Generalized Hypergeometric Functions & Meijer G -Function - 17.2 Calculus

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
17.2.E2 ( a ; q ) - n = 1 ( a ⁒ q - n ; q ) n q-Pochhammer-symbol π‘Ž π‘ž 𝑛 1 q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž 𝑛 {\displaystyle{\displaystyle\left(a;q\right)_{-n}=\frac{1}{\left(aq^{-n};q% \right)_{n}}}}
\qPochhammer{a}{q}{-n} = \frac{1}{\qPochhammer{aq^{-n}}{q}{n}}		 

QPochhammer(a, q, - n) = (1)/(QPochhammer(a*(q)^(- n), q, n))

QPochhammer[a, q, - n] == Divide[1,QPochhammer[a*(q)^(- n), q, n]]
Successful Failure -
Failed [18 / 180]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[q, -1.5]}

... skip entries to safe data
17.2.E2 1 ( a ⁒ q - n ; q ) n = ( - q / a ) n ⁒ q ( n 2 ) ( q / a ; q ) n 1 q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž 𝑛 superscript π‘ž π‘Ž 𝑛 superscript π‘ž binomial 𝑛 2 q-Pochhammer-symbol π‘ž π‘Ž π‘ž 𝑛 {\displaystyle{\displaystyle\frac{1}{\left(aq^{-n};q\right)_{n}}=\frac{(-q/a)^% {n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}}{\left(q/a;q\right)_{n}}}}
\frac{1}{\qPochhammer{aq^{-n}}{q}{n}} = \frac{(-q/a)^{n}q^{\binom{n}{2}}}{\qPochhammer{q/a}{q}{n}}		 

(1)/(QPochhammer(a*(q)^(- n), q, n)) = ((- q/a)^(n)* (q)^(binomial(n,2)))/(QPochhammer(q/a, q, n))

Divide[1,QPochhammer[a*(q)^(- n), q, n]] == Divide[(- q/a)^(n)* (q)^(Binomial[n,2]),QPochhammer[q/a, q, n]]
Successful Failure -
Failed [18 / 180]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[q, -1.5]}

... skip entries to safe data
17.2.E3 ( a ; q ) Ξ½ = ∏ j = 0 ∞ ( 1 - a ⁒ q j 1 - a ⁒ q Ξ½ + j ) q-Pochhammer-symbol π‘Ž π‘ž 𝜈 superscript subscript product 𝑗 0 1 π‘Ž superscript π‘ž 𝑗 1 π‘Ž superscript π‘ž 𝜈 𝑗 {\displaystyle{\displaystyle\left(a;q\right)_{\nu}=\prod_{j=0}^{\infty}\left(% \frac{1-aq^{j}}{1-aq^{\nu+j}}\right)}}
\qPochhammer{a}{q}{\nu} = \prod_{j=0}^{\infty}\left(\frac{1-aq^{j}}{1-aq^{\nu+j}}\right)		 

QPochhammer(a, q, nu) = product((1 - a*(q)^(j))/(1 - a*(q)^(nu + j)), j = 0..infinity)

QPochhammer[a, q, \[Nu]] == Product[Divide[1 - a*(q)^(j),1 - a*(q)^(\[Nu]+ j)], {j, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [33 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.2.E4 ( a ; q ) ∞ = ∏ j = 0 ∞ ( 1 - a ⁒ q j ) q-Pochhammer-symbol π‘Ž π‘ž superscript subscript product 𝑗 0 1 π‘Ž superscript π‘ž 𝑗 {\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{j=0}^{\infty}(1-% aq^{j})}}
\qPochhammer{a}{q}{\infty} = \prod_{j=0}^{\infty}(1-aq^{j})		 

QPochhammer(a, q, infinity) = product(1 - a*(q)^(j), j = 0..infinity)

QPochhammer[a, q, Infinity] == Product[1 - a*(q)^(j), {j, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [48 / 60]
Result: Plus[Times[-1.0, QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
17.2.E7 ( a ; q - 1 ) n = ( a - 1 ; q ) n ⁒ ( - a ) n ⁒ q - ( n 2 ) q-Pochhammer-symbol π‘Ž superscript π‘ž 1 𝑛 q-Pochhammer-symbol superscript π‘Ž 1 π‘ž 𝑛 superscript π‘Ž 𝑛 superscript π‘ž binomial 𝑛 2 {\displaystyle{\displaystyle\left(a;q^{-1}\right)_{n}=\left(a^{-1};q\right)_{n% }(-a)^{n}q^{-\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\qPochhammer{a}{q^{-1}}{n} = \qPochhammer{a^{-1}}{q}{n}(-a)^{n}q^{-\binom{n}{2}}		 

QPochhammer(a, (q)^(- 1), n) = QPochhammer((a)^(- 1), q, n)*(- a)^(n)* (q)^(-binomial(n,2))

QPochhammer[a, (q)^(- 1), n] == QPochhammer[(a)^(- 1), q, n]*(- a)^(n)* (q)^(-Binomial[n,2])
Successful Failure - Successful [Tested: 180]
17.2.E8 ( a ; q - 1 ) n ( b ; q - 1 ) n = ( a - 1 ; q ) n ( b - 1 ; q ) n ⁒ ( a b ) n q-Pochhammer-symbol π‘Ž superscript π‘ž 1 𝑛 q-Pochhammer-symbol 𝑏 superscript π‘ž 1 𝑛 q-Pochhammer-symbol superscript π‘Ž 1 π‘ž 𝑛 q-Pochhammer-symbol superscript 𝑏 1 π‘ž 𝑛 superscript π‘Ž 𝑏 𝑛 {\displaystyle{\displaystyle\frac{\left(a;q^{-1}\right)_{n}}{\left(b;q^{-1}% \right)_{n}}=\frac{\left(a^{-1};q\right)_{n}}{\left(b^{-1};q\right)_{n}}\left(% \frac{a}{b}\right)^{n}}}
\frac{\qPochhammer{a}{q^{-1}}{n}}{\qPochhammer{b}{q^{-1}}{n}} = \frac{\qPochhammer{a^{-1}}{q}{n}}{\qPochhammer{b^{-1}}{q}{n}}\left(\frac{a}{b}\right)^{n}		 

(QPochhammer(a, (q)^(- 1), n))/(QPochhammer(b, (q)^(- 1), n)) = (QPochhammer((a)^(- 1), q, n))/(QPochhammer((b)^(- 1), q, n))*((a)/(b))^(n)

Divide[QPochhammer[a, (q)^(- 1), n],QPochhammer[b, (q)^(- 1), n]] == Divide[QPochhammer[(a)^(- 1), q, n],QPochhammer[(b)^(- 1), q, n]]*(Divide[a,b])^(n)
Successful Failure -
Failed [20 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 3], Rule[q, -1.5]}

