# fractional or multiple

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##### 1: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions
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17.7.6 ${{}_{3}\phi_{2}}\left({q^{-2n},b,c\atop q^{1-2n}/b,q^{1-2n}/c};q,\frac{q^{2-n}% }{bc}\right)=\frac{\left(b,c;q\right)_{n}\left(q,bc;q\right)_{2n}}{\left(q,bc;% q\right)_{n}\left(b,c;q\right)_{2n}}.$
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###### Continued Fractions
βΊFor continued-fraction representations of a ratio of ${{}_{3}\phi_{2}}$ functions, see Cuyt et al. (2008, pp. 399–400). … βΊ
17.7.21 $\sum_{k=0}^{n}\frac{(1-ap^{k}q^{k})(1-bp^{k}q^{-k})}{(1-a)(1-b)}\frac{\left(a,% b;p\right)_{k}\left(c,a/(bc);q\right)_{k}}{\left(q,aq/b;q\right)_{k}\left(ap/c% ,bcp;p\right)_{k}}q^{k}=\frac{\left(ap,bp;p\right)_{n}\left(cq,aq/(bc);q\right% )_{n}}{\left(q,aq/b;q\right)_{n}\left(ap/c,bcp;p\right)_{n}},$
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17.7.22 $\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}% \frac{\left(a,b;p\right)_{k}\left(c,ad^{2}/(bc);q\right)_{k}}{\left(dq,adq/b;q% \right)_{k}\left(adp/c,bcp/d;p\right)_{k}}q^{k}=\frac{(1-a)(1-b)(1-c)(1-(ad^{2% }/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{\left(ap,bp;p% \right)_{n}\left(cq,ad^{2}q/(bc);q\right)_{n}}{\left(dq,adq/b;q\right)_{n}% \left(adp/c,bcp/d;p\right)_{n}}-\frac{\left(c/(ad),d/(bc);p\right)_{m+1}\left(% 1/d,b/(ad);q\right)_{m+1}}{\left(1/c,bc/(ad^{2});q\right)_{m+1}\left(1/a,1/b;p% \right)_{m+1}}\right),$
##### 2: 12.6 Continued Fraction
###### §12.6 Continued Fraction
βΊFor a continued-fraction expansion of the ratio $\ifrac{U\left(a,x\right)}{U\left(a-1,x\right)}$ see Cuyt et al. (2008, pp. 340–341).
##### 3: 17.6 ${{}_{2}\phi_{1}}$ Function
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17.6.13 ${{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{% \left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;% q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{% \infty}},$
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17.6.29 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\left(\frac{-1}{2\pi i}\right)% \frac{\left(a,b;q\right)_{\infty}}{\left(q,c;q\right)_{\infty}}\int_{-i\infty}% ^{i\infty}\frac{\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}}{\left(aq^{% \zeta},bq^{\zeta};q\right)_{\infty}}\frac{\pi(-z)^{\zeta}}{\sin\left(\pi\zeta% \right)}\,\mathrm{d}\zeta,$
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. βΊ
###### §17.6(vi) Continued Fractions
βΊFor continued-fraction representations of the ${{}_{2}\phi_{1}}$ function, see Cuyt et al. (2008, pp. 395–399).
##### 4: 27.19 Methods of Computation: Factorization
βΊThese algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …
##### 5: 17.2 Calculus
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17.2.5 $\left(a_{1},a_{2},\dots,a_{r};q\right)_{n}=\prod_{j=1}^{r}\left(a_{j};q\right)% _{n},$
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17.2.6 $\left(a_{1},a_{2},\dots,a_{r};q\right)_{\infty}=\prod_{j=1}^{r}\left(a_{j};q% \right)_{\infty}.$
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17.2.22 $\frac{\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q\right)_{n}}{\left(a^{\frac{1}% {2}},-a^{\frac{1}{2}};q\right)_{n}}=\frac{\left(aq^{2};q^{2}\right)_{n}}{\left% (a;q^{2}\right)_{n}}=\frac{1-aq^{2n}}{1-a},$
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17.2.23 $\frac{\left(qa^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1% }a^{\frac{1}{k}};q\right)_{n}}{\left(a^{\frac{1}{k}},\omega_{k}a^{\frac{1}{k}}% ,\dots,\omega_{k}^{k-1}a^{\frac{1}{k}};q\right)_{n}}=\frac{\left(aq^{k};q^{k}% \right)_{n}}{\left(a;q^{k}\right)_{n}}=\frac{1-aq^{kn}}{1-a},$
##### 6: 10.55 Continued Fractions
###### §10.55 Continued Fractions
βΊFor continued fractions for $\mathsf{j}_{n+1}\left(z\right)/\mathsf{j}_{n}\left(z\right)$ and ${\mathsf{i}^{(1)}_{n+1}}\left(z\right)/{\mathsf{i}^{(1)}_{n}}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
##### 7: 3.10 Continued Fractions
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###### Stieltjes Fractions
βΊis called a Stieltjes fraction ($S$-fraction). … βΊ
###### Jacobi Fractions
βΊThe continued fraction
##### 8: 17.11 Transformations of $q$-Appell Functions
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17.11.1 $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),$
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17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},$
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17.11.3 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.$
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17.11.4 $\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).$
##### 9: 27.3 Multiplicative Properties
###### §27.3 Multiplicative Properties
βΊIf $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. …Related multiplicative properties are … βΊA function $f$ is completely multiplicative if $f(1)=1$ and … βΊIf $f$ is completely multiplicative, then (27.3.2) becomes …
##### 10: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
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17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$
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17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.$
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17.8.3 $\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$
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17.8.4 ${{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},$
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17.8.5 ${{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},$