Orthogonal Polynomials - 18.33 Polynomials Orthogonal on the Unit Circle

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.33.E2 ϕ n ⁢ ( z ) = κ n ⁢ z n + ∑ ℓ = 1 n κ n , n - ℓ ⁢ z n - ℓ subscript italic-ϕ 𝑛 𝑧 subscript 𝜅 𝑛 superscript 𝑧 𝑛 superscript subscript ℓ 1 𝑛 subscript 𝜅 𝑛 𝑛 ℓ superscript 𝑧 𝑛 ℓ {\displaystyle{\displaystyle\phi_{n}(z)=\kappa_{n}z^{n}+\sum_{\ell=1}^{n}% \kappa_{n,n-\ell}z^{n-\ell}}}
\phi_{n}(z) = \kappa_{n}z^{n}+\sum_{\ell=1}^{n}\kappa_{n,n-\ell}z^{n-\ell}		 

phi[n](z) = kappa[n]*(z)^(n)+ sum(kappa[n , n - ell]*(z)^(n - ell), ell = 1..n)
 
Subscript[\[Phi], n][z] == Subscript[\[Kappa], n]*(z)^(n)+ Sum[Subscript[\[Kappa], n , n - \[ScriptL]]*(z)^(n - \[ScriptL]), {\[ScriptL], 1, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E3 ϕ n * ⁢ ( z ) = z n ⁢ ϕ n ⁢ ( z ¯ - 1 ) ¯ superscript subscript italic-ϕ 𝑛 𝑧 superscript 𝑧 𝑛 subscript italic-ϕ 𝑛 𝑧 1 {\displaystyle{\displaystyle\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline% {z}^{-1}})}}}
\phi_{n}^{*}(z) = z^{n}\conj{\phi_{n}(\conj{z}^{-1})}		 

(phi[n])^(*)(z) = (z)^(n)* conjugate(phi[n]((conjugate(z))^(- 1)))

(Subscript[\[Phi], n])^(*)[z] == (z)^(n)* Conjugate[Subscript[\[Phi], n][(Conjugate[z])^(- 1)]]
Error Failure Skip - symbolical successful subtest Error
18.33.E3 z n ⁢ ϕ n ⁢ ( z ¯ - 1 ) ¯ = κ n + ∑ ℓ = 1 n κ ¯ n , n - ℓ ⁢ z ℓ superscript 𝑧 𝑛 subscript italic-ϕ 𝑛 𝑧 1 subscript 𝜅 𝑛 superscript subscript ℓ 1 𝑛 subscript 𝜅 𝑛 𝑛 ℓ superscript 𝑧 ℓ {\displaystyle{\displaystyle z^{n}\overline{\phi_{n}({\overline{z}^{-1}})}={% \kappa_{n}}+\sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell}}}
z^{n}\conj{\phi_{n}(\conj{z}^{-1})} = {\kappa_{n}}+\sum_{\ell=1}^{n}\conj{\kappa}_{n,n-\ell}z^{\ell}		 

(z)^(n)* conjugate(phi[n]((conjugate(z))^(- 1))) = kappa[n]+ sum(conjugate(kappa)[n , n - ell]*(z)^(ell), ell = 1..n)

(z)^(n)* Conjugate[Subscript[\[Phi], n][(Conjugate[z])^(- 1)]] == Subscript[\[Kappa], n]+ Sum[Subscript[Conjugate[\[Kappa]], n , n - \[ScriptL]]*(z)^\[ScriptL], {\[ScriptL], 1, n}, GenerateConditions->None]
Aborted Failure Error
Failed [300 / 300]
Result: Plus[Complex[0.0, -0.9999999999999999], Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 1, 0]]]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[κ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.1339745962155613, -0.49999999999999994], Times[Complex[-0.5000000000000001, -0.8660254037844386], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2, 0]], Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2, 1]]]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[κ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.33.E4 κ n ⁢ z ⁢ ϕ n ⁢ ( z ) = κ n + 1 ⁢ ϕ n + 1 ⁢ ( z ) - ϕ n + 1 ⁢ ( 0 ) ⁢ ϕ n + 1 * ⁢ ( z ) subscript 𝜅 𝑛 𝑧 subscript italic-ϕ 𝑛 𝑧 subscript 𝜅 𝑛 1 subscript italic-ϕ 𝑛 1 𝑧 subscript italic-ϕ 𝑛 1 0 superscript subscript italic-ϕ 𝑛 1 𝑧 {\displaystyle{\displaystyle\kappa_{n}z\phi_{n}(z)=\kappa_{n+1}\phi_{n+1}(z)-% \phi_{n+1}(0)\phi_{n+1}^{*}(z)}}
\kappa_{n}z\phi_{n}(z) = \kappa_{n+1}\phi_{n+1}(z)-\phi_{n+1}(0)\phi_{n+1}^{*}(z)		 

kappa[n]*z*phi[n](z) = kappa[n + 1]*phi[n + 1](z)- phi[n + 1](0)* (phi[n + 1])^(*)(z)
 