... skip entries to safe data
17.2.E9 ( a ; q ) n = ( q 1 - n / a ; q ) n ⁒ ( - a ) n ⁒ q ( n 2 ) q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 𝑛 π‘Ž π‘ž 𝑛 superscript π‘Ž 𝑛 superscript π‘ž binomial 𝑛 2 {\displaystyle{\displaystyle\left(a;q\right)_{n}=\left(q^{1-n}/a;q\right)_{n}(% -a)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\qPochhammer{a}{q}{n} = \qPochhammer{q^{1-n}/a}{q}{n}(-a)^{n}q^{\binom{n}{2}}		 

QPochhammer(a, q, n) = QPochhammer((q)^(1 - n)/a, q, n)*(- a)^(n)* (q)^(binomial(n,2))

QPochhammer[a, q, n] == QPochhammer[(q)^(1 - n)/a, q, n]*(- a)^(n)* (q)^(Binomial[n,2])
Successful Failure - Successful [Tested: 180]
17.2.E10 ( a ; q ) n ( b ; q ) n = ( q 1 - n / a ; q ) n ( q 1 - n / b ; q ) n ⁒ ( a b ) n q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol 𝑏 π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 𝑛 π‘Ž π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 𝑛 𝑏 π‘ž 𝑛 superscript π‘Ž 𝑏 𝑛 {\displaystyle{\displaystyle\frac{\left(a;q\right)_{n}}{\left(b;q\right)_{n}}=% \frac{\left(q^{1-n}/a;q\right)_{n}}{\left(q^{1-n}/b;q\right)_{n}}\left(\frac{a% }{b}\right)^{n}}}
\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{b}{q}{n}} = \frac{\qPochhammer{q^{1-n}/a}{q}{n}}{\qPochhammer{q^{1-n}/b}{q}{n}}\left(\frac{a}{b}\right)^{n}		 

(QPochhammer(a, q, n))/(QPochhammer(b, q, n)) = (QPochhammer((q)^(1 - n)/a, q, n))/(QPochhammer((q)^(1 - n)/b, q, n))*((a)/(b))^(n)

Divide[QPochhammer[a, q, n],QPochhammer[b, q, n]] == Divide[QPochhammer[(q)^(1 - n)/a, q, n],QPochhammer[(q)^(1 - n)/b, q, n]]*(Divide[a,b])^(n)
Successful Failure -
Failed [12 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[n, 2], Rule[q, -2]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[n, 3], Rule[q, -2]}

... skip entries to safe data
17.2.E11 ( a ⁒ q - n ; q ) n = ( q / a ; q ) n ⁒ ( - a q ) n ⁒ q - ( n 2 ) q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘Ž π‘ž 𝑛 superscript π‘Ž π‘ž 𝑛 superscript π‘ž binomial 𝑛 2 {\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n}=\left(q/a;q\right)_{n}% \left(-\frac{a}{q}\right)^{n}q^{-\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\qPochhammer{aq^{-n}}{q}{n} = \qPochhammer{q/a}{q}{n}\left(-\frac{a}{q}\right)^{n}q^{-\binom{n}{2}}		 

QPochhammer(a*(q)^(- n), q, n) = QPochhammer(q/a, q, n)*(-(a)/(q))^(n)* (q)^(-binomial(n,2))

QPochhammer[a*(q)^(- n), q, n] == QPochhammer[q/a, q, n]*(-Divide[a,q])^(n)* (q)^(-Binomial[n,2])
Successful Failure - Successful [Tested: 180]
17.2.E12 ( a ⁒ q - n ; q ) n ( b ⁒ q - n ; q ) n = ( q / a ; q ) n ( q / b ; q ) n ⁒ ( a b ) n q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž 𝑛 q-Pochhammer-symbol 𝑏 superscript π‘ž 𝑛 π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘Ž π‘ž 𝑛 q-Pochhammer-symbol π‘ž 𝑏 π‘ž 𝑛 superscript π‘Ž 𝑏 𝑛 {\displaystyle{\displaystyle\frac{\left(aq^{-n};q\right)_{n}}{\left(bq^{-n};q% \right)_{n}}=\frac{\left(q/a;q\right)_{n}}{\left(q/b;q\right)_{n}}\left(\frac{% a}{b}\right)^{n}}}
\frac{\qPochhammer{aq^{-n}}{q}{n}}{\qPochhammer{bq^{-n}}{q}{n}} = \frac{\qPochhammer{q/a}{q}{n}}{\qPochhammer{q/b}{q}{n}}\left(\frac{a}{b}\right)^{n}		 

(QPochhammer(a*(q)^(- n), q, n))/(QPochhammer(b*(q)^(- n), q, n)) = (QPochhammer(q/a, q, n))/(QPochhammer(q/b, q, n))*((a)/(b))^(n)

Divide[QPochhammer[a*(q)^(- n), q, n],QPochhammer[b*(q)^(- n), q, n]] == Divide[QPochhammer[q/a, q, n],QPochhammer[q/b, q, n]]*(Divide[a,b])^(n)
Successful Failure -
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[q, -1.5]}

... skip entries to safe data
17.2.E13 ( a ; q ) n - k = ( a ; q ) n ( q 1 - n / a ; q ) k ⁒ ( - q a ) k ⁒ q ( k 2 ) - n ⁒ k q-Pochhammer-symbol π‘Ž π‘ž 𝑛 π‘˜ q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 𝑛 π‘Ž π‘ž π‘˜ superscript π‘ž π‘Ž π‘˜ superscript π‘ž binomial π‘˜ 2 𝑛 π‘˜ {\displaystyle{\displaystyle\left(a;q\right)_{n-k}=\frac{\left(a;q\right)_{n}}% {\left(q^{1-n}/a;q\right)_{k}}\left(-\frac{q}{a}\right)^{k}q^{\genfrac{(}{)}{0% .0pt}{}{k}{2}-nk}}}
\qPochhammer{a}{q}{n-k} = \frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q^{1-n}/a}{q}{k}}\left(-\frac{q}{a}\right)^{k}q^{\binom{k}{2}-nk}		 

QPochhammer(a, q, n - k) = (QPochhammer(a, q, n))/(QPochhammer((q)^(1 - n)/a, q, k))*(-(q)/(a))^(k)* (q)^(binomial(k,2)- n*k)

QPochhammer[a, q, n - k] == Divide[QPochhammer[a, q, n],QPochhammer[(q)^(1 - n)/a, q, k]]*(-Divide[q,a])^(k)* (q)^(Binomial[k,2]- n*k)
Successful Failure -
Failed [14 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[n, 1], Rule[q, -1.5]}

