Orthogonal Polynomials - 18.28 Askey–Wilson Class

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.28.E1 a - n ⁢ ∑ ℓ = 0 n q ℓ ⁢ ( a ⁢ b ⁢ q ℓ , a ⁢ c ⁢ q ℓ , a ⁢ d ⁢ q ℓ ; q ) n - ℓ ⁢ ( q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ; q ) ℓ ( q ; q ) ℓ ⁢ ∏ j = 0 ℓ - 1 ( 1 - 2 ⁢ a ⁢ q j ⁢ cos ⁡ θ + a 2 ⁢ q 2 ⁢ j ) = a - n ⁢ ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n ⁢ ϕ 3 4 ⁡ ( q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ a ⁢ b , a ⁢ c , a ⁢ d ; q , q ) superscript 𝑎 𝑛 superscript subscript ℓ 0 𝑛 superscript 𝑞 ℓ q-multiple-Pochhammer 𝑎 𝑏 superscript 𝑞 ℓ 𝑎 𝑐 superscript 𝑞 ℓ 𝑎 𝑑 superscript 𝑞 ℓ 𝑞 𝑛 ℓ q-multiple-Pochhammer superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ superscript subscript product 𝑗 0 ℓ 1 1 2 𝑎 superscript 𝑞 𝑗 𝜃 superscript 𝑎 2 superscript 𝑞 2 𝑗 superscript 𝑎 𝑛 q-multiple-Pochhammer 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑞 {\displaystyle{\displaystyle a^{-n}\sum_{\ell=0}^{n}q^{\ell}\left(abq^{\ell},% acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},abcdq^{n-1};q% \right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2aq^{j}\cos% \theta+a^{2}q^{2j})}=a^{-n}\left(ab,ac,ad;q\right)_{n}\*{{}_{4}\phi_{3}}\left(% {q^{-n},abcdq^{n-1},ae^{\mathrm{i}\theta},ae^{-\mathrm{i}\theta}\atop ab,ac,ad% };q,q\right)}}
a^{-n}\sum_{\ell=0}^{n}q^{\ell}\qmultiPochhammersym{abq^{\ell},acq^{\ell},adq^{\ell}}{q}{n-\ell}\*\frac{\qmultiPochhammersym{q^{-n},abcdq^{n-1}}{q}{\ell}}{\qPochhammer{q}{q}{\ell}}\prod_{j=0}^{\ell-1}{(1-2aq^{j}\cos@@{\theta}+a^{2}q^{2j})} = a^{-n}\qmultiPochhammersym{ab,ac,ad}{q}{n}\*\qgenhyperphi{4}{3}@@{q^{-n},abcdq^{n-1},ae^{\iunit\theta},ae^{-\iunit\theta}}{ab,ac,ad}{q}{q}		 

Error

(a)^(- n)* Sum[(q)^\[ScriptL]* Product[QPochhammer[Part[{a*b*(q)^\[ScriptL], a*c*(q)^\[ScriptL], a*d*(q)^\[ScriptL]},i],q,n - \[ScriptL]],{i,1,Length[{a*b*(q)^\[ScriptL], a*c*(q)^\[ScriptL], a*d*(q)^\[ScriptL]}]}]*Divide[Product[QPochhammer[Part[{(q)^(- n), a*b*c*d*(q)^(n - 1)},i],q,\[ScriptL]],{i,1,Length[{(q)^(- n), a*b*c*d*(q)^(n - 1)}]}],QPochhammer[q, q, \[ScriptL]]]*Product[1 - 2*a*(q)^(j)* Cos[\[Theta]]+ (a)^(2)* (q)^(2*j), {j, 0, \[ScriptL]- 1}, GenerateConditions->None], {\[ScriptL], 0, n}, GenerateConditions->None] == (a)^(- n)* Product[QPochhammer[Part[{a*b , a*c , a*d},i],q,n],{i,1,Length[{a*b , a*c , a*d}]}]* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , a*c , a*d},q,q]
Missing Macro Error Aborted - Skipped - Because timed out
18.28.E3 2 ⁢ π ⁢ sin ⁡ θ ⁢ w ⁢ ( cos ⁡ θ ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 2 𝜋 𝜃 𝑤 𝜃 q-Pochhammer-symbol superscript 𝑒 2 𝑖 𝜃 𝑞 q-multiple-Pochhammer 𝑎 superscript 𝑒 𝑖 𝜃 𝑏 superscript 𝑒 𝑖 𝜃 𝑐 superscript 𝑒 𝑖 𝜃 𝑑 superscript 𝑒 𝑖 𝜃 𝑞 2 {\displaystyle{\displaystyle 2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left(% e^{2i\theta};q\right)_{\infty}}{\left(ae^{i\theta},be^{i\theta},ce^{i\theta},% de^{i\theta};q\right)_{\infty}}\right|^{2}}}}
2\pi\sin@@{\theta}\,w(\cos@@{\theta}) = \abs{\frac{\qPochhammer{e^{2i\theta}}{q}{\infty}}{\qmultiPochhammersym{ae^{i\theta},be^{i\theta},ce^{i\theta},de^{i\theta}}{q}{\infty}}}^{2}		 