Subscript[\[Kappa], n]*z*Subscript[\[Phi], n][z] == Subscript[\[Kappa], n + 1]*Subscript[\[Phi], n + 1][z]- Subscript[\[Phi], n + 1][0]* (Subscript[\[Phi], n + 1])^(*)[z]
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E5 κ n ⁢ ϕ n + 1 ⁢ ( z ) = κ n + 1 ⁢ z ⁢ ϕ n ⁢ ( z ) + ϕ n + 1 ⁢ ( 0 ) ⁢ ϕ n * ⁢ ( z ) subscript 𝜅 𝑛 subscript italic-ϕ 𝑛 1 𝑧 subscript 𝜅 𝑛 1 𝑧 subscript italic-ϕ 𝑛 𝑧 subscript italic-ϕ 𝑛 1 0 superscript subscript italic-ϕ 𝑛 𝑧 {\displaystyle{\displaystyle\kappa_{n}\phi_{n+1}(z)=\kappa_{n+1}z\phi_{n}(z)+% \phi_{n+1}(0)\phi_{n}^{*}(z)}}
\kappa_{n}\phi_{n+1}(z) = \kappa_{n+1}z\phi_{n}(z)+\phi_{n+1}(0)\phi_{n}^{*}(z)		 

kappa[n]*phi[n + 1](z) = kappa[n + 1]*z*phi[n](z)+ phi[n + 1](0)* (phi[n])^(*)(z)
 
Subscript[\[Kappa], n]*Subscript[\[Phi], n + 1][z] == Subscript[\[Kappa], n + 1]*z*Subscript[\[Phi], n][z]+ Subscript[\[Phi], n + 1][0]* (Subscript[\[Phi], n])^(*)[z]
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E6 κ n ⁢ ϕ n ⁢ ( 0 ) ⁢ ϕ n + 1 ⁢ ( z ) + κ n - 1 ⁢ ϕ n + 1 ⁢ ( 0 ) ⁢ z ⁢ ϕ n - 1 ⁢ ( z ) = ( κ n ⁢ ϕ n + 1 ⁢ ( 0 ) + κ n + 1 ⁢ ϕ n ⁢ ( 0 ) ⁢ z ) ⁢ ϕ n ⁢ ( z ) subscript 𝜅 𝑛 subscript italic-ϕ 𝑛 0 subscript italic-ϕ 𝑛 1 𝑧 subscript 𝜅 𝑛 1 subscript italic-ϕ 𝑛 1 0 𝑧 subscript italic-ϕ 𝑛 1 𝑧 subscript 𝜅 𝑛 subscript italic-ϕ 𝑛 1 0 subscript 𝜅 𝑛 1 subscript italic-ϕ 𝑛 0 𝑧 subscript italic-ϕ 𝑛 𝑧 {\displaystyle{\displaystyle\kappa_{n}\phi_{n}(0)\phi_{n+1}(z)+\kappa_{n-1}% \phi_{n+1}(0)z\phi_{n-1}(z)=\left(\kappa_{n}\phi_{n+1}(0)+\kappa_{n+1}\phi_{n}% (0)z\right)\phi_{n}(z)}}
\kappa_{n}\phi_{n}(0)\phi_{n+1}(z)+\kappa_{n-1}\phi_{n+1}(0)z\phi_{n-1}(z) = \left(\kappa_{n}\phi_{n+1}(0)+\kappa_{n+1}\phi_{n}(0)z\right)\phi_{n}(z)		 

kappa[n]*phi[n](0)* phi[n + 1](z)+ kappa[n - 1]*phi[n + 1](0)* z*phi[n - 1](z) = (kappa[n]*phi[n + 1](0)+ kappa[n + 1]*phi[n](0)* z)*phi[n](z)
 