... skip entries to safe data
17.2.E14 ( a ; q ) n - k ( b ; q ) n - k = ( a ; q ) n ( b ; q ) n ⁒ ( q 1 - n / b ; q ) k ( q 1 - n / a ; q ) k ⁒ ( b a ) k q-Pochhammer-symbol π‘Ž π‘ž 𝑛 π‘˜ q-Pochhammer-symbol 𝑏 π‘ž 𝑛 π‘˜ q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol 𝑏 π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 𝑛 𝑏 π‘ž π‘˜ q-Pochhammer-symbol superscript π‘ž 1 𝑛 π‘Ž π‘ž π‘˜ superscript 𝑏 π‘Ž π‘˜ {\displaystyle{\displaystyle\frac{\left(a;q\right)_{n-k}}{\left(b;q\right)_{n-% k}}=\frac{\left(a;q\right)_{n}}{\left(b;q\right)_{n}}\frac{\left(q^{1-n}/b;q% \right)_{k}}{\left(q^{1-n}/a;q\right)_{k}}\left(\frac{b}{a}\right)^{k}}}
\frac{\qPochhammer{a}{q}{n-k}}{\qPochhammer{b}{q}{n-k}} = \frac{\qPochhammer{a}{q}{n}}{\qPochhammer{b}{q}{n}}\frac{\qPochhammer{q^{1-n}/b}{q}{k}}{\qPochhammer{q^{1-n}/a}{q}{k}}\left(\frac{b}{a}\right)^{k}		 

(QPochhammer(a, q, n - k))/(QPochhammer(b, q, n - k)) = (QPochhammer(a, q, n))/(QPochhammer(b, q, n))*(QPochhammer((q)^(1 - n)/b, q, k))/(QPochhammer((q)^(1 - n)/a, q, k))*((b)/(a))^(k)

Divide[QPochhammer[a, q, n - k],QPochhammer[b, q, n - k]] == Divide[QPochhammer[a, q, n],QPochhammer[b, q, n]]*Divide[QPochhammer[(q)^(1 - n)/b, q, k],QPochhammer[(q)^(1 - n)/a, q, k]]*(Divide[b,a])^(k)
Successful Failure -
Failed [15 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 2], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 3], Rule[n, 1], Rule[q, -1.5]}

... skip entries to safe data
17.2.E15 ( a ⁒ q - n ; q ) k = ( a ; q ) k ⁒ ( q / a ; q ) n ( q 1 - k / a ; q ) n ⁒ q - n ⁒ k q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž π‘˜ q-Pochhammer-symbol π‘Ž π‘ž π‘˜ q-Pochhammer-symbol π‘ž π‘Ž π‘ž 𝑛 q-Pochhammer-symbol superscript π‘ž 1 π‘˜ π‘Ž π‘ž 𝑛 superscript π‘ž 𝑛 π‘˜ {\displaystyle{\displaystyle\left(aq^{-n};q\right)_{k}=\frac{\left(a;q\right)_% {k}\left(q/a;q\right)_{n}}{\left(q^{1-k}/a;q\right)_{n}}q^{-nk}}}
\qPochhammer{aq^{-n}}{q}{k} = \frac{\qPochhammer{a}{q}{k}\qPochhammer{q/a}{q}{n}}{\qPochhammer{q^{1-k}/a}{q}{n}}q^{-nk}		 

QPochhammer(a*(q)^(- n), q, k) = (QPochhammer(a, q, k)*QPochhammer(q/a, q, n))/(QPochhammer((q)^(1 - k)/a, q, n))*(q)^(- n*k)

QPochhammer[a*(q)^(- n), q, k] == Divide[QPochhammer[a, q, k]*QPochhammer[q/a, q, n],QPochhammer[(q)^(1 - k)/a, q, n]]*(q)^(- n*k)
Successful Failure -
Failed [14 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[n, 2], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[n, 3], Rule[q, -1.5]}

... skip entries to safe data
17.2.E16 ( a ⁒ q - n ; q ) n - k = ( q / a ; q ) n ( q / a ; q ) k ⁒ ( - a q ) n - k ⁒ q ( k 2 ) - ( n 2 ) q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž 𝑛 π‘˜ q-Pochhammer-symbol π‘ž π‘Ž π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘Ž π‘ž π‘˜ superscript π‘Ž π‘ž 𝑛 π‘˜ superscript π‘ž binomial π‘˜ 2 binomial 𝑛 2 {\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n-k}=\frac{\left(q/a;q% \right)_{n}}{\left(q/a;q\right)_{k}}\left(-\frac{a}{q}\right)^{n-k}q^{\genfrac% {(}{)}{0.0pt}{}{k}{2}-\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\qPochhammer{aq^{-n}}{q}{n-k} = \frac{\qPochhammer{q/a}{q}{n}}{\qPochhammer{q/a}{q}{k}}\left(-\frac{a}{q}\right)^{n-k}q^{\binom{k}{2}-\binom{n}{2}}		 

QPochhammer(a*(q)^(- n), q, n - k) = (QPochhammer(q/a, q, n))/(QPochhammer(q/a, q, k))*(-(a)/(q))^(n - k)* (q)^(binomial(k,2)-binomial(n,2))

QPochhammer[a*(q)^(- n), q, n - k] == Divide[QPochhammer[q/a, q, n],QPochhammer[q/a, q, k]]*(-Divide[a,q])^(n - k)* (q)^(Binomial[k,2]-Binomial[n,2])
Successful Failure -
Failed [27 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[n, 1], Rule[q, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[n, 2], Rule[q, -1.5]}

... skip entries to safe data
17.2.E17 ( a ⁒ q n ; q ) k = ( a ; q ) k ⁒ ( a ⁒ q k ; q ) n ( a ; q ) n q-Pochhammer-symbol π‘Ž superscript π‘ž 𝑛 π‘ž π‘˜ q-Pochhammer-symbol π‘Ž π‘ž π‘˜ q-Pochhammer-symbol π‘Ž superscript π‘ž π‘˜ π‘ž 𝑛 q-Pochhammer-symbol π‘Ž π‘ž 𝑛 {\displaystyle{\displaystyle\left(aq^{n};q\right)_{k}=\frac{\left(a;q\right)_{% k}\left(aq^{k};q\right)_{n}}{\left(a;q\right)_{n}}}}
\qPochhammer{aq^{n}}{q}{k} = \frac{\qPochhammer{a}{q}{k}\qPochhammer{aq^{k}}{q}{n}}{\qPochhammer{a}{q}{n}}		 

QPochhammer(a*(q)^(n), q, k) = (QPochhammer(a, q, k)*QPochhammer(a*(q)^(k), q, n))/(QPochhammer(a, q, n))

QPochhammer[a*(q)^(n), q, k] == Divide[QPochhammer[a, q, k]*QPochhammer[a*(q)^(k), q, n],QPochhammer[a, q, n]]
Successful Failure -
Failed [6 / 300]
Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 1], Rule[n, 2], Rule[q, -2]}


Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 1], Rule[n, 3], Rule[q, -2]}

