Gauss multiplication formula

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1: 5.5 Functional Relations

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Gauss’s Multiplication Formula

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2: 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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Second $q$-Analog of Bailey’s ${}_4{}^{}F_3^{}\left(1\right)$ Sum

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3: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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$q$-Gauss Sum

… ►Related formulas are (17.7.3), (17.8.8) and … ►For similar formulas see Verma and Jain (1983). … ►

17.6.13 $${}_2{}^{}\varphi _1^{}(a,b;c;q,q)+\frac{{(q/c,a,b;q)}_{\mathrm{\infty }}}{{(c/q,aq/c,bq/c;q)}_{\mathrm{\infty }}}{}_2{}^{}\varphi _1^{}(aq/c,bq/c;q^2/c;q,q)=\frac{{(q/c,abq/c;q)}_{\mathrm{\infty }}}{{(aq/c,bq/c;q)}_{\mathrm{\infty }}},$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

… ►where , , and the contour of integration separates the poles of ${(q^{1+\zeta },c{q}^{\zeta };q)}_{\mathrm{\infty }}/\sin \left(\pi \zeta \right)$ from those of $1/{(a{q}^{\zeta },b{q}^{\zeta };q)}_{\mathrm{\infty }}$, and the infimum of the distances of the poles from the contour is positive. …

4: Errata

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18.28.1 $$p_n(x)=p_n(x;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),$$

18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q)$$

Previously we presented all the information of these formulas in one equation

$$p_n(\cos \theta )=p_n(\cos \theta ;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})=a^{-n}{(ab,ac,ad;q)}_n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,ac,ad};q,q).$$

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17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

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The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

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The wording was changed to make the integration variable more apparent.

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Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F(a,b;a;z)={(1-z)}^{-b}$.

Report by Louis Klauder on 2017-01-01.

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5: Bibliography V

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Multiple Light Scattering. Vol. 1, Academic Press, New York.

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Referenced by:

§6.17.

Encodings:

BibTeX

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Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

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Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:

ISSN 0019-5839, MathReview Entry, ZentralBlatt

Referenced by:

(17.6.4_5), §17.6(i), (17.8.8), §17.8.

Encodings:

BibTeX

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

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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6: Bibliography M

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Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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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

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

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On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

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External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

Encodings:

BibTeX

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Notes:

MPFR is a C library for multiple-precision floating-point computations with correct rounding. The main goal of MPFR is to provide a library for multiple-precision floating-point computation which is both efficient and has well-defined semantics. MPFR includes most of the elementary functions. It also implements the gamma function, and more special functions are planned in the future.

External Links:

MPFR

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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