Gauss sum

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1: 27.10 Periodic Number-Theoretic Functions

… ►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. …In particular, $G(n,{\chi }_1)=c_k\left(n\right)$. ► $G(n,\chi )$ is separable for some $n$ if … ►For a primitive character $\chi (modk)$, $G(n,\chi )$ is separable for every $n$, and … ►Conversely, if $G(n,\chi )$ is separable for every $n$, then $\chi$ is primitive (mod $k$). …

2: 20.11 Generalizations and Analogs

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§20.11(i) Gauss Sum

►For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by ►

20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

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Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

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20.11.2 $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $G(m,n)$: Gauss sum

Permalink:

http://dlmf.nist.gov/20.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

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3: 5.16 Sums

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

4: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Gauss’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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Gasper–Rahman $q$-Analog of Watson’s ${}_3{}^{}F_2^{}$ Sum

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Gasper–Rahman $q$-Analog of Whipple’s ${}_3{}^{}F_2^{}$ Sum

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5: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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$q$-Gauss Sum

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6: 16.4 Argument Unity

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16.4.2_5 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a,1}{-n,c};1)=\sum _{k=0}^n\frac{{\left(a\right)}_k}{{\left(c\right)}_k}=\frac{c-1}{c-a-1}\left(1-\frac{{\left(a\right)}_{n+1}}{{\left(c-1\right)}_{n+1}}\right),$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $a,a_1,\mathrm{\dots },a_p$: real or complex parameters

Source:

Lerch (1905, (3))

Referenced by:

§16.4(ii)

Permalink:

http://dlmf.nist.gov/16.4.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §16.4(ii), §16.4(ii), §16.4 and Ch.16

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7: 18.38 Mathematical Applications

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18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

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8: 34.2 Definition: $\mathit{3}j$ Symbol

… ►where ${}_3{}^{}F_2^{}$ is defined as in §16.2. ►For alternative expressions for the $\mathit{3}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_3{}^{}F_2^{}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

9: 34.4 Definition: $\mathit{6}j$ Symbol

… ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

10: 15.2 Definitions and Analytical Properties

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15.2.1 $$F(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}{z}^2+\mathrm{\cdots }=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+s\right)\mathrm{\Gamma }\left(b+s\right)}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$

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Defines:

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

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.4 $$F(-m,b;c;z)=\sum _{n=0}^m\frac{{\left(-m\right)}_n{\left(b\right)}_n}{{\left(c\right)}_{n}n!}{z}^n=\sum _{n=0}^m{(-1)}^n\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)\frac{{\left(b\right)}_n}{{\left(c\right)}_n}{z}^n.$$

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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), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

A&S Ref:

15.4.1

Permalink:

http://dlmf.nist.gov/15.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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