17.6

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
17.6.E1	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}}}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{\ifrac{c}{(ab)}} = \frac{\qmultiPochhammersym{c/a,c/b}{q}{\infty}}{\qmultiPochhammersym{c,c/(ab)}{q}{\infty}}		Error	QHypergeometricPFQ[{a , b},{c},q,Divide[c,a*b]] == Divide[Product[QPochhammer[Part[{c/a , c/b},i],q,Infinity],{i,1,Length[{c/a , c/b}]}],Product[QPochhammer[Part[{c , c/(a*b)},i],q,Infinity],{i,1,Length[{c , c/(a*b)}]}]]	Missing Macro Error	Failure	-	Failed [262 / 300] Result: Plus[0.0, QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], -0.6666666666666666]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[0.0, QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[-0.4999999999999998, 0.8660254037844387], -0.6666666666666666]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
17.6.E2	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}{a}\right)=\frac{\left(c/a;q\right)_{n}}{\left(c;q\right)_{n}}}}$ \qgenhyperphi{2}{1}@@{a,q^{-n}}{c}{q}{\ifrac{cq^{n}}{a}} = \frac{\qPochhammer{c/a}{q}{n}}{\qPochhammer{c}{q}{n}}		Error	QHypergeometricPFQ[{a , (q)^(- n)},{c},q,Divide[c*(q)^(n),a]] == Divide[QPochhammer[c/a, q, n],QPochhammer[c, q, n]]	Missing Macro Error	Failure	-	Failed [204 / 300] Result: Plus[0.0, QHypergeometricPFQ[{-1.5, Complex[0.8660254037844387, -0.49999999999999994]} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[a, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.0, 0.0], QHypergeometricPFQ[{-1.5, Complex[0.5000000000000001, -0.8660254037844386]} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.5000000000000001, 0.8660254037844386]]], {Rule[a, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
17.6.E3	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,q\right)=\frac{a^{n}\left(c/a;q\right)_{n}}{\left(c;q\right)_{n}}}}$ \qgenhyperphi{2}{1}@@{a,q^{-n}}{c}{q}{q} = \frac{a^{n}\qPochhammer{c/a}{q}{n}}{\qPochhammer{c}{q}{n}}		Error	QHypergeometricPFQ[{a , (q)^(- n)},{c},q,q] == Divide[(a)^(n)* QPochhammer[c/a, q, n],QPochhammer[c, q, n]]	Missing Macro Error	Failure	-	Failed [168 / 300] Result: Plus[0.0, QHypergeometricPFQ[{-1.5, Complex[0.8660254037844387, -0.49999999999999994]} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[a, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.0, 0.0], QHypergeometricPFQ[{-1.5, Complex[0.5000000000000001, -0.8660254037844386]} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[a, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
17.6.E4	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({b^{2},\ifrac{b^{2}}{c}\atop c};q^{2},\ifrac{cq}{b^{2}}\right)=\frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(c,cq/b^{2};q^{2}\right)_{\infty}}\left(\frac{\left(c/b;q\right)_{\infty}}{\left(b;q\right)_{\infty}}+\frac{\left(-c/b;q\right)_{\infty}}{\left(-b;q\right)_{\infty}}\right)}}$ \qgenhyperphi{2}{1}@@{b^{2},\ifrac{b^{2}}{c}}{c}{q^{2}}{\ifrac{cq}{b^{2}}} = \frac{1}{2}\frac{\qmultiPochhammersym{b^{2},q}{q^{2}}{\infty}}{\qmultiPochhammersym{c,cq/b^{2}}{q^{2}}{\infty}}\left(\frac{\qPochhammer{c/b}{q}{\infty}}{\qPochhammer{b}{q}{\infty}}+\frac{\qPochhammer{-c/b}{q}{\infty}}{\qPochhammer{-b}{q}{\infty}}\right)	${\displaystyle{\displaystyle|cq|<|b^{2}|}}$	Error	QHypergeometricPFQ[{(b)^(2),Divide[(b)^(2),c]},{c},(q)^(2),Divide[c*q,(b)^(2)]] == Divide[1,2]*Divide[Product[QPochhammer[Part[{(b)^(2), q},i],(q)^(2),Infinity],{i,1,Length[{(b)^(2), q}]}],Product[QPochhammer[Part[{c , c*q/(b)^(2)},i],(q)^(2),Infinity],{i,1,Length[{c , c*q/(b)^(2)}]}]]*(Divide[QPochhammer[c/b, q, Infinity],QPochhammer[b, q, Infinity]]+Divide[QPochhammer[- c/b, q, Infinity],QPochhammer[- b, q, Infinity]])	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E5	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{\infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q\right)_{\infty}}}}$ \qgenhyperphi{2}{1}@@{a,b}{aq/b}{q}{-q/b} = \frac{\qPochhammer{-q}{q}{\infty}\qmultiPochhammersym{aq,\ifrac{aq^{2}}{b^{2}}}{q^{2}}{\infty}}{\qmultiPochhammersym{-q/b,aq/b}{q}{\infty}}	${\displaystyle{\displaystyle|b|>|q|}}$	Error	QHypergeometricPFQ[{a , b},{a*q/b},q,- q/b] == Divide[QPochhammer[- q, q, Infinity]*Product[QPochhammer[Part[{a*q ,Divide[a*(q)^(2),(b)^(2)]},i],(q)^(2),Infinity],{i,1,Length[{a*q ,Divide[a*(q)^(2),(b)^(2)]}]}],Product[QPochhammer[Part[{- q/b , a*q/b},i],q,Infinity],{i,1,Length[{- q/b , a*q/b}]}]]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E6	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,az;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/b,z\atop az};q,b\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \frac{\qmultiPochhammersym{b,az}{q}{\infty}}{\qmultiPochhammersym{c,z}{q}{\infty}}\qgenhyperphi{2}{1}@@{c/b,z}{az}{q}{b}	${\displaystyle{\displaystyle|z|<1,|b|<1}}$	Error	QHypergeometricPFQ[{a , b},{c},q,z] == Divide[Product[QPochhammer[Part[{b , a*z},i],q,Infinity],{i,1,Length[{b , a*z}]}],Product[QPochhammer[Part[{c , z},i],q,Infinity],{i,1,Length[{c , z}]}]]*QHypergeometricPFQ[{c/b , z},{a*z},q,b]	Missing Macro Error	Failure	-	Skip - No test values generated
17.6.E7	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/b,bz;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({\ifrac{abz}{c},b\atop bz};q,c/b\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \frac{\qmultiPochhammersym{c/b,bz}{q}{\infty}}{\qmultiPochhammersym{c,z}{q}{\infty}}\qgenhyperphi{2}{1}@@{\ifrac{abz}{c},b}{bz}{q}{c/b}	${\displaystyle{\displaystyle|z|<1,|c|<|b|}}$	Error	QHypergeometricPFQ[{a , b},{c},q,z] == Divide[Product[QPochhammer[Part[{c/b , b*z},i],q,Infinity],{i,1,Length[{c/b , b*z}]}],Product[QPochhammer[Part[{c , z},i],q,Infinity],{i,1,Length[{c , z}]}]]*QHypergeometricPFQ[{Divide[a*b*z,c], b},{b*z},q,c/b]	Missing Macro Error	Failure	-	Skip - No test values generated