Error

2*Pi*Sin[\[Theta]]*w[Cos[\[Theta]]] == (Abs[Divide[QPochhammer[Exp[2*I*\[Theta]], q, Infinity],Product[QPochhammer[Part[{a*Exp[I*\[Theta]], b*Exp[I*\[Theta]], c*Exp[I*\[Theta]], d*Exp[I*\[Theta]]},i],q,Infinity],{i,1,Length[{a*Exp[I*\[Theta]], b*Exp[I*\[Theta]], c*Exp[I*\[Theta]], d*Exp[I*\[Theta]]}]}]]])^(2)
Missing Macro Error Failure - Skipped - Because timed out
18.28.E4 h 0 = ( a ⁢ b ⁢ c ⁢ d ; q ) ∞ ( q , a ⁢ b , a ⁢ c , a ⁢ d , b ⁢ c , b ⁢ d , c ⁢ d ; q ) ∞ subscript ℎ 0 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 𝑞 q-multiple-Pochhammer 𝑞 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑏 𝑐 𝑏 𝑑 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle h_{0}=\frac{\left(abcd;q\right)_{\infty}}{\left(q% ,ab,ac,ad,bc,bd,cd;q\right)_{\infty}}}}
h_{0} = \frac{\qPochhammer{abcd}{q}{\infty}}{\qmultiPochhammersym{q,ab,ac,ad,bc,bd,cd}{q}{\infty}}		 

Error

Subscript[h, 0] == Divide[QPochhammer[a*b*c*d, q, Infinity],Product[QPochhammer[Part[{q , a*b , a*c , a*d , b*c , b*d , c*d},i],q,Infinity],{i,1,Length[{q , a*b , a*c , a*d , b*c , b*d , c*d}]}]]
Missing Macro Error Translation Error - -
18.28.E7 a - n ⁢ ∑ ℓ = 0 n q ℓ ⁢ ( a ⁢ b ⁢ q ℓ ; q ) n - ℓ ⁢ ( q - n ; q ) ℓ ( q ; q ) ℓ ⁢ ∏ j = 0 ℓ - 1 ( 1 - 2 ⁢ a ⁢ q j ⁢ cos ⁡ θ + a 2 ⁢ q 2 ⁢ j ) = ( a ⁢ b ; q ) n a n ⁢ ϕ 2 3 ⁡ ( q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ a ⁢ b , 0 ; q , q ) superscript 𝑎 𝑛 superscript subscript ℓ 0 𝑛 superscript 𝑞 ℓ q-Pochhammer-symbol 𝑎 𝑏 superscript 𝑞 ℓ 𝑞 𝑛 ℓ q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ superscript subscript product 𝑗 0 ℓ 1 1 2 𝑎 superscript 𝑞 𝑗 𝜃 superscript 𝑎 2 superscript 𝑞 2 𝑗 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 q-hypergeometric-rphis 3 2 superscript 𝑞 𝑛 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 𝑏 0 𝑞 𝑞 {\displaystyle{\displaystyle a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{% \ell};q\right)_{n-\ell}\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}% \*\prod_{j=0}^{\ell-1}(1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right% )_{n}}{a^{n}}{{}_{3}\phi_{2}}\left({q^{-n},ae^{\mathrm{i}\theta},ae^{-\mathrm{% i}\theta}\atop ab,0};q,q\right)}}
a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\qPochhammer{abq^{\ell}}{q}{n-\ell}\qPochhammer{q^{-n}}{q}{\ell}}{\qPochhammer{q}{q}{\ell}}\*\prod_{j=0}^{\ell-1}(1-2aq^{j}\cos@@{\theta}+a^{2}q^{2j}) = \frac{\qPochhammer{ab}{q}{n}}{a^{n}}\qgenhyperphi{3}{2}@@{q^{-n},ae^{\iunit\theta},ae^{-\iunit\theta}}{ab,0}{q}{q}		 

