q -Hypergeometric and Related Functions - 17.7 Special Cases of Higher Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
17.7.E1 ϕ 2 2 ⁡ ( a , q / a - q , b ; q , - b ) = ( a ⁢ b , b ⁢ q / a ; q 2 ) ∞ ( b ; q ) ∞ q-hypergeometric-rphis 2 2 𝑎 𝑞 𝑎 𝑞 𝑏 𝑞 𝑏 q-multiple-Pochhammer 𝑎 𝑏 𝑏 𝑞 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑏 𝑞 {\displaystyle{\displaystyle{{}_{2}\phi_{2}}\left({a,q/a\atop-q,b};q,-b\right)% =\frac{\left(ab,bq/a;q^{2}\right)_{\infty}}{\left(b;q\right)_{\infty}}}}
\qgenhyperphi{2}{2}@@{a,q/a}{-q,b}{q}{-b} = \frac{\qmultiPochhammersym{ab,bq/a}{q^{2}}{\infty}}{\qPochhammer{b}{q}{\infty}}		 | b | < 1 𝑏 1 {\displaystyle{\displaystyle|b|<1}} 
Error

QHypergeometricPFQ[{a , q/a},{- q , b},q,- b] == Divide[Product[QPochhammer[Part[{a*b , b*q/a},i],(q)^(2),Infinity],{i,1,Length[{a*b , b*q/a}]}],QPochhammer[b, q, Infinity]]
Missing Macro Error Failure -
Failed [96 / 120]
Result: Plus[QHypergeometricPFQ[{-1.5, Complex[-0.5773502691896257, -0.33333333333333326]}
Test Values: {Complex[-0.8660254037844387, -0.49999999999999994], -0.5}, Complex[0.8660254037844387, 0.49999999999999994], 0.5], Times[-1.0, Power[QPochhammer[-0.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.28867513459481287, 0.16666666666666663], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], QPochhammer[0.75, Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]]]], {Rule[a, -1.5], Rule[b, -0.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.7.E2 ϕ 2 2 ⁡ ( a 2 , b 2 a ⁢ b ⁢ q 1 2 , - a ⁢ b ⁢ q 1 2 ; q , - q ) = ( a 2 ⁢ q , b 2 ⁢ q ; q 2 ) ∞ ( q , a 2 ⁢ b 2 ⁢ q ; q 2 ) ∞ q-hypergeometric-rphis 2 2 superscript 𝑎 2 superscript 𝑏 2 𝑎 𝑏 superscript 𝑞 1 2 𝑎 𝑏 superscript 𝑞 1 2 𝑞 𝑞 q-multiple-Pochhammer superscript 𝑎 2 𝑞 superscript 𝑏 2 𝑞 superscript 𝑞 2 q-multiple-Pochhammer 𝑞 superscript 𝑎 2 superscript 𝑏 2 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{{}_{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}}}}
\qgenhyperphi{2}{2}@@{a^{2},b^{2}}{abq^{\frac{1}{2}},-abq^{\frac{1}{2}}}{q}{-q} = \frac{\qmultiPochhammersym{a^{2}q,b^{2}q}{q^{2}}{\infty}}{\qmultiPochhammersym{q,a^{2}b^{2}q}{q^{2}}{\infty}}		 

Error

QHypergeometricPFQ[{(a)^(2), (b)^(2)},{a*b*(q)^(Divide[1,2]), - a*b*(q)^(Divide[1,2])},q,- q] == Divide[Product[QPochhammer[Part[{(a)^(2)* q , (b)^(2)* q},i],(q)^(2),Infinity],{i,1,Length[{(a)^(2)* q , (b)^(2)* q}]}],Product[QPochhammer[Part[{q , (a)^(2)* (b)^(2)* q},i],(q)^(2),Infinity],{i,1,Length[{q , (a)^(2)* (b)^(2)* q}]}]]
Missing Macro Error Failure -
Failed [240 / 300]
Result: Plus[QHypergeometricPFQ[{2.25, 2.25}
Test Values: {Complex[2.173333109150404, 0.5823428514806717], Complex[-2.173333109150404, -0.5823428514806717]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.8660254037844387, -0.49999999999999994]], Times[-1.0, Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], -1], Power[QPochhammer[Complex[1.948557158514987, 1.1249999999999998], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], 2], Power[QPochhammer[Complex[4.384253606658721, 2.5312499999999996], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], -1]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.7.E3 ϕ 2 2 ⁡ ( c 2 / b 2 , b 2 c , c ⁢ q ; q 2 , q ) = 1 2 ⁢ ( b 2 , q ; q 2 ) ∞ ( c , c ⁢ q ; q 2 ) ∞ ⁢ ( ( c / b ; q ) ∞ ( b ; q ) ∞ + ( - c / b ; q ) ∞ ( - b ; q ) ∞ ) q-hypergeometric-rphis 2 2 superscript 𝑐 2 superscript 𝑏 2 superscript 𝑏 2 𝑐 𝑐 𝑞 superscript 𝑞 2 𝑞 1 2 q-multiple-Pochhammer superscript 𝑏 2 𝑞 superscript 𝑞 2 q-multiple-Pochhammer 𝑐 𝑐 𝑞 superscript 𝑞 2 q-Pochhammer-symbol 𝑐 𝑏 𝑞 q-Pochhammer-symbol 𝑏 𝑞 q-Pochhammer-symbol 𝑐 𝑏 𝑞 q-Pochhammer-symbol 𝑏 𝑞 {\displaystyle{\displaystyle{{}_{2}\phi_{2}}\left({\ifrac{c^{2}}{b^{2}},b^{2}% \atop c,cq};q^{2},q\right)=\frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty% }}{\left(c,cq;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}{2}@@{\ifrac{c^{2}}{b^{2}},b^{2}}{c,cq}{q^{2}}{q} = \frac{1}{2}\frac{\qmultiPochhammersym{b^{2},q}{q^{2}}{\infty}}{\qmultiPochhammersym{c,cq}{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)}		 