17.6.E8	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(\ifrac{abz}{c};q\right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,c/b\atop c};q,\ifrac{abz}{c}\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \frac{\qPochhammer{\ifrac{abz}{c}}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}\qgenhyperphi{2}{1}@@{c/a,c/b}{c}{q}{\ifrac{abz}{c}}	${\displaystyle{\displaystyle|z|<1,|abz|<|c|}}$	Error	QHypergeometricPFQ[{a , b},{c},q,z] == Divide[QPochhammer[Divide[a*b*z,c], q, Infinity],QPochhammer[z, q, Infinity]]*QHypergeometricPFQ[{c/a , c/b},{c},q,Divide[a*b*z,c]]	Missing Macro Error	Failure	-	Skip - No test values generated
17.6.E9	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=-\frac{(1-b)(aq/b)}{(1-(\ifrac{aq}{b}))}\sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q\right)_{n}q^{n}}{\left(azq^{2}/b;q\right)_{n}}+\frac{\left(aq,azq/b;q\right)_{\infty}}{\left(aq/b;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({q,0\atop bq};q,z\right)}}$ \qgenhyperphi{2}{1}@@{q,aq}{bq}{q}{z} = -\frac{(1-b)(aq/b)}{(1-(\ifrac{aq}{b}))}\sum_{n=0}^{\infty}\frac{\qmultiPochhammersym{aq,azq/b}{q}{n}q^{n}}{\qPochhammer{azq^{2}/b}{q}{n}}+\frac{\qmultiPochhammersym{aq,azq/b}{q}{\infty}}{\qPochhammer{aq/b}{q}{\infty}}\qgenhyperphi{2}{1}@@{q,0}{bq}{q}{z}	${\displaystyle{\displaystyle|z|<1}}$	Error	QHypergeometricPFQ[{q , a*q},{b*q},q,z] == -Divide[(1 - b)*(a*q/b),1 -(Divide[a*q,b])]*Sum[Divide[Product[QPochhammer[Part[{a*q , a*z*q/b},i],q,n],{i,1,Length[{a*q , a*z*q/b}]}]*(q)^(n),QPochhammer[a*z*(q)^(2)/b, q, n]], {n, 0, Infinity}, GenerateConditions->None]+Divide[Product[QPochhammer[Part[{a*q , a*z*q/b},i],q,Infinity],{i,1,Length[{a*q , a*z*q/b}]}],QPochhammer[a*q/b, q, Infinity]]*QHypergeometricPFQ[{q , 0},{b*q},q,z]	Missing Macro Error	Aborted	-	Skipped - Because timed out
17.6.E10	${\displaystyle{\displaystyle(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{\left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}}}}$ (1-z)\qgenhyperphi{2}{1}@@{q,aq}{bq}{q}{z} = \sum_{n=0}^{\infty}\frac{\qPochhammer{b/a}{q}{n}(-az)^{n}q^{(n^{2}+n)/2}}{\qmultiPochhammersym{bq,zq}{q}{n}}	${\displaystyle{\displaystyle|z|<1}}$	Error	(1 - z)*QHypergeometricPFQ[{q , a*q},{b*q},q,z] == Sum[Divide[QPochhammer[b/a, q, n]*(- a*z)^(n)* (q)^(((n)^(2)+ n)/2),Product[QPochhammer[Part[{b*q , z*q},i],q,n],{i,1,Length[{b*q , z*q}]}]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
17.6.E11	${\displaystyle{\displaystyle\frac{1-z}{1-b}{{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right)_{2n}b^{n}}{\left(zq,aq/b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/b;q\right)_{n+1}}}}$ \frac{1-z}{1-b}\qgenhyperphi{2}{1}@@{q,aq}{bq}{q}{z} = \sum_{n=0}^{\infty}\frac{\qPochhammer{aq}{q}{n}\qPochhammer{azq/b}{q}{2n}b^{n}}{\qmultiPochhammersym{zq,aq/b}{q}{n}}-aq\sum_{n=0}^{\infty}\frac{\qPochhammer{aq}{q}{n}\qPochhammer{azq/b}{q}{2n+1}(bq)^{n}}{\qPochhammer{zq}{q}{n}\qPochhammer{aq/b}{q}{n+1}}	${\displaystyle{\displaystyle|z|<1,|b|<1}}$	Error	Divide[1 - z,1 - b]*QHypergeometricPFQ[{q , a*q},{b*q},q,z] == Sum[Divide[QPochhammer[a*q, q, n]*QPochhammer[a*z*q/b, q, 2*n]*(b)^(n),Product[QPochhammer[Part[{z*q , a*q/b},i],q,n],{i,1,Length[{z*q , a*q/b}]}]], {n, 0, Infinity}, GenerateConditions->None]- a*q*Sum[Divide[QPochhammer[a*q, q, n]*QPochhammer[a*z*q/b, q, 2*n + 1]*(b*q)^(n),QPochhammer[z*q, q, n]*QPochhammer[a*q/b, q, n + 1]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