Error

(a)^(- n)* Sum[(q)^\[ScriptL]*Divide[QPochhammer[a*b*(q)^\[ScriptL], q, n - \[ScriptL]]*QPochhammer[(q)^(- n), q, \[ScriptL]],QPochhammer[q, q, \[ScriptL]]]* Product[1 - 2*a*(q)^(j)* Cos[\[Theta]]+ (a)^(2)* (q)^(2*j), {j, 0, \[ScriptL]- 1}, GenerateConditions->None], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[QPochhammer[a*b, q, n],(a)^(n)]*QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , 0},q,q]
Missing Macro Error Aborted - Skipped - Because timed out
18.28.E7 ( a ⁢ b ; q ) n a n ⁢ ϕ 2 3 ⁡ ( q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ a ⁢ b , 0 ; q , q ) = ( b ⁢ e - i ⁢ θ ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ ϕ 1 2 ⁡ ( q - n , a ⁢ e i ⁢ θ b - 1 ⁢ q 1 - n ⁢ e i ⁢ θ ; q , b - 1 ⁢ q ⁢ e - i ⁢ θ ) q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 q-hypergeometric-rphis 3 2 superscript 𝑞 𝑛 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 𝑏 0 𝑞 𝑞 q-Pochhammer-symbol 𝑏 superscript 𝑒 imaginary-unit 𝜃 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝑎 superscript 𝑒 imaginary-unit 𝜃 superscript 𝑏 1 superscript 𝑞 1 𝑛 superscript 𝑒 imaginary-unit 𝜃 𝑞 superscript 𝑏 1 𝑞 superscript 𝑒 imaginary-unit 𝜃 {\displaystyle{\displaystyle\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}\phi_{2}% }\left({q^{-n},ae^{\mathrm{i}\theta},ae^{-\mathrm{i}\theta}\atop ab,0};q,q% \right)=\left(be^{-\mathrm{i}\theta};q\right)_{n}e^{\mathrm{i}n\theta}{{}_{2}% \phi_{1}}\left({q^{-n},ae^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}e^{\mathrm{i}% \theta}};q,b^{-1}qe^{-\mathrm{i}\theta}\right)}}
\frac{\qPochhammer{ab}{q}{n}}{a^{n}}\qgenhyperphi{3}{2}@@{q^{-n},ae^{\iunit\theta},ae^{-\iunit\theta}}{ab,0}{q}{q} = \qPochhammer{be^{-\iunit\theta}}{q}{n}e^{\iunit n\theta}\qgenhyperphi{2}{1}@@{q^{-n},ae^{\iunit\theta}}{b^{-1}q^{1-n}e^{\iunit\theta}}{q}{b^{-1}qe^{-\iunit\theta}}		 