Error

QHypergeometricPFQ[{Divide[(c)^(2),(b)^(2)], (b)^(2)},{c , c*q},(q)^(2),q] == 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},i],(q)^(2),Infinity],{i,1,Length[{c , c*q}]}]]*(Divide[QPochhammer[c/b, q, Infinity],QPochhammer[b, q, Infinity]]+Divide[QPochhammer[- c/b, q, Infinity],QPochhammer[- b, q, Infinity]])
Missing Macro Error Failure -
Failed [260 / 300]
Result: Plus[QHypergeometricPFQ[{1.0, 2.25}
Test Values: {-1.5, Complex[-1.299038105676658, -0.7499999999999999]}, Complex[0.5000000000000001, 0.8660254037844386], Complex[0.8660254037844387, 0.49999999999999994]], Times[-0.5, Power[QPochhammer[-1.5, Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], -1], Power[QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], Plus[0.0, Times[QPochhammer[-1.0, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], Power[QPochhammer[1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1]]], QPochhammer[2.25, Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]]]], {Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0<syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[b, -1.5], Rule[c, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.7.E4 ϕ 2 3 ⁡ ( a , b , q - n c , a ⁢ b ⁢ q 1 - n / c ; q , q ) = ( c / a , c / b ; q ) n ( c , c / ( a ⁢ b ) ; q ) n q-hypergeometric-rphis 3 2 𝑎 𝑏 superscript 𝑞 𝑛 𝑐 𝑎 𝑏 superscript 𝑞 1 𝑛 𝑐 𝑞 𝑞 q-multiple-Pochhammer 𝑐 𝑎 𝑐 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑐 𝑐 𝑎 𝑏 𝑞 𝑛 {\displaystyle{\displaystyle{{}_{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}}}}
\qgenhyperphi{3}{2}@@{a,b,q^{-n}}{c,abq^{1-n}/c}{q}{q} = \frac{\qmultiPochhammersym{c/a,c/b}{q}{n}}{\qmultiPochhammersym{c,c/(ab)}{q}{n}}		 

Error

QHypergeometricPFQ[{a , b , (q)^(- n)},{c , a*b*(q)^(1 - n)/c},q,q] == Divide[Product[QPochhammer[Part[{c/a , c/b},i],q,n],{i,1,Length[{c/a , c/b}]}],Product[QPochhammer[Part[{c , c/(a*b)},i],q,n],{i,1,Length[{c , c/(a*b)}]}]]
Missing Macro Error Failure -
Failed [196 / 300]
Result: Plus[0.0, QHypergeometricPFQ[{-1.5, -1.5, Complex[0.8660254037844387, -0.49999999999999994]}
Test Values: {-1.5, -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[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.0, 0.0], QHypergeometricPFQ[{-1.5, -1.5, Complex[0.5000000000000001, -0.8660254037844386]}
Test Values: {-1.5, Complex[-1.299038105676658, 0.7499999999999999]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {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]]]}

... skip entries to safe data
17.7.E5 ϕ 2 3 ⁡ ( a , b , c e , f ; q , q ) + ( q / e , a , b , c , q ⁢ f / e ; q ) ∞ ( e / q , a ⁢ q / e , b ⁢ q / e , c ⁢ q / e , f ; q ) ∞ ⁢ ϕ 2 3 ⁡ ( a ⁢ q / e , b ⁢ q / e , c ⁢ q / e q 2 / e , q ⁢ f / e ; q , q ) = ( q / e , f / a , f / b , f / c ; q ) ∞ ( a ⁢ q / e , b ⁢ q / e , c ⁢ q / e , f ; q ) ∞ q-hypergeometric-rphis 3 2 𝑎 𝑏 𝑐 𝑒 𝑓 𝑞 𝑞 q-multiple-Pochhammer 𝑞 𝑒 𝑎 𝑏 𝑐 𝑞 𝑓 𝑒 𝑞 q-multiple-Pochhammer 𝑒 𝑞 𝑎 𝑞 𝑒 𝑏 𝑞 𝑒 𝑐 𝑞 𝑒 𝑓 𝑞 q-hypergeometric-rphis 3 2 𝑎 𝑞 𝑒 𝑏 𝑞 𝑒 𝑐 𝑞 𝑒 superscript 𝑞 2 𝑒 𝑞 𝑓 𝑒 𝑞 𝑞 q-multiple-Pochhammer 𝑞 𝑒 𝑓 𝑎 𝑓 𝑏 𝑓 𝑐 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑒 𝑏 𝑞 𝑒 𝑐 𝑞 𝑒 𝑓 𝑞 {\displaystyle{\displaystyle{{}_{3}\phi_{2}}\left({a,b,c\atop e,f};q,q\right)+% \frac{\left(q/e,a,b,c,qf/e;q\right)_{\infty}}{\left(e/q,aq/e,bq/e,cq/e,f;q% \right)_{\infty}}\*{{}_{3}\phi_{2}}\left({aq/e,bq/e,cq/e\atop q^{2}/e,qf/e};q,% q\right)=\frac{\left(q/e,f/a,f/b,f/c;q\right)_{\infty}}{\left(aq/e,bq/e,cq/e,f% ;q\right)_{\infty}}}}
\qgenhyperphi{3}{2}@@{a,b,c}{e,f}{q}{q}+\frac{\qmultiPochhammersym{q/e,a,b,c,qf/e}{q}{\infty}}{\qmultiPochhammersym{e/q,aq/e,bq/e,cq/e,f}{q}{\infty}}\*\qgenhyperphi{3}{2}@@{aq/e,bq/e,cq/e}{q^{2}/e,qf/e}{q}{q} = \frac{\qmultiPochhammersym{q/e,f/a,f/b,f/c}{q}{\infty}}{\qmultiPochhammersym{aq/e,bq/e,cq/e,f}{q}{\infty}}		 

Error

QHypergeometricPFQ[{a , b , c},{e , f},q,q]+Divide[Product[QPochhammer[Part[{q/e , a , b , c , q*f/e},i],q,Infinity],{i,1,Length[{q/e , a , b , c , q*f/e}]}],Product[QPochhammer[Part[{e/q , a*q/e , b*q/e , c*q/e , f},i],q,Infinity],{i,1,Length[{e/q , a*q/e , b*q/e , c*q/e , f}]}]]* QHypergeometricPFQ[{a*q/e , b*q/e , c*q/e},{(q)^(2)/e , q*f/e},q,q] == Divide[Product[QPochhammer[Part[{q/e , f/a , f/b , f/c},i],q,Infinity],{i,1,Length[{q/e , f/a , f/b , f/c}]}],Product[QPochhammer[Part[{a*q/e , b*q/e , c*q/e , f},i],q,Infinity],{i,1,Length[{a*q/e , b*q/e , c*q/e , f}]}]]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Plus[0.0, QHypergeometricPFQ[{-1.5, -1.5, -1.5}
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Times[QHypergeometricPFQ[{Complex[-1.5, 0.0], Complex[-1.5, 0.0], Complex[-1.5, 0.0]}, {Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Power[QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], 3], Power[QPochhammer[Complex[-1.5, 0.0], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -3]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Co<syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.7.E6 ϕ 2 3 ⁡ ( q - 2 ⁢ n , b , c q 1 - 2 ⁢ n / b , q 1 - 2 ⁢ n / c ; q , q 2 - n b ⁢ c ) = ( b , c ; q ) n ⁢ ( q , b ⁢ c ; q ) 2 ⁢ n ( q , b ⁢ c ; q ) n ⁢ ( b , c ; q ) 2 ⁢ n q-hypergeometric-rphis 3 2 superscript 𝑞 2 𝑛 𝑏 𝑐 superscript 𝑞 1 2 𝑛 𝑏 superscript 𝑞 1 2 𝑛 𝑐 𝑞 superscript 𝑞 2 𝑛 𝑏 𝑐 q-multiple-Pochhammer 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑞 𝑏 𝑐 𝑞 2 𝑛 q-multiple-Pochhammer 𝑞 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑏 𝑐 𝑞 2 𝑛 {\displaystyle{\displaystyle{{}_{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}}}}
\qgenhyperphi{3}{2}@@{q^{-2n},b,c}{q^{1-2n}/b,q^{1-2n}/c}{q}{\frac{q^{2-n}}{bc}} = \frac{\qmultiPochhammersym{b,c}{q}{n}\qmultiPochhammersym{q,bc}{q}{2n}}{\qmultiPochhammersym{q,bc}{q}{n}\qmultiPochhammersym{b,c}{q}{2n}}		 