17.6.E12	${\displaystyle{\displaystyle(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q\right)_{n}}{\left(bq,zq;q\right)_{n}}(1-azq^{2n+1})(bz)^{n}q^{n^{2}}}}$ (1-z)\qgenhyperphi{2}{1}@@{q,aq}{bq}{q}{z} = \sum_{n=0}^{\infty}\frac{\qmultiPochhammersym{aq,azq/b}{q}{n}}{\qmultiPochhammersym{bq,zq}{q}{n}}(1-azq^{2n+1})(bz)^{n}q^{n^{2}}	${\displaystyle{\displaystyle|z|<1}}$	Error	(1 - z)*QHypergeometricPFQ[{q , a*q},{b*q},q,z] == Sum[Divide[Product[QPochhammer[Part[{a*q , a*z*q/b},i],q,n],{i,1,Length[{a*q , a*z*q/b}]}],Product[QPochhammer[Part[{b*q , z*q},i],q,n],{i,1,Length[{b*q , z*q}]}]]*(1 - a*z*(q)^(2*n + 1))*(b*z)^(n)* (q)^((n)^(2)), {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
17.6.E13	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{\left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{\infty}}}}$ \qgenhyperphi{2}{1}@{a,b}{c}{q}{q}+\frac{\qmultiPochhammersym{q/c,a,b}{q}{\infty}}{\qmultiPochhammersym{c/q,aq/c,bq/c}{q}{\infty}}\qgenhyperphi{2}{1}@{aq/c,bq/c}{q^{2}/c}{q}{q} = \frac{\qmultiPochhammersym{q/c,abq/c}{q}{\infty}}{\qmultiPochhammersym{aq/c,bq/c}{q}{\infty}}		Error	QHypergeometricPFQ[{a , b},{c},q,q]+Divide[Product[QPochhammer[Part[{q/c , a , b},i],q,Infinity],{i,1,Length[{q/c , a , b}]}],Product[QPochhammer[Part[{c/q , a*q/c , b*q/c},i],q,Infinity],{i,1,Length[{c/q , a*q/c , b*q/c}]}]]*QHypergeometricPFQ[{a*q/c , b*q/c},{(q)^(2)/c},q,q] == Divide[Product[QPochhammer[Part[{q/c , a*b*q/c},i],q,Infinity],{i,1,Length[{q/c , a*b*q/c}]}],Product[QPochhammer[Part[{a*q/c , b*q/c},i],q,Infinity],{i,1,Length[{a*q/c , b*q/c}]}]]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E14	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}\left(b;q^{2}\right)_{n}z^{n}}{\left(q;q\right)_{n}\left(azb;q^{2}\right)_{n}}=\frac{\left(az,bz;q^{2}\right)_{\infty}}{\left(z,azb;q^{2}\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop bz};q^{2},zq\right)}}$ \sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}\qPochhammer{b}{q^{2}}{n}z^{n}}{\qPochhammer{q}{q}{n}\qPochhammer{azb}{q^{2}}{n}} = \frac{\qmultiPochhammersym{az,bz}{q^{2}}{\infty}}{\qmultiPochhammersym{z,azb}{q^{2}}{\infty}}\qgenhyperphi{2}{1}@@{a,b}{bz}{q^{2}}{zq}		Error	Sum[Divide[QPochhammer[a, q, n]*QPochhammer[b, (q)^(2), n]*(z)^(n),QPochhammer[q, q, n]*QPochhammer[a*z*b, (q)^(2), n]], {n, 0, Infinity}, GenerateConditions->None] == Divide[Product[QPochhammer[Part[{a*z , b*z},i],(q)^(2),Infinity],{i,1,Length[{a*z , b*z}]}],Product[QPochhammer[Part[{z , a*z*b},i],(q)^(2),Infinity],{i,1,Length[{z , a*z*b}]}]]*QHypergeometricPFQ[{a , b},{b*z},(q)^(2),z*q]	Missing Macro Error	Aborted	-	Skipped - Because timed out