... skip entries to safe data
17.2.E18 ( a ⁒ q k ; q ) n - k = ( a ; q ) n ( a ; q ) k q-Pochhammer-symbol π‘Ž superscript π‘ž π‘˜ π‘ž 𝑛 π‘˜ q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol π‘Ž π‘ž π‘˜ {\displaystyle{\displaystyle\left(aq^{k};q\right)_{n-k}=\frac{\left(a;q\right)% _{n}}{\left(a;q\right)_{k}}}}
\qPochhammer{aq^{k}}{q}{n-k} = \frac{\qPochhammer{a}{q}{n}}{\qPochhammer{a}{q}{k}}		 

QPochhammer(a*(q)^(k), q, n - k) = (QPochhammer(a, q, n))/(QPochhammer(a, q, k))

QPochhammer[a*(q)^(k), q, n - k] == Divide[QPochhammer[a, q, n],QPochhammer[a, q, k]]
Successful Failure -
Failed [6 / 300]
Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 2], Rule[n, 1], Rule[q, -2]}


Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 2], Rule[n, 2], Rule[q, -2]}

... skip entries to safe data
17.2.E19 ( a ; q ) 2 ⁒ n = ( a , a ⁒ q ; q 2 ) n q-Pochhammer-symbol π‘Ž π‘ž 2 𝑛 q-multiple-Pochhammer π‘Ž π‘Ž π‘ž superscript π‘ž 2 𝑛 {\displaystyle{\displaystyle\left(a;q\right)_{2n}=\left(a,aq;q^{2}\right)_{n}}}
\qPochhammer{a}{q}{2n} = \qmultiPochhammersym{a,aq}{q^{2}}{n}		 

Error

QPochhammer[a, q, 2*n] == Product[QPochhammer[Part[{a , a*q},i],(q)^(2),n],{i,1,Length[{a , a*q}]}]
Missing Macro Error Failure - Successful [Tested: 180]
17.2.E21 ( a 2 ; q 2 ) n = ( a ; q ) n ⁒ ( - a ; q ) n q-Pochhammer-symbol superscript π‘Ž 2 superscript π‘ž 2 𝑛 q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol π‘Ž π‘ž 𝑛 {\displaystyle{\displaystyle\left(a^{2};q^{2}\right)_{n}=\left(a;q\right)_{n}% \left(-a;q\right)_{n}}}
\qPochhammer{a^{2}}{q^{2}}{n} = \qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}		 

QPochhammer((a)^(2), (q)^(2), n) = QPochhammer(a, q, n)*QPochhammer(- a, q, n)

QPochhammer[(a)^(2), (q)^(2), n] == QPochhammer[a, q, n]*QPochhammer[- a, q, n]
Successful Successful - Successful [Tested: 180]
17.2.E22 ( q ⁒ a 1 2 , - q ⁒ a 1 2 ; q ) n ( a 1 2 , - a 1 2 ; q ) n = ( a ⁒ q 2 ; q 2 ) n ( a ; q 2 ) n q-multiple-Pochhammer π‘ž superscript π‘Ž 1 2 π‘ž superscript π‘Ž 1 2 π‘ž 𝑛 q-multiple-Pochhammer superscript π‘Ž 1 2 superscript π‘Ž 1 2 π‘ž 𝑛 q-Pochhammer-symbol π‘Ž superscript π‘ž 2 superscript π‘ž 2 𝑛 q-Pochhammer-symbol π‘Ž superscript π‘ž 2 𝑛 {\displaystyle{\displaystyle\frac{\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q% \right)_{n}}{\left(a^{\frac{1}{2}},-a^{\frac{1}{2}};q\right)_{n}}=\frac{\left(% aq^{2};q^{2}\right)_{n}}{\left(a;q^{2}\right)_{n}}}}
\frac{\qmultiPochhammersym{qa^{\frac{1}{2}},-qa^{\frac{1}{2}}}{q}{n}}{\qmultiPochhammersym{a^{\frac{1}{2}},-a^{\frac{1}{2}}}{q}{n}} = \frac{\qPochhammer{aq^{2}}{q^{2}}{n}}{\qPochhammer{a}{q^{2}}{n}}		 

Error

Divide[Product[QPochhammer[Part[{q*(a)^(Divide[1,2]), - q*(a)^(Divide[1,2])},i],q,n],{i,1,Length[{q*(a)^(Divide[1,2]), - q*(a)^(Divide[1,2])}]}],Product[QPochhammer[Part[{(a)^(Divide[1,2]), - (a)^(Divide[1,2])},i],q,n],{i,1,Length[{(a)^(Divide[1,2]), - (a)^(Divide[1,2])}]}]] == Divide[QPochhammer[a*(q)^(2), (q)^(2), n],QPochhammer[a, (q)^(2), n]]
Missing Macro Error Successful - Successful [Tested: 180]
17.2.E22 ( a ⁒ q 2 ; q 2 ) n ( a ; q 2 ) n = 1 - a ⁒ q 2 ⁒ n 1 - a q-Pochhammer-symbol π‘Ž superscript π‘ž 2 superscript π‘ž 2 𝑛 q-Pochhammer-symbol π‘Ž superscript π‘ž 2 𝑛 1 π‘Ž superscript π‘ž 2 𝑛 1 π‘Ž {\displaystyle{\displaystyle\frac{\left(aq^{2};q^{2}\right)_{n}}{\left(a;q^{2}% \right)_{n}}=\frac{1-aq^{2n}}{1-a}}}
\frac{\qPochhammer{aq^{2}}{q^{2}}{n}}{\qPochhammer{a}{q^{2}}{n}} = \frac{1-aq^{2n}}{1-a}		 

(QPochhammer(a*(q)^(2), (q)^(2), n))/(QPochhammer(a, (q)^(2), n)) = (1 - a*(q)^(2*n))/(1 - a)

Divide[QPochhammer[a*(q)^(2), (q)^(2), n],QPochhammer[a, (q)^(2), n]] == Divide[1 - a*(q)^(2*n),1 - a]
Successful Failure - Successful [Tested: 180]
17.2.E23 ( a ⁒ q k ; q k ) n ( a ; q k ) n = 1 - a ⁒ q k ⁒ n 1 - a q-Pochhammer-symbol π‘Ž superscript π‘ž π‘˜ superscript π‘ž π‘˜ 𝑛 q-Pochhammer-symbol π‘Ž superscript π‘ž π‘˜ 𝑛 1 π‘Ž superscript π‘ž π‘˜ 𝑛 1 π‘Ž {\displaystyle{\displaystyle\frac{\left(aq^{k};q^{k}\right)_{n}}{\left(a;q^{k}% \right)_{n}}=\frac{1-aq^{kn}}{1-a}}}
\frac{\qPochhammer{aq^{k}}{q^{k}}{n}}{\qPochhammer{a}{q^{k}}{n}} = \frac{1-aq^{kn}}{1-a}		 

(QPochhammer(a*(q)^(k), (q)^(k), n))/(QPochhammer(a, (q)^(k), n)) = (1 - a*(q)^(k*n))/(1 - a)

