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##### 1: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
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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 $\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).$
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17.10.6 $\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\*{{}_{10}\phi_{9}% }\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,% qg/h,qg/k};q,\frac{q^{2}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k% \right).$
##### 2: 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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17.9.14 ${{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$
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17.9.16 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{% bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{\infty}}{\left(% aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}\phi_{3}}\left({aq/(bc),d,e,f% \atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2% }q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}\phi_{3}}\left({aq/(de),aq/% (df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2% }/(def)};q,q\right).$
##### 3: 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).
##### 4: 26.18 Counting Techniques
βΊLet $A_{1},A_{2},\ldots,A_{n}$ be subsets of a set $S$ that are not necessarily disjoint. Then the number of elements in the set $S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})$ is βΊ
26.18.1 $\left|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right|=\left|S\right|+% \sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}
##### 5: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
βΊ
17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$
βΊ
17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.$
βΊ
17.8.4 ${{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},$
βΊ
17.8.6 ${{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},$
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17.8.7 ${{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}}.$
##### 6: 26.2 Basic Definitions
βΊA permutation is a one-to-one and onto function from a non-empty set to itself. If the set consists of the integers 1 through $n$, a permutation $\sigma$ can be thought of as a rearrangement of these integers where the integer in position $j$ is $\sigma(j)$. … βΊGiven a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\ell=\ell(j,k)$ such that $j=\sigma^{\ell}(k)$, where $\sigma^{1}=\sigma$ and $\sigma^{\ell}$ is the composition of $\sigma$ with $\sigma^{\ell-1}$. … βΊ
###### Partition
βΊA partition of a set $S$ is an unordered collection of pairwise disjoint nonempty sets whose union is $S$. …
##### 7: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions
βΊwhere $\lambda=-c(ab/q)^{\frac{1}{2}}$. … βΊ
17.7.10 ${{}_{8}\phi_{7}}\left({-c,q(-c)^{\frac{1}{2}},-q(-c)^{\frac{1}{2}},a,q/a,c,-d,% -q/d\atop(-c)^{\frac{1}{2}},-(-c)^{\frac{1}{2}},-cq/a,-ac,-q,cq/d,cd};q,c% \right)=\frac{\left(-c,-cq;q\right)_{\infty}\left(acd,acq/d,cdq/a,cq^{2}/(ad);% q^{2}\right)_{\infty}}{\left(cd,cq/d,-ac,-cq/a;q\right)_{\infty}}.$
βΊ
17.7.14 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}% \atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,q\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{n}}{\left(aq/b,aq/c,aq/d,aq% /(bcd);q\right)_{n}},$
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17.7.15 ${{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{aq}{bcd}\right)=\frac{% \left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/(% bcd);q\right)_{\infty}},$
βΊ
17.7.17 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,q\right)-\frac{b}{a}% \frac{\left(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f;q\right)_{\infty}}{\left(aq/b,% aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b^{2}q/a;q\right)_{\infty}}\*{{}_{8}% \phi_{7}}\left({b^{2}/a,qba^{-\frac{1}{2}},-qba^{-\frac{1}{2}},b,bc/a,bd/a,be/% a,bf/a\atop ba^{-\frac{1}{2}},-ba^{-\frac{1}{2}},bq/a,bq/c,bq/d,bq/e,bq/f};q,q% \right)=\frac{\left(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q% \right)_{\infty}}{\left(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q\right)_{% \infty}},$
##### 8: 26.1 Special Notation
βΊ βΊβΊβΊ
 $x$ real variable. … number of elements of a finite set $A$. …
βΊ
##### 10: 17.6 ${{}_{2}\phi_{1}}$ Function
βΊ βΊ
17.6.2 ${{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}{a}\right)=\frac{\left% (c/a;q\right)_{n}}{\left(c;q\right)_{n}}.$
βΊ
17.6.15 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c,q/c;q\right)_{% \infty}}{\left(az/c,q/a;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,cq/(abz)% \atop cq/(az)};q,bq/c\right)-\frac{\left(b,q/c,c/a,az/q,q^{2}/(az);q\right)_{% \infty}}{\left(c/q,bq/c,q/a,az/c,cq/(az);q\right)_{\infty}}{{}_{2}\phi_{1}}% \left({aq/c,bq/c\atop q^{2}/c};q,z\right),$ $|z|<1,|bq|<|c|$.
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17.6.16 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,c/a,az,q/(az);q% \right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a% ,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{% \infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({b,bq/c% \atop bq/a};q,cq/(abz)\right),$ $|z|<1$, $|cq|<|abz|$.
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. …