Error

QHypergeometricPFQ[{(q)^(- 2*n), b , c},{(q)^(1 - 2*n)/b , (q)^(1 - 2*n)/c},q,Divide[(q)^(2 - n),b*c]] == Divide[Product[QPochhammer[Part[{b , c},i],q,n],{i,1,Length[{b , c}]}]*Product[QPochhammer[Part[{q , b*c},i],q,2*n],{i,1,Length[{q , b*c}]}],Product[QPochhammer[Part[{q , b*c},i],q,n],{i,1,Length[{q , b*c}]}]*Product[QPochhammer[Part[{b , c},i],q,2*n],{i,1,Length[{b , c}]}]]
Missing Macro Error Failure - Skipped - Because timed out
17.7.E7 ϕ 3 4 ⁡ ( a , - q ⁢ a 1 2 , b , c - a 1 2 , a ⁢ q / b , a ⁢ q / c ; q , q ⁢ a 1 2 b ⁢ c ) = ( a ⁢ q , q ⁢ a 1 2 / b , q ⁢ a 1 2 / c , a ⁢ q / ( b ⁢ c ) ; q ) ∞ ( a ⁢ q / b , a ⁢ q / c , q ⁢ a 1 2 , q ⁢ a 1 2 / ( b ⁢ c ) ; q ) ∞ q-hypergeometric-rphis 4 3 𝑎 𝑞 superscript 𝑎 1 2 𝑏 𝑐 superscript 𝑎 1 2 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑞 𝑞 superscript 𝑎 1 2 𝑏 𝑐 q-multiple-Pochhammer 𝑎 𝑞 𝑞 superscript 𝑎 1 2 𝑏 𝑞 superscript 𝑎 1 2 𝑐 𝑎 𝑞 𝑏 𝑐 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑞 superscript 𝑎 1 2 𝑞 superscript 𝑎 1 2 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{{}_{4}\phi_{3}}\left({a,-qa^{\frac{1}{2}},b,c% \atop-a^{\frac{1}{2}},aq/b,aq/c};q,\frac{qa^{\frac{1}{2}}}{bc}\right)=\frac{% \left(aq,qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,aq/(bc);q\right)_{\infty}}{% \left(aq/b,aq/c,qa^{\frac{1}{2}},qa^{\frac{1}{2}}/(bc);q\right)_{\infty}}}}
\qgenhyperphi{4}{3}@@{a,-qa^{\frac{1}{2}},b,c}{-a^{\frac{1}{2}},aq/b,aq/c}{q}{\frac{qa^{\frac{1}{2}}}{bc}} = \frac{\qmultiPochhammersym{aq,qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,aq/(bc)}{q}{\infty}}{\qmultiPochhammersym{aq/b,aq/c,qa^{\frac{1}{2}},qa^{\frac{1}{2}}/(bc)}{q}{\infty}}		 

Error

QHypergeometricPFQ[{a , - q*(a)^(Divide[1,2]), b , c},{- (a)^(Divide[1,2]), a*q/b , a*q/c},q,Divide[q*(a)^(Divide[1,2]),b*c]] == Divide[Product[QPochhammer[Part[{a*q , q*(a)^(Divide[1,2])/b , q*(a)^(Divide[1,2])/c , a*q/(b*c)},i],q,Infinity],{i,1,Length[{a*q , q*(a)^(Divide[1,2])/b , q*(a)^(Divide[1,2])/c , a*q/(b*c)}]}],Product[QPochhammer[Part[{a*q/b , a*q/c , q*(a)^(Divide[1,2]), q*(a)^(Divide[1,2])/(b*c)},i],q,Infinity],{i,1,Length[{a*q/b , a*q/c , q*(a)^(Divide[1,2]), q*(a)^(Divide[1,2])/(b*c)}]}]]
Missing Macro Error Failure -
Failed [248 / 300]
Result: Plus[QHypergeometricPFQ[{-1.5, Complex[0.6123724356957944, -1.0606601717798212], -1.5, -1.5}
Test Values: {Complex[0.0, -1.224744871391589], Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.27216552697590857, 0.4714045207910316]], Times[-1.0, QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], Power[QPochhammer[Complex[-0.6123724356957944, 1.0606601717798212], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[-0.5773502691896257, -0.33333333333333326], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], Power[QPochhammer[Complex[-0.27216552697590857, 0.4714045207910316], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], Power[QPochhammer[Complex[0.4082482904638<syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {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
17.7.E8 ϕ 7 8 ⁡ ( λ , q ⁢ λ 1 2 , - q ⁢ λ 1 2 , a , b , c , - c , λ ⁢ q / c 2 λ 1 2 , - λ 1 2 , λ ⁢ q / a , λ ⁢ q / b , λ ⁢ q / c , - λ ⁢ q / c , c 2 ; q , - λ ⁢ q a ⁢ b ) = ( λ ⁢ q , c 2 / λ ; q ) ∞ ⁢ ( a ⁢ q , b ⁢ q , c 2 ⁢ q / a , c 2 ⁢ q / b ; q 2 ) ∞ ( λ ⁢ q / a , λ ⁢ q / b ; q ) ∞ ⁢ ( q , a ⁢ b ⁢ q , c 2 ⁢ q , c 2 ⁢ q / ( a ⁢ b ) ; q 2 ) ∞ q-hypergeometric-rphis 8 7 𝜆 𝑞 superscript 𝜆 1 2 𝑞 superscript 𝜆 1 2 𝑎 𝑏 𝑐 𝑐 𝜆 𝑞 superscript 𝑐 2 superscript 𝜆 1 2 superscript 𝜆 1 2 𝜆 𝑞 𝑎 𝜆 𝑞 𝑏 𝜆 𝑞 𝑐 𝜆 𝑞 𝑐 superscript 𝑐 2 𝑞 𝜆 𝑞 𝑎 𝑏 q-multiple-Pochhammer 𝜆 𝑞 superscript 𝑐 2 𝜆 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑞 superscript 𝑐 2 𝑞 𝑎 superscript 𝑐 2 𝑞 𝑏 superscript 𝑞 2 q-multiple-Pochhammer 𝜆 𝑞 𝑎 𝜆 𝑞 𝑏 𝑞 q-multiple-Pochhammer 𝑞 𝑎 𝑏 𝑞 superscript 𝑐 2 𝑞 superscript 𝑐 2 𝑞 𝑎 𝑏 superscript 𝑞 2 {\displaystyle{\displaystyle{{}_{8}\phi_{7}}\left({\lambda,q\lambda^{\frac{1}{% 2}},-q\lambda^{\frac{1}{2}},a,b,c,-c,\lambda q/c^{2}\atop\lambda^{\frac{1}{2}}% ,-\lambda^{\frac{1}{2}},\lambda q/a,\lambda q/b,\lambda q/c,-\lambda q/c,c^{2}% };q,-\frac{\lambda q}{ab}\right)=\frac{\left(\lambda q,c^{2}/\lambda;q\right)_% {\infty}\left(aq,bq,c^{2}q/a,c^{2}q/b;q^{2}\right)_{\infty}}{\left(\lambda q/a% ,\lambda q/b;q\right)_{\infty}\left(q,abq,c^{2}q,c^{2}q/(ab);q^{2}\right)_{% \infty}}}}
\qgenhyperphi{8}{7}@@{\lambda,q\lambda^{\frac{1}{2}},-q\lambda^{\frac{1}{2}},a,b,c,-c,\lambda q/c^{2}}{\lambda^{\frac{1}{2}},-\lambda^{\frac{1}{2}},\lambda q/a,\lambda q/b,\lambda q/c,-\lambda q/c,c^{2}}{q}{-\frac{\lambda q}{ab}} = \frac{\qmultiPochhammersym{\lambda q,c^{2}/\lambda}{q}{\infty}\qmultiPochhammersym{aq,bq,c^{2}q/a,c^{2}q/b}{q^{2}}{\infty}}{\qmultiPochhammersym{\lambda q/a,\lambda q/b}{q}{\infty}\qmultiPochhammersym{q,abq,c^{2}q,c^{2}q/(ab)}{q^{2}}{\infty}}		 