Error

Divide[QPochhammer[a*b, q, n],(a)^(n)]*QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , 0},q,q] == QPochhammer[b*Exp[- I*\[Theta]], q, n]*Exp[I*n*\[Theta]]*QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]]},{(b)^(- 1)* (q)^(1 - n)* Exp[I*\[Theta]]},q,(b)^(- 1)* q*Exp[- I*\[Theta]]]
Missing Macro Error Failure -
Failed [240 / 300]
Result: Plus[Times[Complex[-1.8929465558343552, -0.4620307840711053], QHypergeometricPFQ[{Complex[0.8660254037844387, -0.49999999999999994], Complex[-0.5894198337515327, -0.693046176106658]}
Test Values: {Complex[-0.2619643705562368, -0.3080205227140702]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[-1.0353339124695373, 0.3690649628228472]]], Times[0.8333333333333333, QHypergeometricPFQ[{Complex[0.8660254037844387, -0.49999999999999994], Complex[-0.5894198337515327, -0.693046176106658], Complex[-1.6022092234201426, 1.8838948267937556]}, {2.25, 0.0}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[-2.642841004049141, -3.076058498066829], QHypergeometricPFQ[{Complex[0.5000000000000001, -0.8660254037844386], Complex[-0.5894198337515327, -0.693046176106658]}
Test Values: {Complex[-0.38087806114513634, -0.13577141227922815]}, Complex[0.8660254037844387, 0.49999999999999994], Complex[-1.0353339124695373, 0.3690649628228472]]], Times[Complex[0.5269761991749927, 0.6249999999999999], QHypergeometricPFQ[{Complex[0.5000000000000001, -0.8660254037844386], Complex[-0.5894198337515327, -0.693046176106658], Complex[-1.6022092234201426, 1.8838948267937556]}, {2.25, 0.0}, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.28.E11 0 < q 0 𝑞 {\displaystyle{\displaystyle 0<q}}
0 < q		 

0 < q

0 < q
Failure Failure
Failed [3 / 10]
Result: 0. < -1.500000000
Test Values: {q = -3/2}


Result: 0. < -.5000000000
Test Values: {q = -1/2}

... skip entries to safe data
Failed [7 / 10]
Result: Less[0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
18.28.E11 q < 1 , a , b ∈ ℝ , a ⁢ b formulae-sequence 𝑞 1 𝑎 𝑏 𝑎 𝑏 {\displaystyle{\displaystyle q<1,a,b\in\mathbb{R},ab}}
q < 1,a,b\in\Reals,ab		 

q < 1; a , b in real , a*b

q < 1
 a , b \[Element]Reals , a*b
Failure Failure Error Error
18.28.E11 1 , a , b ∈ ℝ , a ⁢ b > 1 , a - 1 ⁢ b formulae-sequence 1 𝑎 𝑏 𝑎 𝑏 1 superscript 𝑎 1 𝑏 {\displaystyle{\displaystyle 1,a,b\in\mathbb{R},ab>1,a^{-1}b}}
1,a,b\in\Reals,ab > 1,a^{-1}b		 

1 , a , b in real; a*b > 1 , (a)^(- 1)* b

1 , a , b \[Element]Reals
 a*b > 1 , (a)^(- 1)* b
Error Failure Skip - symbolical successful subtest Error
18.28.E11 1 , a - 1 ⁢ b < q - 1 1 superscript 𝑎 1 𝑏 superscript 𝑞 1 {\displaystyle{\displaystyle 1,a^{-1}b<q^{-1}}}
1,a^{-1}b < q^{-1}		 

1 , (a)^(- 1)* b < (q)^(- 1)

1 , (a)^(- 1)* b < (q)^(- 1)
Failure Failure Error Error
18.28.E12 0 < q 0 𝑞 {\displaystyle{\displaystyle 0<q}}
0 < q		 

0 < q

0 < q
Failure Failure
Failed [3 / 10]
Result: 0. < -1.500000000
Test Values: {q = -3/2}


