Orthogonal Polynomials - 18.27 -Hahn Class

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.27.E1 A ⁢ ( x ) ⁢ p n ⁢ ( q ⁢ x ) + B ⁢ ( x ) ⁢ p n ⁢ ( x ) + C ⁢ ( x ) ⁢ p n ⁢ ( q - 1 ⁢ x ) = λ n ⁢ p n ⁢ ( x ) 𝐴 𝑥 subscript 𝑝 𝑛 𝑞 𝑥 𝐵 𝑥 subscript 𝑝 𝑛 𝑥 𝐶 𝑥 subscript 𝑝 𝑛 superscript 𝑞 1 𝑥 subscript 𝜆 𝑛 subscript 𝑝 𝑛 𝑥 {\displaystyle{\displaystyle A(x)p_{n}(qx)+B(x)p_{n}(x)+C(x)p_{n}(q^{-1}x)=% \lambda_{n}p_{n}(x)}}
A(x)p_{n}(qx)+B(x)p_{n}(x)+C(x)p_{n}(q^{-1}x) = \lambda_{n}p_{n}(x)		 

A(x)* p[n](q*x)+ B(x)* p[n](x)+ C(x)* p[n]((q)^(- 1)* x) = lambda[n]*p[n](x)
 
A[x]* Subscript[p, n][q*x]+ B[x]* Subscript[p, n][x]+ C[x]* Subscript[p, n][(q)^(- 1)* x] == Subscript[\[Lambda], n]*Subscript[p, n][x]
 Skipped - no semantic math Skipped - no semantic math - -
18.27.E9 v x = ( a - 1 ⁢ x , c - 1 ⁢ x ; q ) ∞ ( x , b ⁢ c - 1 ⁢ x ; q ) ∞ subscript 𝑣 𝑥 subscript superscript 𝑎 1 𝑥 superscript 𝑐 1 𝑥 𝑞 subscript 𝑥 𝑏 superscript 𝑐 1 𝑥 𝑞 {\displaystyle{\displaystyle v_{x}=\frac{(a^{-1}x,c^{-1}x;q)_{\infty}}{(x,bc^{% -1}x;q)_{\infty}}}}
v_{x} = \frac{(a^{-1}x,c^{-1}x;q)_{\infty}}{(x,bc^{-1}x;q)_{\infty}}		 0 < a , a < q - 1 , 0 < b , b < q - 1 , c < 0 formulae-sequence 0 𝑎 formulae-sequence 𝑎 superscript 𝑞 1 formulae-sequence 0 𝑏 formulae-sequence 𝑏 superscript 𝑞 1 𝑐 0 {\displaystyle{\displaystyle 0<a,a<q^{-1},0<b,b<q^{-1},c<0}} 
v[x] = ((a)^(- 1)* x , (c)^(- 1)* x ; q[infinity])/(x , b*(c)^(- 1)* x ; q[infinity])
 
Subscript[v, x] == Divide[Subscript[(a)^(- 1)* x , (c)^(- 1)* x ; q, Infinity],Subscript[x , b*(c)^(- 1)* x ; q, Infinity]]
 Skipped - no semantic math Skipped - no semantic math - -
18.27.E12 v x = ( q ⁢ x / c , - q ⁢ x / d ; q ) ∞ ( q α + 1 ⁢ x / c , - q β + 1 ⁢ x / d ; q ) ∞ subscript 𝑣 𝑥 subscript 𝑞 𝑥 𝑐 𝑞 𝑥 𝑑 𝑞 subscript superscript 𝑞 𝛼 1 𝑥 𝑐 superscript 𝑞 𝛽 1 𝑥 𝑑 𝑞 {\displaystyle{\displaystyle v_{x}=\frac{(qx/c,-qx/d;q)_{\infty}}{(q^{\alpha+1% }x/c,-q^{\beta+1}x/d;q)_{\infty}}}}
v_{x} = \frac{(qx/c,-qx/d;q)_{\infty}}{(q^{\alpha+1}x/c,-q^{\beta+1}x/d;q)_{\infty}}		 α > - 1 , β > - 1 , c > 0 , d > 0 formulae-sequence 𝛼 1 formulae-sequence 𝛽 1 formulae-sequence 𝑐 0 𝑑 0 {\displaystyle{\displaystyle\alpha>-1,\beta>-1,c>0,d>0}} 
v[x] = (q*x/c , - q*x/d ; q[infinity])/((q)^(alpha + 1)* x/c , - (q)^(beta + 1)* x/d ; q[infinity])
 