Divide[QPochhammer[a*(q)^(k), (q)^(k), n],QPochhammer[a, (q)^(k), n]] == Divide[1 - a*(q)^(k*n),1 - a]
Error Failure -
Failed [2 / 300]
Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 1], Rule[n, 2], Rule[q, -2]}


Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[k, 1], Rule[n, 3], Rule[q, -2]}

17.2.E24 lim Ο„ β†’ 0 ⁑ ( a / Ο„ ; q ) n ⁒ Ο„ n = lim Οƒ β†’ ∞ ⁑ ( a ⁒ Οƒ ; q ) n ⁒ Οƒ - n subscript β†’ 𝜏 0 q-Pochhammer-symbol π‘Ž 𝜏 π‘ž 𝑛 superscript 𝜏 𝑛 subscript β†’ 𝜎 q-Pochhammer-symbol π‘Ž 𝜎 π‘ž 𝑛 superscript 𝜎 𝑛 {\displaystyle{\displaystyle\lim_{\tau\to 0}\left(a/\tau;q\right)_{n}\tau^{n}=% \lim_{\sigma\to\infty}\left(a\sigma;q\right)_{n}\sigma^{-n}}}
\lim_{\tau\to 0}\qPochhammer{a/\tau}{q}{n}\tau^{n} = \lim_{\sigma\to\infty}\qPochhammer{a\sigma}{q}{n}\sigma^{-n}		 

limit(QPochhammer(a/tau, q, n)*(tau)^(n), tau = 0) = limit(QPochhammer(a*sigma, q, n)*(sigma)^(- n), sigma = infinity)

Limit[QPochhammer[a/\[Tau], q, n]*\[Tau]^(n), \[Tau] -> 0, GenerateConditions->None] == Limit[QPochhammer[a*\[Sigma], q, n]*\[Sigma]^(- n), \[Sigma] -> Infinity, GenerateConditions->None]
Failure Failure Error Successful [Tested: 180]
17.2.E24 lim Οƒ β†’ ∞ ⁑ ( a ⁒ Οƒ ; q ) n ⁒ Οƒ - n = ( - a ) n ⁒ q ( n 2 ) subscript β†’ 𝜎 q-Pochhammer-symbol π‘Ž 𝜎 π‘ž 𝑛 superscript 𝜎 𝑛 superscript π‘Ž 𝑛 superscript π‘ž binomial 𝑛 2 {\displaystyle{\displaystyle\lim_{\sigma\to\infty}\left(a\sigma;q\right)_{n}% \sigma^{-n}=(-a)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\lim_{\sigma\to\infty}\qPochhammer{a\sigma}{q}{n}\sigma^{-n} = (-a)^{n}q^{\binom{n}{2}}		 

limit(QPochhammer(a*sigma, q, n)*(sigma)^(- n), sigma = infinity) = (- a)^(n)* (q)^(binomial(n,2))

Limit[QPochhammer[a*\[Sigma], q, n]*\[Sigma]^(- n), \[Sigma] -> Infinity, GenerateConditions->None] == (- a)^(n)* (q)^(Binomial[n,2])
Failure Failure Error Successful [Tested: 180]
17.2.E25 lim Ο„ β†’ 0 ⁑ ( a / Ο„ ; q ) n ( b / Ο„ ; q ) n = lim Οƒ β†’ ∞ ⁑ ( a ⁒ Οƒ ; q ) n ( b ⁒ Οƒ ; q ) n subscript β†’ 𝜏 0 q-Pochhammer-symbol π‘Ž 𝜏 π‘ž 𝑛 q-Pochhammer-symbol 𝑏 𝜏 π‘ž 𝑛 subscript β†’ 𝜎 q-Pochhammer-symbol π‘Ž 𝜎 π‘ž 𝑛 q-Pochhammer-symbol 𝑏 𝜎 π‘ž 𝑛 {\displaystyle{\displaystyle\lim_{\tau\to 0}\frac{\left(a/\tau;q\right)_{n}}{% \left(b/\tau;q\right)_{n}}=\lim_{\sigma\to\infty}\frac{\left(a\sigma;q\right)_% {n}}{\left(b\sigma;q\right)_{n}}}}
\lim_{\tau\to 0}\frac{\qPochhammer{a/\tau}{q}{n}}{\qPochhammer{b/\tau}{q}{n}} = \lim_{\sigma\to\infty}\frac{\qPochhammer{a\sigma}{q}{n}}{\qPochhammer{b\sigma}{q}{n}}		 

limit((QPochhammer(a/tau, q, n))/(QPochhammer(b/tau, q, n)), tau = 0) = limit((QPochhammer(a*sigma, q, n))/(QPochhammer(b*sigma, q, n)), sigma = infinity)

Limit[Divide[QPochhammer[a/\[Tau], q, n],QPochhammer[b/\[Tau], q, n]], \[Tau] -> 0, GenerateConditions->None] == Limit[Divide[QPochhammer[a*\[Sigma], q, n],QPochhammer[b*\[Sigma], q, n]], \[Sigma] -> Infinity, GenerateConditions->None]
Failure Failure Error Successful [Tested: 300]
17.2.E25 lim Οƒ β†’ ∞ ⁑ ( a ⁒ Οƒ ; q ) n ( b ⁒ Οƒ ; q ) n = ( a b ) n subscript β†’ 𝜎 q-Pochhammer-symbol π‘Ž 𝜎 π‘ž 𝑛 q-Pochhammer-symbol 𝑏 𝜎 π‘ž 𝑛 superscript π‘Ž 𝑏 𝑛 {\displaystyle{\displaystyle\lim_{\sigma\to\infty}\frac{\left(a\sigma;q\right)% _{n}}{\left(b\sigma;q\right)_{n}}=\left(\frac{a}{b}\right)^{n}}}
\lim_{\sigma\to\infty}\frac{\qPochhammer{a\sigma}{q}{n}}{\qPochhammer{b\sigma}{q}{n}} = \left(\frac{a}{b}\right)^{n}		 

limit((QPochhammer(a*sigma, q, n))/(QPochhammer(b*sigma, q, n)), sigma = infinity) = ((a)/(b))^(n)
 