Result: 0. < -.5000000000
Test Values: {q = -1/2}

... skip entries to safe data
Failed [7 / 10]
Result: Less[0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
18.28.E12 q < 1 , a / i , b / i ∈ ℝ , ( ℑ ⁡ a ) ⁢ ( ℑ ⁡ b ) formulae-sequence 𝑞 1 𝑎 imaginary-unit 𝑏 imaginary-unit 𝑎 𝑏 {\displaystyle{\displaystyle q<1,\ifrac{a}{\mathrm{i}},\ifrac{b}{\mathrm{i}}% \in\mathbb{R},(\Im a)(\Im b)}}
q < 1,\ifrac{a}{\iunit},\ifrac{b}{\iunit}\in\Reals,(\imagpart@@{a})(\imagpart@@{b})		 

q < 1; (a)/(I),(b)/(I) in real ,Im(a)*Im(b)

q < 1
 Divide[a,I],Divide[b,I] \[Element]Reals ,Im[a]*Im[b]
Failure Failure Error Error
18.28.E12 1 , a / i , b / i ∈ ℝ , ( ℑ ⁡ a ) ⁢ ( ℑ ⁡ b ) > 0 , a - 1 ⁢ b formulae-sequence 1 𝑎 imaginary-unit 𝑏 imaginary-unit 𝑎 𝑏 0 superscript 𝑎 1 𝑏 {\displaystyle{\displaystyle 1,\ifrac{a}{\mathrm{i}},\ifrac{b}{\mathrm{i}}\in% \mathbb{R},(\Im a)(\Im b)>0,a^{-1}b}}
1,\ifrac{a}{\iunit},\ifrac{b}{\iunit}\in\Reals,(\imagpart@@{a})(\imagpart@@{b}) > 0,a^{-1}b		 

1 ,(a)/(I),(b)/(I) in real; Im(a)*Im(b) > 0 , (a)^(- 1)* b

1 ,Divide[a,I],Divide[b,I] \[Element]Reals
 Im[a]*Im[b] > 0 , (a)^(- 1)* b
Error Failure Skip - symbolical successful subtest Error
18.28.E12 0 , a - 1 ⁢ b < q - 1 0 superscript 𝑎 1 𝑏 superscript 𝑞 1 {\displaystyle{\displaystyle 0,a^{-1}b<q^{-1}}}
0,a^{-1}b < q^{-1}		 

0 , (a)^(- 1)* b < (q)^(- 1)

0 , (a)^(- 1)* b < (q)^(- 1)
Failure Failure Error Error
18.28.E13 ∑ ℓ = 0 n ( β ; q ) ℓ ⁢ ( β ; q ) n - ℓ ( q ; q ) ℓ ⁢ ( q ; q ) n - ℓ ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ = ( β ; q ) n ( q ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ ϕ 1 2 ⁡ ( q - n , β β - 1 ⁢ q 1 - n ; q , β - 1 ⁢ q ⁢ e - 2 ⁢ i ⁢ θ ) superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝛽 𝑞 ℓ q-Pochhammer-symbol 𝛽 𝑞 𝑛 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝛽 superscript 𝛽 1 superscript 𝑞 1 𝑛 𝑞 superscript 𝛽 1 𝑞 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}% \left(\beta;q\right)_{n-\ell}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell% }}e^{\mathrm{i}(n-2\ell)\theta}=\frac{\left(\beta;q\right)_{n}}{\left(q;q% \right)_{n}}e^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},\beta\atop\beta% ^{-1}q^{1-n}};q,\beta^{-1}qe^{-2\mathrm{i}\theta}\right)}}
\sum_{\ell=0}^{n}\frac{\qPochhammer{\beta}{q}{\ell}\qPochhammer{\beta}{q}{n-\ell}}{\qPochhammer{q}{q}{\ell}\qPochhammer{q}{q}{n-\ell}}e^{\iunit(n-2\ell)\theta} = \frac{\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}e^{\iunit n\theta}\qgenhyperphi{2}{1}@@{q^{-n},\beta}{\beta^{-1}q^{1-n}}{q}{\beta^{-1}qe^{-2\iunit\theta}}		 