Error

QHypergeometricPFQ[{\[Lambda], q*\[Lambda]^(Divide[1,2]), - q*\[Lambda]^(Divide[1,2]), a , b , c , - c , \[Lambda]*q/(c)^(2)},{\[Lambda]^(Divide[1,2]), - \[Lambda]^(Divide[1,2]), \[Lambda]*q/a , \[Lambda]*q/b , \[Lambda]*q/c , - \[Lambda]*q/c , (c)^(2)},q,-Divide[\[Lambda]*q,a*b]] == Divide[Product[QPochhammer[Part[{\[Lambda]*q , (c)^(2)/\[Lambda]},i],q,Infinity],{i,1,Length[{\[Lambda]*q , (c)^(2)/\[Lambda]}]}]*Product[QPochhammer[Part[{a*q , b*q , (c)^(2)* q/a , (c)^(2)* q/b},i],(q)^(2),Infinity],{i,1,Length[{a*q , b*q , (c)^(2)* q/a , (c)^(2)* q/b}]}],Product[QPochhammer[Part[{\[Lambda]*q/a , \[Lambda]*q/b},i],q,Infinity],{i,1,Length[{\[Lambda]*q/a , \[Lambda]*q/b}]}]*Product[QPochhammer[Part[{q , a*b*q , (c)^(2)* q , (c)^(2)* q/(a*b)},i],(q)^(2),Infinity],{i,1,Length[{q , a*b*q , (c)^(2)* q , (c)^(2)* q/(a*b)}]}]]
Missing Macro Error Failure - Skipped - Because timed out
17.7.E11 ϕ 3 4 ⁡ ( q - n , q n + 1 , c , - c e , c 2 ⁢ q / e , - q ; q , q ) = ( e ⁢ q - n , e ⁢ q n + 1 , c 2 ⁢ q 1 - n / e , c 2 ⁢ q n + 2 / e ; q 2 ) ∞ ( e , c 2 ⁢ q / e ; q ) ∞ q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 𝑐 𝑐 𝑒 superscript 𝑐 2 𝑞 𝑒 𝑞 𝑞 𝑞 q-multiple-Pochhammer 𝑒 superscript 𝑞 𝑛 𝑒 superscript 𝑞 𝑛 1 superscript 𝑐 2 superscript 𝑞 1 𝑛 𝑒 superscript 𝑐 2 superscript 𝑞 𝑛 2 𝑒 superscript 𝑞 2 q-multiple-Pochhammer 𝑒 superscript 𝑐 2 𝑞 𝑒 𝑞 {\displaystyle{\displaystyle{{}_{4}\phi_{3}}\left({q^{-n},q^{n+1},c,-c\atop e,% c^{2}q/e,-q};q,q\right)=\frac{\left(eq^{-n},eq^{n+1},c^{2}q^{1-n}/e,c^{2}q^{n+% 2}/e;q^{2}\right)_{\infty}}{\left(e,c^{2}q/e;q\right)_{\infty}}}}
\qgenhyperphi{4}{3}@@{q^{-n},q^{n+1},c,-c}{e,c^{2}q/e,-q}{q}{q} = \frac{\qmultiPochhammersym{eq^{-n},eq^{n+1},c^{2}q^{1-n}/e,c^{2}q^{n+2}/e}{q^{2}}{\infty}}{\qmultiPochhammersym{e,c^{2}q/e}{q}{\infty}}		 

