# Gauss multiplication formula

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##### 3: Errata
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• Equation (18.28.1)
18.28.1 $p_{n}(x)=p_{n}\left(x;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}% \left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},% abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2% aq^{j}x+a^{2}q^{2j})},$
18.28.1_5 $R_{n}(z)=R_{n}(z;a,b,c,d\,|\,q)=\frac{p_{n}\left(\frac{1}{2}(z+z^{-1});a,b,c,d% \,|\,q\right)}{a^{-n}\left(ab,ac,ad;q\right)_{n}}={{}_{4}\phi_{3}}\left({q^{-n% },abcdq^{n-1},az,az^{-1}\atop ab,ac,ad};q,q\right)$

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

$p_{n}(\cos\theta)=p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=% 0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{% \left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^% {\ell-1}{(1-2aq^{j}\cos\theta+a^{2}q^{2j})}=a^{-n}\left(ab,ac,ad;q\right)_{n}% \*{{}_{4}\phi_{3}}\left({q^{-n},abcdq^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta},a% {\mathrm{e}}^{-\mathrm{i}\theta}\atop ab,ac,ad};q,q\right).$
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• Equation (17.11.2)
17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}}$

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

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• Equation (17.4.6)

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

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• Paragraph Inversion Formula (in §35.2)

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

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• Subsections 15.4(i), 15.4(ii)

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 $F\left(a,b;a;z\right)=(1-z)^{-b}$.

Report by Louis Klauder on 2017-01-01.

• ##### 4: Bibliography V
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• H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
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• A. J. van der Poorten (1980) Some Wonderful Formulas $\ldots$ 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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• A. Verma and V. K. Jain (1983) Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
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• R. VidΕ«nas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
• ##### 5: Bibliography M
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• T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
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• J. P. McClure and R. Wong (1987) Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.
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• 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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• D. S. Moak (1984) The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
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• L. J. Mordell (1958) On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.
• ##### 6: 17.6 ${{}_{2}\phi_{1}}$ Function
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###### $q$-Gauss Sum
βΊRelated formulas are (17.7.3), (17.8.8) and … βΊFor similar formulas see Verma and Jain (1983). … βΊ
17.6.13 ${{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{% \left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;% q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{% \infty}},$
βΊwhere $|z|<1$, $|\operatorname{ph}\left(-z\right)|<\pi$, and the contour of integration separates the poles of $\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\sin\left(\pi\zeta\right)$ from those of $1/\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}$, and the infimum of the distances of the poles from the contour is positive. …