Error

Sum[Divide[QPochhammer[\[Beta], q, \[ScriptL]]*QPochhammer[\[Beta], q, n - \[ScriptL]],QPochhammer[q, q, \[ScriptL]]*QPochhammer[q, q, n - \[ScriptL]]]*Exp[I*(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[QPochhammer[\[Beta], q, n],QPochhammer[q, q, n]]*Exp[I*n*\[Theta]]*QHypergeometricPFQ[{(q)^(- n), \[Beta]},{\[Beta]^(- 1)* (q)^(1 - n)},q,\[Beta]^(- 1)* q*Exp[- 2*I*\[Theta]]]
Missing Macro Error Aborted - Skipped - Because timed out
18.28.E16 ∑ ℓ = 0 n ( q ; q ) n ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ ( q ; q ) ℓ ⁢ ( q ; q ) n - ℓ = e i ⁢ n ⁢ θ ⁢ ϕ 0 2 ⁡ ( q - n , 0 - ; q , q n ⁢ e - 2 ⁢ i ⁢ θ ) superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 q-Pochhammer-symbol 𝑞 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 0 superscript 𝑞 𝑛 0 𝑞 superscript 𝑞 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{\left(q;q\right)_{n}e^{% \mathrm{i}(n-2\ell)\theta}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}=% e^{\mathrm{i}n\theta}{{}_{2}\phi_{0}}\left({q^{-n},0\atop-};q,q^{n}e^{-2% \mathrm{i}\theta}\right)}}
\sum_{\ell=0}^{n}\frac{\qPochhammer{q}{q}{n}e^{\iunit(n-2\ell)\theta}}{\qPochhammer{q}{q}{\ell}\qPochhammer{q}{q}{n-\ell}} = e^{\iunit n\theta}\qgenhyperphi{2}{0}@@{q^{-n},0}{-}{q}{q^{n}e^{-2\iunit\theta}}		 

Error

Sum[Divide[QPochhammer[q, q, n]*Exp[I*(n - 2*\[ScriptL])*\[Theta]],QPochhammer[q, q, \[ScriptL]]*QPochhammer[q, q, n - \[ScriptL]]], {\[ScriptL], 0, n}, GenerateConditions->None] == Exp[I*n*\[Theta]]*QHypergeometricPFQ[{(q)^(- n), 0},{-},q,(q)^(n)* Exp[- 2*I*\[Theta]]]
Missing Macro Error Failure - Error
18.28.E18 ∑ ℓ = 0 n q 1 2 ⁢ ℓ ⁢ ( ℓ + 1 ) ⁢ ( q - n ; q ) ℓ ( q ; q ) ℓ ⁢ e ( n - 2 ⁢ ℓ ) ⁢ t = e n ⁢ t ⁢ ϕ 1 1 ⁡ ( q - n 0 ; q , - q ⁢ e - 2 ⁢ t ) superscript subscript ℓ 0 𝑛 superscript 𝑞 1 2 ℓ ℓ 1 q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ superscript 𝑒 𝑛 2 ℓ 𝑡 superscript 𝑒 𝑛 𝑡 q-hypergeometric-rphis 1 1 superscript 𝑞 𝑛 0 𝑞 𝑞 superscript 𝑒 2 𝑡 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}\frac{% \left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}e^{(n-2\ell)t}=e^{nt}{{}% _{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t}\right)}}
\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}\frac{\qPochhammer{q^{-n}}{q}{\ell}}{\qPochhammer{q}{q}{\ell}}e^{(n-2\ell)t} = e^{nt}\qgenhyperphi{1}{1}@@{q^{-n}}{0}{q}{-qe^{-2t}}		 

Error

Sum[(q)^(Divide[1,2]*\[ScriptL]*(\[ScriptL]+ 1))*Divide[QPochhammer[(q)^(- n), q, \[ScriptL]],QPochhammer[q, q, \[ScriptL]]]*Exp[(n - 2*\[ScriptL])*t], {\[ScriptL], 0, n}, GenerateConditions->None] == Exp[n*t]*QHypergeometricPFQ[{(q)^(- n)},{0},q,- q*Exp[- 2*t]]
Missing Macro Error Aborted - Skipped - Because timed out
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