Error

QHypergeometricPFQ[{(q)^(- n), (q)^(n + 1), c , - c},{e , (c)^(2)* q/e , - q},q,q] == Divide[Product[QPochhammer[Part[{e*(q)^(- n), e*(q)^(n + 1), (c)^(2)* (q)^(1 - n)/e , (c)^(2)* (q)^(n + 2)/e},i],(q)^(2),Infinity],{i,1,Length[{e*(q)^(- n), e*(q)^(n + 1), (c)^(2)* (q)^(1 - n)/e , (c)^(2)* (q)^(n + 2)/e}]}],Product[QPochhammer[Part[{e , (c)^(2)* q/e},i],q,Infinity],{i,1,Length[{e , (c)^(2)* q/e}]}]]
Missing Macro Error Failure -
Failed [296 / 300]
Result: Plus[0.0, QHypergeometricPFQ[{Complex[0.8660254037844387, -0.49999999999999994], Complex[0.5000000000000001, 0.8660254037844386], -1.5, 1.5}
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Complex[2.25, 0.0], Complex[-0.8660254037844387, -0.49999999999999994]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], {Rule[c, -1.5], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[QHypergeometricPFQ[{Complex[0.5000000000000001, -0.8660254037844386], Complex[0.0, 1.0], -1.5, 1.5}
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Complex[2.25, 0.0], Complex[-0.8660254037844387, -0.49999999999999994]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Times[-1.0, QPochhammer[Complex[-0.49999999999999994, 0.8660254037844387], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], QPochhammer[Complex[3.3306690738754696*^-16, 2.25], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], QPochhammer[Complex[0.8660254037844387, -0.49999999999999994], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[1.1250000000000004, -1.9485571585149868], Complex[0.5000000000000001, 0.8660254037844386], DirectedInfinity[1]], Power[QPochhammer[Complex[2.25, 0.0], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1]]], {Rule[c, -1.5], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.7.E12 ϕ 3 4 ⁡ ( a , a ⁢ q , b 2 ⁢ q 2 ⁢ n , q - 2 ⁢ n b , b ⁢ q , a 2 ⁢ q 2 ; q 2 , q 2 ) = a n ⁢ ( - q , b / a ; q ) n ( - a ⁢ q , b ; q ) n q-hypergeometric-rphis 4 3 𝑎 𝑎 𝑞 superscript 𝑏 2 superscript 𝑞 2 𝑛 superscript 𝑞 2 𝑛 𝑏 𝑏 𝑞 superscript 𝑎 2 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑎 𝑛 q-multiple-Pochhammer 𝑞 𝑏 𝑎 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑞 𝑛 {\displaystyle{\displaystyle{{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n},q^{-2n}% \atop b,bq,a^{2}q^{2}};q^{2},q^{2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}% }{\left(-aq,b;q\right)_{n}}}}
\qgenhyperphi{4}{3}@@{a,aq,b^{2}q^{2n},q^{-2n}}{b,bq,a^{2}q^{2}}{q^{2}}{q^{2}} = \frac{a^{n}\qmultiPochhammersym{-q,b/a}{q}{n}}{\qmultiPochhammersym{-aq,b}{q}{n}}		 

Error

QHypergeometricPFQ[{a , a*q , (b)^(2)* (q)^(2*n), (q)^(- 2*n)},{b , b*q , (a)^(2)* (q)^(2)},(q)^(2),(q)^(2)] == Divide[(a)^(n)* Product[QPochhammer[Part[{- q , b/a},i],q,n],{i,1,Length[{- q , b/a}]}],Product[QPochhammer[Part[{- a*q , b},i],q,n],{i,1,Length[{- a*q , b}]}]]
Missing Macro Error Failure -
Failed [246 / 300]
Result: Plus[Complex[0.0, 0.0], QHypergeometricPFQ[{-1.5, Complex[-1.299038105676658, -0.7499999999999999], Complex[1.1250000000000002, 1.9485571585149868], Complex[0.5000000000000001, -0.8660254037844386]}
Test Values: {-1.5, Complex[-1.299038105676658, -0.7499999999999999], Complex[1.1250000000000002, 1.9485571585149868]}, Complex[0.5000000000000001, 0.8660254037844386], Complex[0.5000000000000001, 0.8660254037844386]]], {Rule[a, -1.5], Rule[b, -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[-1.299038105676658, -0.7499999999999999], Complex[-1.1249999999999996, 1.948557158514987], Complex[-0.4999999999999998, -0.8660254037844387]}
Test Values: {-1.5, Complex[-1.299038105676658, -0.7499999999999999], Complex[1.1250000000000002, 1.9485571585149868]}, Complex[0.5000000000000001, 0.8660254037844386], Complex[0.5000000000000001, 0.8660254037844386]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.7.E13 ϕ 3 4 ⁡ ( a , a ⁢ q , b 2 ⁢ q 2 ⁢ n - 2 , q - 2 ⁢ n b , b ⁢ q , a 2 ; q 2 , q 2 ) = a n ⁢ ( - q , b / a ; q ) n ⁢ ( 1 - b ⁢ q n - 1 ) ( - a , b ; q ) n ⁢ ( 1 - b ⁢ q 2 ⁢ n - 1 ) q-hypergeometric-rphis 4 3 𝑎 𝑎 𝑞 superscript 𝑏 2 superscript 𝑞 2 𝑛 2 superscript 𝑞 2 𝑛 𝑏 𝑏 𝑞 superscript 𝑎 2 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑎 𝑛 q-multiple-Pochhammer 𝑞 𝑏 𝑎 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 1 q-multiple-Pochhammer 𝑎 𝑏 𝑞 𝑛 1 𝑏 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n-2},q^{-2n}% \atop b,bq,a^{2}};q^{2},q^{2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}(1-bq% ^{n-1})}{\left(-a,b;q\right)_{n}(1-bq^{2n-1})}}}
\qgenhyperphi{4}{3}@@{a,aq,b^{2}q^{2n-2},q^{-2n}}{b,bq,a^{2}}{q^{2}}{q^{2}} = \frac{a^{n}\qmultiPochhammersym{-q,b/a}{q}{n}(1-bq^{n-1})}{\qmultiPochhammersym{-a,b}{q}{n}(1-bq^{2n-1})}		 