Subscript[\[Kappa], n]*Subscript[\[Phi], n][0]* Subscript[\[Phi], n + 1][z]+ Subscript[\[Kappa], n - 1]*Subscript[\[Phi], n + 1][0]* z*Subscript[\[Phi], n - 1][z] == (Subscript[\[Kappa], n]*Subscript[\[Phi], n + 1][0]+ Subscript[\[Kappa], n + 1]*Subscript[\[Phi], n][0]* z)*Subscript[\[Phi], n][z]
 Skipped - no semantic math Skipped - no semantic math - -
18.33#Ex1 w 1 ⁢ ( x ) = ( 1 - x 2 ) - 1 2 ⁢ w ⁢ ( x + i ⁢ ( 1 - x 2 ) 1 2 ) subscript 𝑤 1 𝑥 superscript 1 superscript 𝑥 2 1 2 𝑤 𝑥 imaginary-unit superscript 1 superscript 𝑥 2 1 2 {\displaystyle{\displaystyle w_{1}(x)=(1-x^{2})^{-\frac{1}{2}}w\left(x+\mathrm% {i}(1-x^{2})^{\frac{1}{2}}\right)}}
w_{1}(x) = (1-x^{2})^{-\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)		 

w[1](x) = (1 - (x)^(2))^(-(1)/(2))* w(x + I*(1 - (x)^(2))^((1)/(2)))

Subscript[w, 1][x] == (1 - (x)^(2))^(-Divide[1,2])* w[x + I*(1 - (x)^(2))^(Divide[1,2])]
Failure Failure
Failed [300 / 300]
Result: 1.128217713+1.045869600*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I}


Result: -.9208203932+1.594907706*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.1282177124267212, 1.0458696000777863]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9208203932499366, 1.5949077057544443]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
18.33#Ex2 w 2 ⁢ ( x ) = ( 1 - x 2 ) 1 2 ⁢ w ⁢ ( x + i ⁢ ( 1 - x 2 ) 1 2 ) subscript 𝑤 2 𝑥 superscript 1 superscript 𝑥 2 1 2 𝑤 𝑥 imaginary-unit superscript 1 superscript 𝑥 2 1 2 {\displaystyle{\displaystyle w_{2}(x)=(1-x^{2})^{\frac{1}{2}}w\left(x+\mathrm{% i}(1-x^{2})^{\frac{1}{2}}\right)}}
w_{2}(x) = (1-x^{2})^{\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)		 

w[2](x) = (1 - (x)^(2))^((1)/(2))* w(x + I*(1 - (x)^(2))^((1)/(2)))

Subscript[w, 2][x] == (1 - (x)^(2))^(Divide[1,2])* w[x + I*(1 - (x)^(2))^(Divide[1,2])]
Failure Failure
Failed [300 / 300]
Result: 1.512563597+.3801630000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[2] = 1/2*3^(1/2)+1/2*I}


Result: -.5364745086+.9292011060*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.5125635972390792, 0.38016299990276686]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5364745084375786, 0.9292011055794249]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
18.33.E10 z - n ⁢ ϕ 2 ⁢ n ⁢ ( z ) = A n ⁢ p n ⁢ ( 1 2 ⁢ ( z + z - 1 ) ) + B n ⁢ ( z - z - 1 ) ⁢ q n - 1 ⁢ ( 1 2 ⁢ ( z + z - 1 ) ) superscript 𝑧 𝑛 subscript italic-ϕ 2 𝑛 𝑧 subscript 𝐴 𝑛 subscript 𝑝 𝑛 1 2 𝑧 superscript 𝑧 1 subscript 𝐵 𝑛 𝑧 superscript 𝑧 1 subscript 𝑞 𝑛 1 1 2 𝑧 superscript 𝑧 1 {\displaystyle{\displaystyle z^{-n}\phi_{2n}(z)={A_{n}p_{n}\left(\tfrac{1}{2}(% z+z^{-1})\right)+B_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}}}
z^{-n}\phi_{2n}(z) = {A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+B_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}		 

(z)^(- n)* phi[2*n](z) = A[n]*p[n]*((1)/(2)*(z + (z)^(- 1)))+ B[n]*(z - (z)^(- 1))*q[n - 1]*((1)/(2)*(z + (z)^(- 1)))
 
