalnins (1986) Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.

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External Links:: ISBN 0-582-98807-1, MathReview (H. Kilp), ZentralBlatt
Referenced by:: §31.17(i).
Encodings:: BibTeX

… ►

N. M. Korobov (1958) Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).

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External Links:: MathReview (R. A. Rankin), ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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