Error

QHypergeometricPFQ[{a , a*q , (b)^(2)* (q)^(2*n - 2), (q)^(- 2*n)},{b , b*q , (a)^(2)},(q)^(2),(q)^(2)] == Divide[(a)^(n)* Product[QPochhammer[Part[{- q , b/a},i],q,n],{i,1,Length[{- q , b/a}]}]*(1 - b*(q)^(n - 1)),Product[QPochhammer[Part[{- a , b},i],q,n],{i,1,Length[{- a , b}]}]*(1 - b*(q)^(2*n - 1))]
Missing Macro Error Failure -
Failed [210 / 300]
Result: Plus[Complex[0.0, 0.0], QHypergeometricPFQ[{-1.5, Complex[-1.299038105676658, -0.7499999999999999], 2.25, Complex[0.5000000000000001, -0.8660254037844386]}
Test Values: {-1.5, Complex[-1.299038105676658, -0.7499999999999999], 2.25}, Complex[0.5000000000000001, 0.8660254037844386], Complex[0.5000000000000001, 0.8660254037844386]]], {Rule[a, -1.5], Rule[b, -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[-1.299038105676658, -0.7499999999999999], Complex[1.1250000000000002, 1.9485571585149868], Complex[-0.4999999999999998, -0.8660254037844387]}
Test Values: {-1.5, Complex[-1.299038105676658, -0.7499999999999999], 2.25}, Complex[0.5000000000000001, 0.8660254037844386], Complex[0.5000000000000001, 0.8660254037844386]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.7.E16 ϕ 5 6 ⁡ ( a , q ⁢ a 1 2 , - q ⁢ a 1 2 , b , c , q - n a 1 2 , - a 1 2 , a ⁢ q / b , a ⁢ q / c , a ⁢ q n + 1 ; q , a ⁢ q n + 1 b ⁢ c ) = ( a ⁢ q , a ⁢ q / ( b ⁢ c ) ; q ) n ( a ⁢ q / b , a ⁢ q / c ; q ) n q-hypergeometric-rphis 6 5 𝑎 𝑞 superscript 𝑎 1 2 𝑞 superscript 𝑎 1 2 𝑏 𝑐 superscript 𝑞 𝑛 superscript 𝑎 1 2 superscript 𝑎 1 2 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑎 superscript 𝑞 𝑛 1 𝑞 𝑎 superscript 𝑞 𝑛 1 𝑏 𝑐 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑞 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{% \frac{1}{2}},b,c,q^{-n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq^{n+% 1}};q,\frac{aq^{n+1}}{bc}\right)=\frac{\left(aq,aq/(bc);q\right)_{n}}{\left(aq% /b,aq/c;q\right)_{n}}}}
\qgenhyperphi{6}{5}@@{a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq^{n+1}}{q}{\frac{aq^{n+1}}{bc}} = \frac{\qmultiPochhammersym{aq,aq/(bc)}{q}{n}}{\qmultiPochhammersym{aq/b,aq/c}{q}{n}}		 

Error

QHypergeometricPFQ[{a , q*(a)^(Divide[1,2]), - q*(a)^(Divide[1,2]), b , c , (q)^(- n)},{(a)^(Divide[1,2]), - (a)^(Divide[1,2]), a*q/b , a*q/c , a*(q)^(n + 1)},q,Divide[a*(q)^(n + 1),b*c]] == Divide[Product[QPochhammer[Part[{a*q , a*q/(b*c)},i],q,n],{i,1,Length[{a*q , a*q/(b*c)}]}],Product[QPochhammer[Part[{a*q/b , a*q/c},i],q,n],{i,1,Length[{a*q/b , a*q/c}]}]]
Missing Macro Error Failure -
Failed [240 / 300]
Result: Plus[Complex[14.55021169820366, 2.220446049250313*^-16], QHypergeometricPFQ[{-1.5, Complex[-0.6123724356957944, 1.0606601717798212], Complex[0.6123724356957944, -1.0606601717798212], -1.5, -1.5, Complex[0.8660254037844387, -0.49999999999999994]}
Test Values: {Complex[0.0, 1.224744871391589], Complex[0.0, -1.224744871391589], Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.7500000000000002, -1.299038105676658]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.33333333333333337, -0.5773502691896257]]], {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]]]}


Result: Plus[Complex[-46.07567037764495, -8.881784197001252*^-15], QHypergeometricPFQ[{-1.5, Complex[-0.6123724356957944, 1.0606601717798212], Complex[0.6123724356957944, -1.0606601717798212], -1.5, -1.5, Complex[0.5000000000000001, -0.8660254037844386]}
Test Values: {Complex[0.0, 1.224744871391589], Complex[0.0, -1.224744871391589], Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], Complex[0.0, -1.5]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.0, -0.6666666666666666]]], {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]]]}

... skip entries to safe data
17.7.E20 ∑ k = 0 n 1 - a ⁢ p k ⁢ q k 1 - a ⁢ ( a ; p ) k ⁢ ( c ; q ) k ( q ; q ) k ⁢ ( a ⁢ p / c ; p ) k ⁢ c - k = ( a ⁢ p ; p ) n ⁢ ( c ⁢ q ; q ) n ( q ; q ) n ⁢ ( a ⁢ p / c ; p ) n ⁢ c - n superscript subscript 𝑘 0 𝑛 1 𝑎 superscript 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑎 q-Pochhammer-symbol 𝑎 𝑝 𝑘 q-Pochhammer-symbol 𝑐 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑎 𝑝 𝑐 𝑝 𝑘 superscript 𝑐 𝑘 q-Pochhammer-symbol 𝑎 𝑝 𝑝 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑝 𝑐 𝑝 𝑛 superscript 𝑐 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\left% (a;p\right)_{k}\left(c;q\right)_{k}}{\left(q;q\right)_{k}\left(ap/c;p\right)_{% k}}c^{-k}=\frac{\left(ap;p\right)_{n}\left(cq;q\right)_{n}}{\left(q;q\right)_{% n}\left(ap/c;p\right)_{n}}c^{-n}}}
\sum_{k=0}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\qPochhammer{a}{p}{k}\qPochhammer{c}{q}{k}}{\qPochhammer{q}{q}{k}\qPochhammer{ap/c}{p}{k}}c^{-k} = \frac{\qPochhammer{ap}{p}{n}\qPochhammer{cq}{q}{n}}{\qPochhammer{q}{q}{n}\qPochhammer{ap/c}{p}{n}}c^{-n}		 

sum((1 - a*(p)^(k)* (q)^(k))/(1 - a)*(QPochhammer(a, p, k)*QPochhammer(c, q, k))/(QPochhammer(q, q, k)*QPochhammer(a*p/c, p, k))*(c)^(- k), k = 0..n) = (QPochhammer(a*p, p, n)*QPochhammer(c*q, q, n))/(QPochhammer(q, q, n)*QPochhammer(a*p/c, p, n))*(c)^(- n)