(z)^(- n)* Subscript[\[Phi], 2*n][z] == Subscript[A, n]*Subscript[p, n]*(Divide[1,2]*(z + (z)^(- 1)))+ Subscript[B, n]*(z - (z)^(- 1))*Subscript[q, n - 1]*(Divide[1,2]*(z + (z)^(- 1)))
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E11 z - n + 1 ⁢ ϕ 2 ⁢ n - 1 ⁢ ( z ) = C n ⁢ p n ⁢ ( 1 2 ⁢ ( z + z - 1 ) ) + D n ⁢ ( z - z - 1 ) ⁢ q n - 1 ⁢ ( 1 2 ⁢ ( z + z - 1 ) ) superscript 𝑧 𝑛 1 subscript italic-ϕ 2 𝑛 1 𝑧 subscript 𝐶 𝑛 subscript 𝑝 𝑛 1 2 𝑧 superscript 𝑧 1 subscript 𝐷 𝑛 𝑧 superscript 𝑧 1 subscript 𝑞 𝑛 1 1 2 𝑧 superscript 𝑧 1 {\displaystyle{\displaystyle z^{-n+1}\phi_{2n-1}(z)={C_{n}p_{n}\left(\tfrac{1}% {2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)% }}}
z^{-n+1}\phi_{2n-1}(z) = {C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}		 

(z)^(- n + 1)* phi[2*n - 1](z) = C[n]*p[n]*((1)/(2)*(z + (z)^(- 1)))+ D[n]*(z - (z)^(- 1))*q[n - 1]*((1)/(2)*(z + (z)^(- 1)))
 
(z)^(- n + 1)* Subscript[\[Phi], 2*n - 1][z] == Subscript[C, n]*Subscript[p, n]*(Divide[1,2]*(z + (z)^(- 1)))+ Subscript[D, n]*(z - (z)^(- 1))*Subscript[q, n - 1]*(Divide[1,2]*(z + (z)^(- 1)))
 Skipped - no semantic math Skipped - no semantic math - -
18.33#Ex3 ϕ n ⁢ ( z ) = z n subscript italic-ϕ 𝑛 𝑧 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\phi_{n}(z)=z^{n}}}
\phi_{n}(z) = z^{n}		 

phi[n](z) = (z)^(n)
 
Subscript[\[Phi], n][z] == (z)^(n)
 Skipped - no semantic math Skipped - no semantic math - -
18.33#Ex4 w ⁢ ( z ) = 1 𝑤 𝑧 1 {\displaystyle{\displaystyle w(z)=1}}
w(z) = 1		 

w(z) = 1
 
w[z] == 1
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E13 ϕ n ⁢ ( z ) = ∑ ℓ = 0 n ( λ + 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ z ℓ subscript italic-ϕ 𝑛 𝑧 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ superscript 𝑧 ℓ {\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+% 1\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}}}
\phi_{n}(z) = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell}		 

phi[n](z) = sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n)

Subscript[\[Phi], n][z] == Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Aborted Failure
Failed [299 / 300]
Result: -1.732050808-1.000000000*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.9330127026-4.482050809*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.7320508075688772, -1.0]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9330127018922204, -4.482050807568885]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.33.E13 ∑ ℓ = 0 n ( λ + 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ z ℓ = ( λ ) n n ! ⁢ F 1 2 ⁡ ( - n , λ + 1 - λ - n + 1 ; z ) superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ superscript 𝑧 ℓ Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 1 𝜆 𝑛 1 𝑧 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{% \ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{% \left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+% 1};z\right)}}
\sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell} = \frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda+1}{-\lambda-n+1}{z}		 

sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n) = (pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda + 1], [- lambda - n + 1], z)

Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]+ 1}, {- \[Lambda]- n + 1}, z]
Aborted Successful Successful [Tested: 0]
Failed [21 / 210]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}

... skip entries to safe data
18.33#Ex5 w ⁢ ( z ) = ( 1 - 1 2 ⁢ ( z + z - 1 ) ) λ 𝑤 𝑧 superscript 1 1 2 𝑧 superscript 𝑧 1 𝜆 {\displaystyle{\displaystyle w(z)=\left(1-\tfrac{1}{2}(z+z^{-1})\right)^{% \lambda}}}
w(z) = \left(1-\tfrac{1}{2}(z+z^{-1})\right)^{\lambda}		 

w(z) = (1 -(1)/(2)*(z + (z)^(- 1)))^(lambda)
 
w[z] == (1 -Divide[1,2]*(z + (z)^(- 1)))^\[Lambda]
 Skipped - no semantic math Skipped - no semantic math - -
18.33#Ex6 w 1 ⁢ ( x ) = ( 1 - x ) λ - 1 2 ⁢ ( 1 + x ) - 1 2 subscript 𝑤 1 𝑥 superscript 1 𝑥 𝜆 1 2 superscript 1 𝑥 1 2 {\displaystyle{\displaystyle w_{1}(x)=(1-x)^{\lambda-\frac{1}{2}}(1+x)^{-\frac% {1}{2}}}}
w_{1}(x) = (1-x)^{\lambda-\frac{1}{2}}(1+x)^{-\frac{1}{2}}		 

w[1](x) = (1 - x)^(lambda -(1)/(2))*(1 + x)^(-(1)/(2))
 
