multiplicative

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1: 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 $\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).$
2: 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 …
3: 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 ${{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},$
4: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions
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17.7.2 ${{}_{2}\phi_{2}}\left({a^{2},b^{2}\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}}};% q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}\right)_{\infty}}{\left(q,a^{2}b^{2% }q;q^{2}\right)_{\infty}}.$
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17.7.4 ${{}_{3}\phi_{2}}\left({a,b,q^{-n}\atop c,abq^{1-n}/c};q,q\right)=\frac{\left(c% /a,c/b;q\right)_{n}}{\left(c,c/(ab);q\right)_{n}}.$
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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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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 $\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}% \frac{\left(a,b;p\right)_{k}\left(c,ad^{2}/(bc);q\right)_{k}}{\left(dq,adq/b;q% \right)_{k}\left(adp/c,bcp/d;p\right)_{k}}q^{k}=\frac{(1-a)(1-b)(1-c)(1-(ad^{2% }/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{\left(ap,bp;p% \right)_{n}\left(cq,ad^{2}q/(bc);q\right)_{n}}{\left(dq,adq/b;q\right)_{n}% \left(adp/c,bcp/d;p\right)_{n}}-\frac{\left(c/(ad),d/(bc);p\right)_{m+1}\left(% 1/d,b/(ad);q\right)_{m+1}}{\left(1/c,bc/(ad^{2});q\right)_{m+1}\left(1/a,1/b;p% \right)_{m+1}}\right),$
5: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
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18.29.1 $\left(bc,bd,cd;q\right)_{n}\*\left(Q_{n}({\mathrm{e}}^{\mathrm{i}\theta};a,b,c% ,d\mid q)+Q_{n}({\mathrm{e}}^{-\mathrm{i}\theta};a,b,c,d\mid q)\right),$
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18.29.2 $Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},$ $n\to\infty$; $z,a,b,c,d,q$ fixed.
6: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
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17.10.1 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),$
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17.10.2 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).$
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17.10.3 ${{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),$
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17.10.4 ${{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.$
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17.10.5 $\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).$
7: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions
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17.9.3_5 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/a,c/b;q\right)_{% \infty}}{\left(c,c/(ab);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c% \atop qab/c,0};q,q\right)+\frac{\left(a,b,abz/c;q\right)_{\infty}}{\left(c,ab/% c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,c/b,z\atop qc/(ab),0};q,q% \right),$
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17.9.6 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(e/a,de/(% bc);q\right)_{\infty}}{\left(e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),$
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17.9.7 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(b,de/(ab)% ,de/(bc);q\right)_{\infty}}{\left(d,e,de/(abc);q\right)_{\infty}}\*{{}_{3}\phi% _{2}}\left({d/b,e/b,de/(abc)\atop de/(ab),de/(bc)};q,b\right),$
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17.9.13 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).$
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17.9.14 ${{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$
8: 27.20 Methods of Computation: Other Number-Theoretic Functions
βΊTo calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). …
9: 17.6 ${{}_{2}\phi_{1}}$ Function
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17.6.5 ${{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},$ $|b|>|q|$.
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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. …
10: 17.4 Basic Hypergeometric Functions
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17.4.3 ${{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.$
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17.4.6 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a;q\right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y% ^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}},$
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17.4.7 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a,b;q\right)_{m}\left(a^{\prime},b^{\prime};q\right)_{n}x^{m}y^{n}}{% \left(q;q\right)_{m}\left(q;q\right)_{n}\left(c;q\right)_{m+n}},$
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17.4.8 $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a,b;% q\right)_{m+n}x^{m}y^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n% }}.$