Sum[Divide[1 - a*(p)^(k)* (q)^(k),1 - a]*Divide[QPochhammer[a, p, k]*QPochhammer[c, q, k],QPochhammer[q, q, k]*QPochhammer[a*p/c, p, k]]*(c)^(- k), {k, 0, n}, GenerateConditions->None] == Divide[QPochhammer[a*p, p, n]*QPochhammer[c*q, q, n],QPochhammer[q, q, n]*QPochhammer[a*p/c, p, n]]*(c)^(- n)
Failure Aborted Error Skipped - Because timed out
17.7.E21 ∑ k = 0 n ( 1 - a ⁢ p k ⁢ q k ) ⁢ ( 1 - b ⁢ p k ⁢ q - k ) ( 1 - a ) ⁢ ( 1 - b ) ⁢ ( a , b ; p ) k ⁢ ( c , a / ( b ⁢ c ) ; q ) k ( q , a ⁢ q / b ; q ) k ⁢ ( a ⁢ p / c , b ⁢ c ⁢ p ; p ) k ⁢ q k = ( a ⁢ p , b ⁢ p ; p ) n ⁢ ( c ⁢ q , a ⁢ q / ( b ⁢ c ) ; q ) n ( q , a ⁢ q / b ; q ) n ⁢ ( a ⁢ p / c , b ⁢ c ⁢ p ; p ) n superscript subscript 𝑘 0 𝑛 1 𝑎 superscript 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑏 superscript 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑎 1 𝑏 q-multiple-Pochhammer 𝑎 𝑏 𝑝 𝑘 q-multiple-Pochhammer 𝑐 𝑎 𝑏 𝑐 𝑞 𝑘 q-multiple-Pochhammer 𝑞 𝑎 𝑞 𝑏 𝑞 𝑘 q-multiple-Pochhammer 𝑎 𝑝 𝑐 𝑏 𝑐 𝑝 𝑝 𝑘 superscript 𝑞 𝑘 q-multiple-Pochhammer 𝑎 𝑝 𝑏 𝑝 𝑝 𝑛 q-multiple-Pochhammer 𝑐 𝑞 𝑎 𝑞 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑞 𝑎 𝑞 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑝 𝑐 𝑏 𝑐 𝑝 𝑝 𝑛 {\displaystyle{\displaystyle\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}}}}
\sum_{k=0}^{n}\frac{(1-ap^{k}q^{k})(1-bp^{k}q^{-k})}{(1-a)(1-b)}\frac{\qmultiPochhammersym{a,b}{p}{k}\qmultiPochhammersym{c,a/(bc)}{q}{k}}{\qmultiPochhammersym{q,aq/b}{q}{k}\qmultiPochhammersym{ap/c,bcp}{p}{k}}q^{k} = \frac{\qmultiPochhammersym{ap,bp}{p}{n}\qmultiPochhammersym{cq,aq/(bc)}{q}{n}}{\qmultiPochhammersym{q,aq/b}{q}{n}\qmultiPochhammersym{ap/c,bcp}{p}{n}}		 

Error

Sum[Divide[(1 - a*(p)^(k)* (q)^(k))*(1 - b*(p)^(k)* (q)^(- k)),(1 - a)*(1 - b)]*Divide[Product[QPochhammer[Part[{a , b},i],p,k],{i,1,Length[{a , b}]}]*Product[QPochhammer[Part[{c , a/(b*c)},i],q,k],{i,1,Length[{c , a/(b*c)}]}],Product[QPochhammer[Part[{q , a*q/b},i],q,k],{i,1,Length[{q , a*q/b}]}]*Product[QPochhammer[Part[{a*p/c , b*c*p},i],p,k],{i,1,Length[{a*p/c , b*c*p}]}]]*(q)^(k), {k, 0, n}, GenerateConditions->None] == Divide[Product[QPochhammer[Part[{a*p , b*p},i],p,n],{i,1,Length[{a*p , b*p}]}]*Product[QPochhammer[Part[{c*q , a*q/(b*c)},i],q,n],{i,1,Length[{c*q , a*q/(b*c)}]}],Product[QPochhammer[Part[{q , a*q/b},i],q,n],{i,1,Length[{q , a*q/b}]}]*Product[QPochhammer[Part[{a*p/c , b*c*p},i],p,n],{i,1,Length[{a*p/c , b*c*p}]}]]
Missing Macro Error Aborted - Skipped - Because timed out
17.7.E22 ∑ k = - m n ( 1 - a ⁢ d ⁢ p k ⁢ q k ) ⁢ ( 1 - b ⁢ p k / ( d ⁢ q k ) ) ( 1 - a ⁢ d ) ⁢ ( 1 - ( b / d ) ) ⁢ ( a , b ; p ) k ⁢ ( c , a ⁢ d 2 / ( b ⁢ c ) ; q ) k ( d ⁢ q , a ⁢ d ⁢ q / b ; q ) k ⁢ ( a ⁢ d ⁢ p / c , b ⁢ c ⁢ p / d ; p ) k ⁢ q k = ( 1 - a ) ⁢ ( 1 - b ) ⁢ ( 1 - c ) ⁢ ( 1 - ( a ⁢ d 2 / ( b ⁢ c ) ) ) d ⁢ ( 1 - a ⁢ d ) ⁢ ( 1 - ( b / d ) ) ⁢ ( 1 - ( c / d ) ) ⁢ ( 1 - ( a ⁢ d / ( b ⁢ c ) ) ) ⁢ ( ( a ⁢ p , b ⁢ p ; p ) n ⁢ ( c ⁢ q , a ⁢ d 2 ⁢ q / ( b ⁢ c ) ; q ) n ( d ⁢ q , a ⁢ d ⁢ q / b ; q ) n ⁢ ( a ⁢ d ⁢ p / c , b ⁢ c ⁢ p / d ; p ) n - ( c / ( a ⁢ d ) , d / ( b ⁢ c ) ; p ) m + 1 ⁢ ( 1 / d , b / ( a ⁢ d ) ; q ) m + 1 ( 1 / c , b ⁢ c / ( a ⁢ d 2 ) ; q ) m + 1 ⁢ ( 1 / a , 1 / b ; p ) m + 1 ) superscript subscript 𝑘 𝑚 𝑛 1 𝑎 𝑑 superscript 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑏 superscript 𝑝 𝑘 𝑑 superscript 𝑞 𝑘 1 𝑎 𝑑 1 𝑏 𝑑 q-multiple-Pochhammer 𝑎 𝑏 𝑝 𝑘 q-multiple-Pochhammer 𝑐 𝑎 superscript 𝑑 2 𝑏 𝑐 𝑞 𝑘 q-multiple-Pochhammer 𝑑 𝑞 𝑎 𝑑 𝑞 𝑏 𝑞 𝑘 q-multiple-Pochhammer 𝑎 𝑑 𝑝 𝑐 𝑏 𝑐 𝑝 𝑑 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑎 1 𝑏 1 𝑐 1 𝑎 superscript 𝑑 2 𝑏 𝑐 𝑑 1 𝑎 𝑑 1 𝑏 𝑑 1 𝑐 𝑑 1 𝑎 𝑑 𝑏 𝑐 q-multiple-Pochhammer 𝑎 𝑝 𝑏 𝑝 𝑝 𝑛 q-multiple-Pochhammer 𝑐 𝑞 𝑎 superscript 𝑑 2 𝑞 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑑 𝑞 𝑎 𝑑 𝑞 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑑 𝑝 𝑐 𝑏 𝑐 𝑝 𝑑 𝑝 𝑛 q-multiple-Pochhammer 𝑐 𝑎 𝑑 𝑑 𝑏 𝑐 𝑝 𝑚 1 q-multiple-Pochhammer 1 𝑑 𝑏 𝑎 𝑑 𝑞 𝑚 1 q-multiple-Pochhammer 1 𝑐 𝑏 𝑐 𝑎 superscript 𝑑 2 𝑞 𝑚 1 q-multiple-Pochhammer 1 𝑎 1 𝑏 𝑝 𝑚 1 {\displaystyle{\displaystyle\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)}}
\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}\frac{\qmultiPochhammersym{a,b}{p}{k}\qmultiPochhammersym{c,ad^{2}/(bc)}{q}{k}}{\qmultiPochhammersym{dq,adq/b}{q}{k}\qmultiPochhammersym{adp/c,bcp/d}{p}{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{\qmultiPochhammersym{ap,bp}{p}{n}\qmultiPochhammersym{cq,ad^{2}q/(bc)}{q}{n}}{\qmultiPochhammersym{dq,adq/b}{q}{n}\qmultiPochhammersym{adp/c,bcp/d}{p}{n}}-\frac{\qmultiPochhammersym{c/(ad),d/(bc)}{p}{m+1}\qmultiPochhammersym{1/d,b/(ad)}{q}{m+1}}{\qmultiPochhammersym{1/c,bc/(ad^{2})}{q}{m+1}\qmultiPochhammersym{1/a,1/b}{p}{m+1}}\right)		 

