Gauss%20sums

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1: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes

p\leq x

: … ►Gauss and Legendre conjectured that

\pi\left(x\right)

is asymptotic to

x/\ln x

x\to\infty

: …(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. …

2: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by ►

20.11.1

G(m,n)=\sum\limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n};

ⓘ

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

… ► … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …

3: 18.5 Explicit Representations

… ►For the definitions of

{{}_{2}F_{1}}

{{}_{1}F_{1}}

, and

{{}_{2}F_{0}}

see §16.2. … ►

18.5.7

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

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.42
Referenced by:: §18.17(iv), §18.17(vii), (18.27.12_5), Table 18.3.1, §18.5(iii), §18.5(iii), §18.6(ii), §18.9(iii)
Permalink:: http://dlmf.nist.gov/18.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.11_1

T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}% _{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}% {x^{2}}\right),

n\geq 1

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(22))
Referenced by:: §18.5(iii), Erratum (V1.2.0) Section 18.5
Permalink:: http://dlmf.nist.gov/18.5.E11_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.11_3

U_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{% \ell}(n-\ell)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=\left(2x\right)^{n}{{}_{2}F_{% 1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-n};\frac{1}{x^{2}}% \right),

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(23))
Permalink:: http://dlmf.nist.gov/18.5.E11_3
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.13

H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^% {\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1% }{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).

ⓘ

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, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.10
Referenced by:: §18.17(v), §18.17(vii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

…

4: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►The step from

n

n+1

is an ascending Landen transformation if

\theta=1

(leading ultimately to a hyperbolic case of

R_{C}

) or a descending Gauss transformation if

\theta=-1

(leading to a circular case of

R_{C}

). … ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

5: Bibliography V

… ►

J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

ⓘ

External Links:: ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt
Referenced by:: §19.35(i).
Encodings:: BibTeX

… ►

R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

ⓘ

External Links:: ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

… ►

I. M. Vinogradov (1937) Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

ⓘ

External Links:: ZentralBlatt
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

…

6: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(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.

ⓘ

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

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

7: Bibliography G

… ►

M. J. Gander and A. H. Karp (2001) Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

ⓘ

Referenced by:: §18.39(iii).
Encodings:: BibTeX

… ►

C. F. Gauss (1863) 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

… ►

W. Gautschi (1994) 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

… ►

W. Gautschi (1968) 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

… ►

A. Gil, J. Segura, and N. M. Temme (2014) 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

…

8: Bibliography K

… ►

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

ⓘ

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

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
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.

ⓘ

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

…

9: 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)”.

… ►

Subsection 15.19(v)

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

… ►

Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

…