17.6.E15	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c,q/c;q\right)_{\infty}}{\left(az/c,q/a;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,cq/(abz)\atop cq/(az)};q,bq/c\right)-\frac{\left(b,q/c,c/a,az/q,q^{2}/(az);q\right)_{\infty}}{\left(c/q,bq/c,q/a,az/c,cq/(az);q\right)_{\infty}}{{}_{2}\phi_{1}}\left({aq/c,bq/c\atop q^{2}/c};q,z\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \frac{\qmultiPochhammersym{abz/c,q/c}{q}{\infty}}{\qmultiPochhammersym{az/c,q/a}{q}{\infty}}\qgenhyperphi{2}{1}@@{c/a,cq/(abz)}{cq/(az)}{q}{bq/c}-\frac{\qmultiPochhammersym{b,q/c,c/a,az/q,q^{2}/(az)}{q}{\infty}}{\qmultiPochhammersym{c/q,bq/c,q/a,az/c,cq/(az)}{q}{\infty}}\qgenhyperphi{2}{1}@@{aq/c,bq/c}{q^{2}/c}{q}{z}	${\displaystyle{\displaystyle|z|<1,|bq|<|c|}}$	Error	QHypergeometricPFQ[{a , b},{c},q,z] == Divide[Product[QPochhammer[Part[{a*b*z/c , q/c},i],q,Infinity],{i,1,Length[{a*b*z/c , q/c}]}],Product[QPochhammer[Part[{a*z/c , q/a},i],q,Infinity],{i,1,Length[{a*z/c , q/a}]}]]*QHypergeometricPFQ[{c/a , c*q/(a*b*z)},{c*q/(a*z)},q,b*q/c]-Divide[Product[QPochhammer[Part[{b , q/c , c/a , a*z/q , (q)^(2)/(a*z)},i],q,Infinity],{i,1,Length[{b , q/c , c/a , a*z/q , (q)^(2)/(a*z)}]}],Product[QPochhammer[Part[{c/q , b*q/c , q/a , a*z/c , c*q/(a*z)},i],q,Infinity],{i,1,Length[{c/q , b*q/c , q/a , a*z/c , c*q/(a*z)}]}]]*QHypergeometricPFQ[{a*q/c , b*q/c},{(q)^(2)/c},q,z]	Missing Macro Error	Failure	-	Skip - No test values generated
17.6.E16	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,c/a,az,q/(az);q\right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{\infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({b,bq/c\atop bq/a};q,cq/(abz)\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \frac{\qmultiPochhammersym{b,c/a,az,q/(az)}{q}{\infty}}{\qmultiPochhammersym{c,b/a,z,q/z}{q}{\infty}}\qgenhyperphi{2}{1}@@{a,aq/c}{aq/b}{q}{cq/(abz)}+\frac{\qmultiPochhammersym{a,c/b,bz,q/(bz)}{q}{\infty}}{\qmultiPochhammersym{c,a/b,z,q/z}{q}{\infty}}\qgenhyperphi{2}{1}@@{b,bq/c}{bq/a}{q}{cq/(abz)}	${\displaystyle{\displaystyle|z|<1,|abz|<|cq|}}$	Error	QHypergeometricPFQ[{a , b},{c},q,z] == Divide[Product[QPochhammer[Part[{b , c/a , a*z , q/(a*z)},i],q,Infinity],{i,1,Length[{b , c/a , a*z , q/(a*z)}]}],Product[QPochhammer[Part[{c , b/a , z , q/z},i],q,Infinity],{i,1,Length[{c , b/a , z , q/z}]}]]*QHypergeometricPFQ[{a , a*q/c},{a*q/b},q,c*q/(a*b*z)]+Divide[Product[QPochhammer[Part[{a , c/b , b*z , q/(b*z)},i],q,Infinity],{i,1,Length[{a , c/b , b*z , q/(b*z)}]}],Product[QPochhammer[Part[{c , a/b , z , q/z},i],q,Infinity],{i,1,Length[{c , a/b , z , q/z}]}]]*QHypergeometricPFQ[{b , b*q/c},{b*q/a},q,c*q/(a*b*z)]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E17	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=cz\frac{(1-a)(1-b)}{(q-c)(1-c)}{{}_{2}\phi_{1}}\left({aq,bq\atop cq};q,z\right)}}$ \qgenhyperphi{2}{1}@@{a,b}{c/q}{q}{z}-\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = cz\frac{(1-a)(1-b)}{(q-c)(1-c)}\qgenhyperphi{2}{1}@@{aq,bq}{cq}{q}{z}		Error	QHypergeometricPFQ[{a , b},{c/q},q,z]- QHypergeometricPFQ[{a , b},{c},q,z] == c*z*Divide[(1 - a)*(1 - b),(q - c)*(1 - c)]*QHypergeometricPFQ[{a*q , b*q},{c*q},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E18	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=az\frac{1-b}{1-c}{{}_{2}\phi_{1}}\left({aq,bq\atop