Limit[Divide[QPochhammer[a*\[Sigma], q, n],QPochhammer[b*\[Sigma], q, n]], \[Sigma] -> Infinity, GenerateConditions->None] == (Divide[a,b])^(n)
 Failure Failure Error Successful [Tested: 300]
17.2.E26 lim Ο„ β†’ 0 ⁑ ( a / Ο„ ; q ) n ⁒ ( b / Ο„ ; q ) n ( c / Ο„ 2 ; q ) n = ( - 1 ) n ⁒ ( a ⁒ b c ) n ⁒ q ( n 2 ) subscript β†’ 𝜏 0 q-Pochhammer-symbol π‘Ž 𝜏 π‘ž 𝑛 q-Pochhammer-symbol 𝑏 𝜏 π‘ž 𝑛 q-Pochhammer-symbol 𝑐 superscript 𝜏 2 π‘ž 𝑛 superscript 1 𝑛 superscript π‘Ž 𝑏 𝑐 𝑛 superscript π‘ž binomial 𝑛 2 {\displaystyle{\displaystyle\lim_{\tau\to 0}\frac{\left(a/\tau;q\right)_{n}% \left(b/\tau;q\right)_{n}}{\left(c/\tau^{2};q\right)_{n}}=(-1)^{n}\left(\frac{% ab}{c}\right)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}}}
\lim_{\tau\to 0}\frac{\qPochhammer{a/\tau}{q}{n}\qPochhammer{b/\tau}{q}{n}}{\qPochhammer{c/\tau^{2}}{q}{n}} = (-1)^{n}\left(\frac{ab}{c}\right)^{n}q^{\binom{n}{2}}		 

limit((QPochhammer(a/tau, q, n)*QPochhammer(b/tau, q, n))/(QPochhammer(c/(tau)^(2), q, n)), tau = 0) = (- 1)^(n)*((a*b)/(c))^(n)* (q)^(binomial(n,2))
 
Limit[Divide[QPochhammer[a/\[Tau], q, n]*QPochhammer[b/\[Tau], q, n],QPochhammer[c/\[Tau]^(2), q, n]], \[Tau] -> 0, GenerateConditions->None] == (- 1)^(n)*(Divide[a*b,c])^(n)* (q)^(Binomial[n,2])
 Error Failure - Successful [Tested: 300]
17.2.E27 [ n m ] q = ( q ; q ) n ( q ; q ) m ⁒ ( q ; q ) n - m q-binomial 𝑛 π‘š π‘ž q-Pochhammer-symbol π‘ž π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘ž π‘š q-Pochhammer-symbol π‘ž π‘ž 𝑛 π‘š {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{\left(q;q% \right)_{n}}{\left(q;q\right)_{m}\left(q;q\right)_{n-m}}\\ }}
\qbinom{n}{m}{q} = \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{m}\qPochhammer{q}{q}{n-m}}\\		 

QBinomial(n, m, q) = (QPochhammer(q, q, n))/(QPochhammer(q, q, m)*QPochhammer(q, q, n - m))
 
QBinomial[n,m,q] == Divide[QPochhammer[q, q, n],QPochhammer[q, q, m]*QPochhammer[q, q, n - m]]
 Successful Failure -
Failed [1 / 90]
Result: Complex[-0.058394160583941646, 0.1605839416058394]
Test Values: {Rule[m, 3], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

17.2.E27 ( q ; q ) n ( q ; q ) m ⁒ ( q ; q ) n - m = ( q - n ; q ) m ⁒ ( - 1 ) m ⁒ q n ⁒ m - ( m 2 ) ( q ; q ) m q-Pochhammer-symbol π‘ž π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘ž π‘š q-Pochhammer-symbol π‘ž π‘ž 𝑛 π‘š q-Pochhammer-symbol superscript π‘ž 𝑛 π‘ž π‘š superscript 1 π‘š superscript π‘ž 𝑛 π‘š binomial π‘š 2 q-Pochhammer-symbol π‘ž π‘ž π‘š {\displaystyle{\displaystyle\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{m}% \left(q;q\right)_{n-m}}\\ =\frac{\left(q^{-n};q\right)_{m}(-1)^{m}q^{nm-\genfrac{(}{)}{0.0pt}{}{m}{2}}}{% \left(q;q\right)_{m}}}}
\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{m}\qPochhammer{q}{q}{n-m}}\\ = \frac{\qPochhammer{q^{-n}}{q}{m}(-1)^{m}q^{nm-\binom{m}{2}}}{\qPochhammer{q}{q}{m}}		 

(QPochhammer(q, q, n))/(QPochhammer(q, q, m)*QPochhammer(q, q, n - m)) = (QPochhammer((q)^(- n), q, m)*(- 1)^(m)* (q)^(n*m -binomial(m,2)))/(QPochhammer(q, q, m))
 
Divide[QPochhammer[q, q, n],QPochhammer[q, q, m]*QPochhammer[q, q, n - m]] == Divide[QPochhammer[(q)^(- n), q, m]*(- 1)^(m)* (q)^(n*m -Binomial[m,2]),QPochhammer[q, q, m]]
 Successful Failure -
Failed [3 / 90]
Result: Complex[0.11678832116788332, -0.3211678832116788]
Test Values: {Rule[m, 3], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[1.0, 0.0]
Test Values: {Rule[m, 3], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.2.E28 lim q β†’ 1 ⁑ [ n m ] q = ( n m ) subscript β†’ π‘ž 1 q-binomial 𝑛 π‘š π‘ž binomial 𝑛 π‘š {\displaystyle{\displaystyle\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=% \genfrac{(}{)}{0.0pt}{}{n}{m}}}
\lim_{q\to 1}\qbinom{n}{m}{q} = \binom{n}{m}		 

limit(QBinomial(n, m, q), q = 1) = binomial(n,m)
 
Limit[QBinomial[n,m,q], q -> 1, GenerateConditions->None] == Binomial[n,m]
 Failure Aborted Error Skipped - Because timed out
17.2.E28 ( n m ) = n ! m ! ⁒ ( n - m ) ! binomial 𝑛 π‘š 𝑛 π‘š 𝑛 π‘š {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{m}=\frac{n!}{m!(n-m)!}}}
\binom{n}{m} = \frac{n!}{m!(n-m)!}		 

binomial(n,m) = (factorial(n))/(factorial(m)*factorial(n - m))
 
Binomial[n,m] == Divide[(n)!,(m)!*(n - m)!]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 9]
17.2.E29 [ m + n m ] q = ( q n + 1 ; q ) m ( q ; q ) m q-binomial π‘š 𝑛 π‘š π‘ž q-Pochhammer-symbol superscript π‘ž 𝑛 1 π‘ž π‘š q-Pochhammer-symbol π‘ž π‘ž π‘š {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{q}=\frac{\left(q^% {n+1};q\right)_{m}}{\left(q;q\right)_{m}}}}
\qbinom{m+n}{m}{q} = \frac{\qPochhammer{q^{n+1}}{q}{m}}{\qPochhammer{q}{q}{m}}		 

QBinomial(m + n, m, q) = (QPochhammer((q)^(n + 1), q, m))/(QPochhammer(q, q, m))
 