Subscript[v, x] == Divide[Subscript[q*x/c , - q*x/d ; q, Infinity],Subscript[(q)^(\[Alpha]+ 1)* x/c , - (q)^(\[Beta]+ 1)* x/d ; q, Infinity]]
 Skipped - no semantic math Skipped - no semantic math - -
18.27.E21 ( q ; q ) n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ q ℓ ⁢ ( ℓ - 1 ) ⁢ x n - 2 ⁢ ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q ; q ) n - 2 ⁢ ℓ = x n ⁢ ϕ 0 2 ⁡ ( q - n , q - n + 1 - ; q 2 , x - 2 ⁢ q 2 ⁢ n - 1 ) q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 𝑞 ℓ ℓ 1 superscript 𝑥 𝑛 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 2 ℓ superscript 𝑥 𝑛 q-hypergeometric-rphis 2 0 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 superscript 𝑥 2 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\left(q;q\right)_{n}\sum_{\ell=0}^{\left\lfloor n/% 2\right\rfloor}\frac{(-1)^{\ell}q^{\ell(\ell-1)}x^{n-2\ell}}{\left(q^{2};q^{2}% \right)_{\ell}\left(q;q\right)_{n-2\ell}}=x^{n}{{}_{2}\phi_{0}}\left({q^{-n},q% ^{-n+1}\atop-};q^{2},x^{-2}q^{2n-1}\right)}}
\qPochhammer{q}{q}{n}\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}q^{\ell(\ell-1)}x^{n-2\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q}{q}{n-2\ell}} = x^{n}\qgenhyperphi{2}{0}@@{q^{-n},q^{-n+1}}{-}{q^{2}}{x^{-2}q^{2n-1}}		 

Error

QPochhammer[q, q, n]*Sum[Divide[(- 1)^\[ScriptL]* (q)^(\[ScriptL]*(\[ScriptL]- 1))* (x)^(n - 2*\[ScriptL]),QPochhammer[(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[q, q, n - 2*\[ScriptL]]], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (x)^(n)* QHypergeometricPFQ[{(q)^(- n), (q)^(- n + 1)},{-},(q)^(2),(x)^(- 2)* (q)^(2*n - 1)]
Missing Macro Error Failure - Error
18.27.E23 ( q ; q ) n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ q - 2 ⁢ n ⁢ ℓ ⁢ q ℓ ⁢ ( 2 ⁢ ℓ + 1 ) ⁢ x n - 2 ⁢ ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q ; q ) n - 2 ⁢ ℓ = x n ⁢ ϕ 1 2 ⁡ ( q - n , q - n + 1 0 ; q 2 , - x - 2 ⁢ q 2 ) q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 𝑞 2 𝑛 ℓ superscript 𝑞 ℓ 2 ℓ 1 superscript 𝑥 𝑛 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 2 ℓ superscript 𝑥 𝑛 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 0 superscript 𝑞 2 superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle\left(q;q\right)_{n}\sum_{\ell=0}^{\left\lfloor n/% 2\right\rfloor}\frac{(-1)^{\ell}q^{-2n\ell}q^{\ell(2\ell+1)}x^{n-2\ell}}{\left% (q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-2\ell}}=x^{n}{{}_{2}\phi_{1}}% \left({q^{-n},q^{-n+1}\atop 0};q^{2},-x^{-2}q^{2}\right)}}
\qPochhammer{q}{q}{n}\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}q^{-2n\ell}q^{\ell(2\ell+1)}x^{n-2\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q}{q}{n-2\ell}} = x^{n}\qgenhyperphi{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^{2}}{-x^{-2}q^{2}}		 

Error

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