cq};q,z\right)}}$ \qgenhyperphi{2}{1}@@{aq,b}{c}{q}{z}-\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = az\frac{1-b}{1-c}\qgenhyperphi{2}{1}@@{aq,bq}{cq}{q}{z}		Error	QHypergeometricPFQ[{a*q , b},{c},q,z]- QHypergeometricPFQ[{a , b},{c},q,z] == a*z*Divide[1 - b,1 - c]*QHypergeometricPFQ[{a*q , b*q},{c*q},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E19	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({aq,b\atop cq};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}{{}_{2}\phi_{1}}\left({aq,bq\atop cq^{2}};q,z\right)}}$ \qgenhyperphi{2}{1}@@{aq,b}{cq}{q}{z}-\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}\qgenhyperphi{2}{1}@@{aq,bq}{cq^{2}}{q}{z}		Error	QHypergeometricPFQ[{a*q , b},{c*q},q,z]- QHypergeometricPFQ[{a , b},{c},q,z] == a*z*Divide[(1 - b)*(1 -(c/a)),(1 - c)*(1 - c*q)]*QHypergeometricPFQ[{a*q , b*q},{c*(q)^(2)},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E20	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({aq,b/q\atop c};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=az\frac{(1-b/(aq))}{1-c}{{}_{2}\phi_{1}}\left({aq,b\atop cq};q,z\right)}}$ \qgenhyperphi{2}{1}@@{aq,b/q}{c}{q}{z}-\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = az\frac{(1-b/(aq))}{1-c}\qgenhyperphi{2}{1}@@{aq,b}{cq}{q}{z}		Error	QHypergeometricPFQ[{a*q , b/q},{c},q,z]- QHypergeometricPFQ[{a , b},{c},q,z] == a*z*Divide[1 - b/(a*q),1 - c]*QHypergeometricPFQ[{a*q , b},{c*q},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E21	${\displaystyle{\displaystyle b(1-a){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-a(1-b){{}_{2}\phi_{1}}\left({a,bq\atop c};q,z\right)=(b-a){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)}}$ b(1-a)\qgenhyperphi{2}{1}@@{aq,b}{c}{q}{z}-a(1-b)\qgenhyperphi{2}{1}@@{a,bq}{c}{q}{z} = (b-a)\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}		Error	b*(1 - a)*QHypergeometricPFQ[{a*q , b},{c},q,z]- a*(1 - b)*QHypergeometricPFQ[{a , b*q},{c},q,z] == (b - a)*QHypergeometricPFQ[{a , b},{c},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E22	${\displaystyle{\displaystyle a\left(1-\frac{b}{c}\right){{}_{2}\phi_{1}}\left({a,b/q\atop c};q,z\right)-b\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)=(a-b)\left(1-\frac{abz}{cq}\right){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)}}$ a\left(1-\frac{b}{c}\right)\qgenhyperphi{2}{1}@@{a,b/q}{c}{q}{z}-b\left(1-\frac{a}{c}\right)\qgenhyperphi{2}{1}@@{a/q,b}{c}{q}{z} = (a-b)\left(1-\frac{abz}{cq}\right)\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}		Error	a*(1 -Divide[b,c])*QHypergeometricPFQ[{a , b/q},{c},q,z]- b*(1 -Divide[a,c])*QHypergeometricPFQ[{a/q , b},{c},q,z] == (a - b)*(1 -Divide[a*b*z,c*q])*QHypergeometricPFQ[{a , b},{c},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E23	${\displaystyle{\displaystyle q\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)+(1-a)\left(1-\frac{abz}{c}\right){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)=\left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac{abz}{c}\right){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)}}$ q\left(1-\frac{a}{c}\right)\qgenhyperphi{2}{1}@@{a/q,b}{c}{q}{z}+(1-a)\left(1-\frac{abz}{c}\right)\qgenhyperphi{2}{1}@@{aq,b}{c}{q}{z} = \left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac{abz}{c}\right)\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}		Error	q*(1 -Divide[a,c])*QHypergeometricPFQ[{a/q , b},{c},q,z]+(1 - a)*(1 -Divide[a*b*z,c])*QHypergeometricPFQ[{a*q , b},{c},q,z] == (1 + q - a -Divide[a*q,c]+Divide[(a)^(2)* z,c]-Divide[a*b*z,c])*QHypergeometricPFQ[{a , b},{c},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E24	${\displaystyle{\displaystyle(1-c)(q-c)(abz-c){{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)+z(c-a)(c-b){{}_{2}\phi_{1}}\left({a,b\atop cq};q,z\right)=(c-1)(c(q-c)+z(ca+cb-ab-abq)){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)}}$ (1-c)(q-c)(abz-c)\qgenhyperphi{2}{1}@@{a,b}{c/q}{q}{z}+z(c-a)(c-b)\qgenhyperphi{2}{1}@@{a,b}{cq}{q}{z} = (c-1)(c(q-c)+z(ca+cb-ab-abq))\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}		Error	(1 - c)*(q - c)*(a*b*z - c)*QHypergeometricPFQ[{a , b},{c/q},q,z]+ z*(c - a)*(c - b)*QHypergeometricPFQ[{a , b},{c*q},q,z] == (c - 1)*(c*(q - c)+ z*(c*a + c*b - a*b - a*b*q))*QHypergeometricPFQ[{a , b},{c},q,z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E25	${\displaystyle{\displaystyle\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq^{n},bq^{n}\atop cq^{n}};q,dz\right)}}$ \mathcal{D}_{q}^{n}\qgenhyperphi{2}{1}@@{a,b}{c}{q}{zd} = \frac{\qmultiPochhammersym{a,b}{q}{n}d^{n}}{\qPochhammer{c}{q}{n}(1-q)^{n}}\qgenhyperphi{2}{1}@@{aq^{n},bq^{n}}{cq^{n}}{q}{dz}		Error	(Subscript[D, q])^(n)*QHypergeometricPFQ[{a , b},{c},q,z*d] == Divide[Product[QPochhammer[Part[{a , b},i],q,n],{i,1,Length[{a , b}]}]*(d)^(n),QPochhammer[c, q, n]*(1 - q)^(n)]*QHypergeometricPFQ[{a*(q)^(n), b*(q)^(n)},{c*(q)^(n)},q,d*z]	Missing Macro Error	Failure	-	Skipped - Because timed out
17.6.E26	${\displaystyle{\displaystyle\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}\frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right)}}$ \mathcal{D}_{q}^{n}\left(\frac{\qPochhammer{z}{q}{\infty}}{\qPochhammer{abz/c}{q}{\infty}}\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}\right) = \frac{\qmultiPochhammersym{c/a,c/b}{q}{n}}{\qPochhammer{c}{q}{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}\frac{\qPochhammer{zq^{n}}{q}{\infty}}{\qPochhammer{abz/c}{q}{\infty}}\qgenhyperphi{2}{1}@@{a,b}{cq^{n}}{q}{zq^{n}}		Error	(Subscript[D, q])^(n)[Divide[QPochhammer[z, q, Infinity],QPochhammer[a*b*z/c, q, Infinity]]*QHypergeometricPFQ[{a , b},{c},q,z]] == Divide[Product[QPochhammer[Part[{c/a , c/b},i],q,n],{i,1,Length[{c/a , c/b}]}],QPochhammer[c, q, n]*(1 - q)^(n)]*(Divide[a*b,c])^(n)*Divide[QPochhammer[z*(q)^(n), q, Infinity],QPochhammer[a*b*z/c, q, Infinity]]*QHypergeometricPFQ[{a , b},{c*(q)^(n)},q,z*(q)^(n)]	Missing Macro Error	Failure	-	Failed [264 / 300] Result: Plus[0.0, Times[Complex[0.8660254037844387, 0.49999999999999994], QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Power[QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[D, q], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.0, 0.0], Times[Complex[0.5000000000000001, 0.8660254037844386], QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Power[QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[D, q], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