QBinomial[m + n,m,q] == Divide[QPochhammer[(q)^(n + 1), q, m],QPochhammer[q, q, m]]
 Successful Failure -
Failed [3 / 90]
Result: Complex[0.9416058394160581, 0.1605839416058394]
Test Values: {Rule[m, 3], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[1.0, 0.0]
Test Values: {Rule[m, 3], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.2.E30 [ - n m ] q = [ m + n - 1 m ] q ⁒ ( - 1 ) m ⁒ q - m ⁒ n - ( m 2 ) q-binomial 𝑛 π‘š π‘ž q-binomial π‘š 𝑛 1 π‘š π‘ž superscript 1 π‘š superscript π‘ž π‘š 𝑛 binomial π‘š 2 {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}=\genfrac{[}{]}{% 0.0pt}{}{m+n-1}{m}_{q}(-1)^{m}q^{-mn-\genfrac{(}{)}{0.0pt}{}{m}{2}}}}
\qbinom{-n}{m}{q} = \qbinom{m+n-1}{m}{q}(-1)^{m}q^{-mn-\binom{m}{2}}		 

QBinomial(- n, m, q) = QBinomial(m + n - 1, m, q)*(- 1)^(m)* (q)^(- m*n -binomial(m,2))
 
QBinomial[- n,m,q] == QBinomial[m + n - 1,m,q]*(- 1)^(m)* (q)^(- m*n -Binomial[m,2])
 Failure Failure Error
Failed [84 / 90]
Result: Complex[0.7320508075688774, 0.0]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-2.3660254037844384, -1.3660254037844386]
Test Values: {Rule[m, 1], Rule[n, 3], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.2.E31 [ n m ] q = [ n - 1 m - 1 ] q + q m ⁒ [ n - 1 m ] q q-binomial 𝑛 π‘š π‘ž q-binomial 𝑛 1 π‘š 1 π‘ž superscript π‘ž π‘š q-binomial 𝑛 1 π‘š π‘ž {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{[}{]}{0% .0pt}{}{n-1}{m-1}_{q}+q^{m}\genfrac{[}{]}{0.0pt}{}{n-1}{m}_{q}}}
\qbinom{n}{m}{q} = \qbinom{n-1}{m-1}{q}+q^{m}\qbinom{n-1}{m}{q}		 

QBinomial(n, m, q) = QBinomial(n - 1, m - 1, q)+ (q)^(m)* QBinomial(n - 1, m, q)
 
QBinomial[n,m,q] == QBinomial[n - 1,m - 1,q]+ (q)^(m)* QBinomial[n - 1,m,q]
 Successful Failure - Successful [Tested: 90]
17.2.E32 [ n m ] q = [ n - 1 m ] q + q n - m ⁒ [ n - 1 m - 1 ] q q-binomial 𝑛 π‘š π‘ž q-binomial 𝑛 1 π‘š π‘ž superscript π‘ž 𝑛 π‘š q-binomial 𝑛 1 π‘š 1 π‘ž {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{[}{]}{0% .0pt}{}{n-1}{m}_{q}+q^{n-m}\genfrac{[}{]}{0.0pt}{}{n-1}{m-1}_{q}}}
\qbinom{n}{m}{q} = \qbinom{n-1}{m}{q}+q^{n-m}\qbinom{n-1}{m-1}{q}		 

QBinomial(n, m, q) = QBinomial(n - 1, m, q)+ (q)^(n - m)* QBinomial(n - 1, m - 1, q)
 
QBinomial[n,m,q] == QBinomial[n - 1,m,q]+ (q)^(n - m)* QBinomial[n - 1,m - 1,q]
 Successful Failure - Successful [Tested: 90]
17.2.E33 lim n β†’ ∞ ⁑ [ n m ] q = 1 ( q ; q ) m subscript β†’ 𝑛 q-binomial 𝑛 π‘š π‘ž 1 q-Pochhammer-symbol π‘ž π‘ž π‘š {\displaystyle{\displaystyle\lim_{n\to\infty}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}% =\frac{1}{\left(q;q\right)_{m}}}}
\lim_{n\to\infty}\qbinom{n}{m}{q} = \frac{1}{\qPochhammer{q}{q}{m}}		 

limit(QBinomial(n, m, q), n = infinity) = (1)/(QPochhammer(q, q, m))
 
Limit[QBinomial[n,m,q], n -> Infinity, GenerateConditions->None] == Divide[1,QPochhammer[q, q, m]]
 Failure Failure Error
Failed [24 / 30]
Result: Plus[Complex[-0.5, -1.866025403784439], Times[Complex[-0.5, -1.866025403784439], Plus[-1.0, Power[2.718281828459045, Times[Complex[0.0, 2.0], Interval[{-2.2250738585072014*^-308, 3.1415926535897936}]]]]]]
Test Values: {Rule[m, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[1.3660254037844395, -1.3660254037844388], Times[Complex[0.5000000000000009, -1.866025403784439], Plus[-1.0, Power[2.718281828459045, Times[Complex[0.0, 2.0], Interval[{-2.2250738585072014*^-308, 3.1415926535897936}]]]], Plus[Complex[0.8660254037844387, 0.49999999999999994], Power[2.718281828459045, Times[Complex[0.0, 2.0], Interval[{-2.2250738585072014*^-308, 3.1415926535897936}]]]]]]
Test Values: {Rule[m, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.2.E34 lim n β†’ ∞ ⁑ [ r ⁒ n + u s ⁒ n + t ] q = 1 ( q ; q ) ∞ subscript β†’ 𝑛 q-binomial π‘Ÿ 𝑛 𝑒 𝑠 𝑛 𝑑 π‘ž 1 q-Pochhammer-symbol π‘ž π‘ž {\displaystyle{\displaystyle\lim_{n\to\infty}\genfrac{[}{]}{0.0pt}{}{rn+u}{sn+% t}_{q}=\frac{1}{\left(q;q\right)_{\infty}}}}
\lim_{n\to\infty}\qbinom{rn+u}{sn+t}{q} = \frac{1}{\qPochhammer{q}{q}{\infty}}		 

limit(QBinomial(r*n + u, sn(+)*t, q), n = infinity) = (1)/(QPochhammer(q, q, infinity))
 
Limit[QBinomial[r*n + u,sn[+]*t,q], n -> Infinity, GenerateConditions->None] == Divide[1,QPochhammer[q, q, Infinity]]
 Error Failure Skip - symbolical successful subtest Error
17.2.E34 1 ( q ; q ) ∞ = ∏ j = 1 ∞ 1 ( 1 - q j ) 1 q-Pochhammer-symbol π‘ž π‘ž superscript subscript product 𝑗 1 1 1 superscript π‘ž 𝑗 {\displaystyle{\displaystyle\frac{1}{\left(q;q\right)_{\infty}}=\prod_{j=1}^{% \infty}\frac{1}{(1-q^{j})}}}
\frac{1}{\qPochhammer{q}{q}{\infty}} = \prod_{j=1}^{\infty}\frac{1}{(1-q^{j})}		 

(1)/(QPochhammer(q, q, infinity)) = product((1)/(1 - (q)^(j)), j = 1..infinity)
 
