# separable Gauss sum

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## 5 matching pages

##### 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 $n$ if $\left(n,k\right)=1$, 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

Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F\left(a,a;a+1;\tfrac{1}{2}\right)$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

• Equation (35.7.3)

Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

• 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.
• W. Miller (1974) Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.
• W. Miller (1977) Symmetry and Separation of Variables. Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam.
• 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.
• 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.
• ##### 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 ${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\sum_% {k=1}^{m}A_{p,q,k}^{m,n}(z){{}_{p}F_{q-1}}\left({1+b_{k}-a_{1},\dots,1+b_{k}-a% _{p}\atop 1+b_{k}-b_{1},\ldots*\dots,1+b_{k}-b_{q}};(-1)^{p-m-n}z\right),$
##### 5: Bibliography K
• S. L. Kalla (1992) On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.
• 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.
• E. G. Kalnins (1986) Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.
• N. M. Korobov (1958) Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).
• 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.