17.6.E27	${\displaystyle{\displaystyle z(c-abqz)\mathcal{D}_{q}^{2}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)+\left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right)\mathcal{D}_{q}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)-\frac{(1-a)(1-b)}{(1-q)^{2}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=0}}$ z(c-abqz)\mathcal{D}_{q}^{2}\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}+\left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right)\mathcal{D}_{q}\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z}-\frac{(1-a)(1-b)}{(1-q)^{2}}\qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = 0		Error	z*(c - a*b*q*z)*(Subscript[D, q])^(2)*QHypergeometricPFQ[{a , b},{c},q,z]+(Divide[1 - c,1 - q]+Divide[(1 - a)*(1 - b)-(1 - a*b*q),1 - q]*z)*Subscript[D, q]*QHypergeometricPFQ[{a , b},{c},q,z]-Divide[(1 - a)*(1 - b),(1 - q)^(2)]*QHypergeometricPFQ[{a , b},{c},q,z] == 0	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Times[Complex[9.528684177437189, -1.3259618943233384], QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[D, q], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Times[Complex[5.290063509461103, -21.657849302036027], QHypergeometricPFQ[{-1.5, -1.5} Test Values: {-1.5}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[D, q], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
17.6.E29	${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\left(\frac{-1}{2\pi i}\right)\frac{\left(a,b;q\right)_{\infty}}{\left(q,c;q\right)_{\infty}}\int_{-i\infty}^{i\infty}\frac{\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}}{\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}}\frac{\pi(-z)^{\zeta}}{\sin\left(\pi\zeta\right)}\mathrm{d}\zeta}}$ \qgenhyperphi{2}{1}@@{a,b}{c}{q}{z} = \left(\frac{-1}{2\pi i}\right)\frac{\qmultiPochhammersym{a,b}{q}{\infty}}{\qmultiPochhammersym{q,c}{q}{\infty}}\int_{-i\infty}^{i\infty}\frac{\qmultiPochhammersym{q^{1+\zeta},cq^{\zeta}}{q}{\infty}}{\qmultiPochhammersym{aq^{\zeta},bq^{\zeta}}{q}{\infty}}\frac{\pi(-z)^{\zeta}}{\sin@{\pi\zeta}}\diff{\zeta}		Error	QHypergeometricPFQ[{a , b},{c},q,z] == (Divide[- 1,2*Pi*I])*Divide[Product[QPochhammer[Part[{a , b},i],q,Infinity],{i,1,Length[{a , b}]}],Product[QPochhammer[Part[{q , c},i],q,Infinity],{i,1,Length[{q , c}]}]]*Integrate[Divide[Product[QPochhammer[Part[{(q)^(1 + \[Zeta]), c*(q)^\[Zeta]},i],q,Infinity],{i,1,Length[{(q)^(1 + \[Zeta]), c*(q)^\[Zeta]}]}],Product[QPochhammer[Part[{a*(q)^\[Zeta], b*(q)^\[Zeta]},i],q,Infinity],{i,1,Length[{a*(q)^\[Zeta], b*(q)^\[Zeta]}]}]]*Divide[Pi*(- z)^\[Zeta],Sin[Pi*\[Zeta]]], {\[Zeta], - I*Infinity, I*Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out