Divide[1,QPochhammer[q, q, Infinity]] == Product[Divide[1,1 - (q)^(j)], {j, 1, Infinity}, GenerateConditions->None]
 Failure Failure Error
Failed [7 / 10]
Result: Plus[Times[-1.0, Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -1]], Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Times[-1.0, Power[QPochhammer[Complex[0.5000000000000001, -0.8660254037844386], Complex[0.5000000000000001, -0.8660254037844386]], -1]], Power[QPochhammer[Complex[0.5000000000000001, -0.8660254037844386], Complex[0.5000000000000001, -0.8660254037844386], DirectedInfinity[1]], -1]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
17.2.E35 βˆ‘ j = 0 n [ n j ] q ⁒ ( - z ) j ⁒ q ( j 2 ) = ( z ; q ) n superscript subscript 𝑗 0 𝑛 q-binomial 𝑛 𝑗 π‘ž superscript 𝑧 𝑗 superscript π‘ž binomial 𝑗 2 q-Pochhammer-symbol 𝑧 π‘ž 𝑛 {\displaystyle{\displaystyle\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}(-z% )^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}=\left(z;q\right)_{n}}}
\sum_{j=0}^{n}\qbinom{n}{j}{q}(-z)^{j}q^{\binom{j}{2}} = \qPochhammer{z}{q}{n}		 

sum(QBinomial(n, j, q)*(- z)^(j)* (q)^(binomial(j,2)), j = 0..n) = QPochhammer(z, q, n)
 
Sum[QBinomial[n,j,q]*(- z)^(j)* (q)^(Binomial[j,2]), {j, 0, n}, GenerateConditions->None] == QPochhammer[z, q, n]
 Failure Successful Error Successful [Tested: 210]
17.2.E36 βˆ‘ j = 0 n ( n j ) ⁒ ( - z ) j = ( 1 - z ) n superscript subscript 𝑗 0 𝑛 binomial 𝑛 𝑗 superscript 𝑧 𝑗 superscript 1 𝑧 𝑛 {\displaystyle{\displaystyle\sum_{j=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}(-z)^{j% }=(1-z)^{n}}}
\sum_{j=0}^{n}\binom{n}{j}(-z)^{j} = (1-z)^{n}		 

sum(binomial(n,j)*(- z)^(j), j = 0..n) = (1 - z)^(n)
 
Sum[Binomial[n,j]*(- z)^(j), {j, 0, n}, GenerateConditions->None] == (1 - z)^(n)
 Successful Successful - Successful [Tested: 21]
17.2.E37 βˆ‘ n = 0 ∞ ( a ; q ) n ( q ; q ) n ⁒ z n = ( a ⁒ z ; q ) ∞ ( z ; q ) ∞ superscript subscript 𝑛 0 q-Pochhammer-symbol π‘Ž π‘ž 𝑛 q-Pochhammer-symbol π‘ž π‘ž 𝑛 superscript 𝑧 𝑛 q-Pochhammer-symbol π‘Ž 𝑧 π‘ž q-Pochhammer-symbol 𝑧 π‘ž {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}}{% \left(q;q\right)_{n}}z^{n}=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_% {\infty}}}}
\sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q}{q}{n}}z^{n} = \frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}		 

sum((QPochhammer(a, q, n))/(QPochhammer(q, q, n))*(z)^(n), n = 0..infinity) = (QPochhammer(a*z, q, infinity))/(QPochhammer(z, q, infinity))
 
Sum[Divide[QPochhammer[a, q, n],QPochhammer[q, q, n]]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[QPochhammer[a*z, q, Infinity],QPochhammer[z, q, Infinity]]
 Failure Failure Error
Failed [240 / 300]
Result: Plus[Times[QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.8660254037844387, 0.49999999999999994]], Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[-1.0, QPochhammer[Complex[-1.299038105676658, -0.7499999999999999], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], Power[QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1]]]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Times[Power[QPochhammer[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], -1], QPochhammer[Complex[0.7499999999999997, -1.299038105676658], Complex[0.8660254037844387, 0.49999999999999994]]], Times[-1.0, Power[QPochhammer[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.7499999999999997, -1.299038105676658], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
17.2.E38 βˆ‘ n = 0 ∞ [ n + m n ] q ⁒ z n = 1 ( z ; q ) m + 1 superscript subscript 𝑛 0 q-binomial 𝑛 π‘š 𝑛 π‘ž superscript 𝑧 𝑛 1 q-Pochhammer-symbol 𝑧 π‘ž π‘š 1 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\genfrac{[}{]}{0.0pt}{}{n+m}{n}% _{q}z^{n}=\frac{1}{\left(z;q\right)_{m+1}}}}
\sum_{n=0}^{\infty}\qbinom{n+m}{n}{q}z^{n} = \frac{1}{\qPochhammer{z}{q}{m+1}}		 

sum(QBinomial(n + m, n, q)*(z)^(n), n = 0..infinity) = (1)/(QPochhammer(z, q, m + 1))
 
Sum[QBinomial[n + m,n,q]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,QPochhammer[z, q, m + 1]]
 Failure Aborted Error Skipped - Because timed out
17.2.E39 βˆ‘ j = 0 n [ n j ] q 2 ⁒ q j = ( - q ; q ) n superscript subscript 𝑗 0 𝑛 q-binomial 𝑛 𝑗 superscript π‘ž 2 superscript π‘ž 𝑗 q-Pochhammer-symbol π‘ž π‘ž 𝑛 {\displaystyle{\displaystyle\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q^{2}% }q^{j}=\left(-q;q\right)_{n}}}
\sum_{j=0}^{n}\qbinom{n}{j}{q^{2}}q^{j} = \qPochhammer{-q}{q}{n}		 

sum(QBinomial(n, j, (q)^(2))*(q)^(j), j = 0..n) = QPochhammer(- q, q, n)
 
Sum[QBinomial[n,j,(q)^(2)]*(q)^(j), {j, 0, n}, GenerateConditions->None] == QPochhammer[- q, q, n]
 Failure Aborted Error Successful [Tested: 30]
17.2.E40 βˆ‘ j = 0 2 ⁒ n ( - 1 ) j ⁒ [ 2 ⁒ n j ] q = ( q ; q 2 ) n superscript subscript 𝑗 0 2 𝑛 superscript 1 𝑗 q-binomial 2 𝑛 𝑗 π‘ž q-Pochhammer-symbol π‘ž superscript π‘ž 2 𝑛 {\displaystyle{\displaystyle\sum_{j=0}^{2n}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{2n}% {j}_{q}=\left(q;q^{2}\right)_{n}}}
\sum_{j=0}^{2n}(-1)^{j}\qbinom{2n}{j}{q} = \qPochhammer{q}{q^{2}}{n}		 

sum((- 1)^(j)* QBinomial(2*n, j, q), j = 0..2*n) = QPochhammer(q, (q)^(2), n)
 
Sum[(- 1)^(j)* QBinomial[2*n,j,q], {j, 0, 2*n}, GenerateConditions->None] == QPochhammer[q, (q)^(2), n]
 Failure Successful Error Successful [Tested: 30]
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