Error

Sum[Divide[(1 - a*d*(p)^(k)* (q)^(k))*(1 - b*(p)^(k)/(d*(q)^(k))),(1 - a*d)*(1 -(b/d))]*Divide[Product[QPochhammer[Part[{a , b},i],p,k],{i,1,Length[{a , b}]}]*Product[QPochhammer[Part[{c , a*(d)^(2)/(b*c)},i],q,k],{i,1,Length[{c , a*(d)^(2)/(b*c)}]}],Product[QPochhammer[Part[{d*q , a*d*q/b},i],q,k],{i,1,Length[{d*q , a*d*q/b}]}]*Product[QPochhammer[Part[{a*d*p/c , b*c*p/d},i],p,k],{i,1,Length[{a*d*p/c , b*c*p/d}]}]]*(q)^(k), {k, - m, n}, GenerateConditions->None] == Divide[(1 - a)*(1 - b)*(1 - c)*(1 -(a*(d)^(2)/(b*c))),d*(1 - a*d)*(1 -(b/d))*(1 -(c/d))*(1 -(a*d/(b*c)))]*(Divide[Product[QPochhammer[Part[{a*p , b*p},i],p,n],{i,1,Length[{a*p , b*p}]}]*Product[QPochhammer[Part[{c*q , a*(d)^(2)* q/(b*c)},i],q,n],{i,1,Length[{c*q , a*(d)^(2)* q/(b*c)}]}],Product[QPochhammer[Part[{d*q , a*d*q/b},i],q,n],{i,1,Length[{d*q , a*d*q/b}]}]*Product[QPochhammer[Part[{a*d*p/c , b*c*p/d},i],p,n],{i,1,Length[{a*d*p/c , b*c*p/d}]}]]-Divide[Product[QPochhammer[Part[{c/(a*d), d/(b*c)},i],p,m + 1],{i,1,Length[{c/(a*d), d/(b*c)}]}]*Product[QPochhammer[Part[{1/d , b/(a*d)},i],q,m + 1],{i,1,Length[{1/d , b/(a*d)}]}],Product[QPochhammer[Part[{1/c , b*c/(a*(d)^(2))},i],q,m + 1],{i,1,Length[{1/c , b*c/(a*(d)^(2))}]}]*Product[QPochhammer[Part[{1/a , 1/b},i],p,m + 1],{i,1,Length[{1/a , 1/b}]}]])
Missing Macro Error Aborted - Skipped - Because timed out
17.7.E23 ( 1 - a q ) ⁢ ( 1 - b q ) ⁢ ∑ k = 0 n ( a ⁢ p k , b ⁢ p - k ; q ) n - 1 ⁢ ( 1 - ( a ⁢ p 2 ⁢ k / b ) ) ( p ; p ) n ⁢ ( p ; p ) n - k ⁢ ( a ⁢ p k / b ; q ) n + 1 ⁢ ( - 1 ) k ⁢ p ( k 2 ) = δ n , 0 1 𝑎 𝑞 1 𝑏 𝑞 superscript subscript 𝑘 0 𝑛 q-multiple-Pochhammer 𝑎 superscript 𝑝 𝑘 𝑏 superscript 𝑝 𝑘 𝑞 𝑛 1 1 𝑎 superscript 𝑝 2 𝑘 𝑏 q-Pochhammer-symbol 𝑝 𝑝 𝑛 q-Pochhammer-symbol 𝑝 𝑝 𝑛 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑝 𝑘 𝑏 𝑞 𝑛 1 superscript 1 𝑘 superscript 𝑝 binomial 𝑘 2 Kronecker 𝑛 0 {\displaystyle{\displaystyle\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}% \right)\sum_{k=0}^{n}\frac{\left(ap^{k},bp^{-k};q\right)_{n-1}(1-(ap^{2k}/b))}% {\left(p;p\right)_{n}\left(p;p\right)_{n-k}\left(ap^{k}/b;q\right)_{n+1}}(-1)^% {k}p^{\genfrac{(}{)}{0.0pt}{}{k}{2}}=\delta_{n,0}}}
\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}\right)\sum_{k=0}^{n}\frac{\qmultiPochhammersym{ap^{k},bp^{-k}}{q}{n-1}(1-(ap^{2k}/b))}{\qPochhammer{p}{p}{n}\qPochhammer{p}{p}{n-k}\qPochhammer{ap^{k}/b}{q}{n+1}}(-1)^{k}p^{\binom{k}{2}} = \Kroneckerdelta{n}{0}		 

Error

(1 -Divide[a,q])*(1 -Divide[b,q])*Sum[Divide[Product[QPochhammer[Part[{a*(p)^(k), b*(p)^(- k)},i],q,n - 1],{i,1,Length[{a*(p)^(k), b*(p)^(- k)}]}]*(1 -(a*(p)^(2*k)/b)),QPochhammer[p, p, n]*QPochhammer[p, p, n - k]*QPochhammer[a*(p)^(k)/b, q, n + 1]]*(- 1)^(k)* (p)^(Binomial[k,2]), {k, 0, n}, GenerateConditions->None] == KroneckerDelta[n, 0]
Missing Macro Error Aborted - Skipped - Because timed out
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