Subscript[w, 1][x] == (1 - x)^(\[Lambda]-Divide[1,2])*(1 + x)^(-Divide[1,2])
 Skipped - no semantic math Skipped - no semantic math - -
18.33#Ex7 w 2 ⁢ ( x ) = ( 1 - x ) λ + 1 2 ⁢ ( 1 + x ) 1 2 subscript 𝑤 2 𝑥 superscript 1 𝑥 𝜆 1 2 superscript 1 𝑥 1 2 {\displaystyle{\displaystyle w_{2}(x)=(1-x)^{\lambda+\frac{1}{2}}(1+x)^{\frac{% 1}{2}}}}
w_{2}(x) = (1-x)^{\lambda+\frac{1}{2}}(1+x)^{\frac{1}{2}}		 λ > - 1 2 𝜆 1 2 {\displaystyle{\displaystyle\lambda>-\tfrac{1}{2}}} 
w[2](x) = (1 - x)^(lambda +(1)/(2))*(1 + x)^((1)/(2))
 
Subscript[w, 2][x] == (1 - x)^(\[Lambda]+Divide[1,2])*(1 + x)^(Divide[1,2])
 Skipped - no semantic math Skipped - no semantic math - -
18.33.E15 ϕ n ⁢ ( z ) = ∑ ℓ = 0 n ( a ⁢ q 2 ; q 2 ) ℓ ⁢ ( a ; q 2 ) n - ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q 2 ; q 2 ) n - ℓ ⁢ ( q - 1 ⁢ z ) ℓ subscript italic-ϕ 𝑛 𝑧 superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 ℓ superscript superscript 𝑞 1 𝑧 ℓ {\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^% {2}\right)_{\ell}\left(a;q^{2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell% }\left(q^{2};q^{2}\right)_{n-\ell}}(q^{-1}z)^{\ell}}}
\phi_{n}(z) = \sum_{\ell=0}^{n}\frac{\qPochhammer{aq^{2}}{q^{2}}{\ell}\qPochhammer{a}{q^{2}}{n-\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q^{2}}{q^{2}}{n-\ell}}(q^{-1}z)^{\ell}		 

phi[n](z) = sum((QPochhammer(a*(q)^(2), (q)^(2), ell)*QPochhammer(a, (q)^(2), n - ell))/(QPochhammer((q)^(2), (q)^(2), ell)*QPochhammer((q)^(2), (q)^(2), n - ell))*((q)^(- 1)* z)^(ell), ell = 0..n)

Subscript[\[Phi], n][z] == Sum[Divide[QPochhammer[a*(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[a, (q)^(2), n - \[ScriptL]],QPochhammer[(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[(q)^(2), (q)^(2), n - \[ScriptL]]]*((q)^(- 1)* z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Aborted Aborted Error Skipped - Because timed out
18.33.E15 ∑ ℓ = 0 n ( a ⁢ q 2 ; q 2 ) ℓ ⁢ ( a ; q 2 ) n - ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q 2 ; q 2 ) n - ℓ ⁢ ( q - 1 ⁢ z ) ℓ = ( a ; q 2 ) n ( q 2 ; q 2 ) n ⁢ ϕ 1 2 ⁡ ( a ⁢ q 2 , q - 2 ⁢ n a - 1 ⁢ q 2 - 2 ⁢ n ; q 2 , q ⁢ z a ) superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 ℓ superscript superscript 𝑞 1 𝑧 ℓ q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 q-hypergeometric-rphis 2 1 𝑎 superscript 𝑞 2 superscript 𝑞 2 𝑛 superscript 𝑎 1 superscript 𝑞 2 2 𝑛 superscript 𝑞 2 𝑞 𝑧 𝑎 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^{2}\right)_{% \ell}\left(a;q^{2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q^{2}% ;q^{2}\right)_{n-\ell}}(q^{-1}z)^{\ell}=\frac{\left(a;q^{2}\right)_{n}}{\left(% q^{2};q^{2}\right)_{n}}{{}_{2}\phi_{1}}\left({aq^{2},q^{-2n}\atop a^{-1}q^{2-2% n}};q^{2},\frac{qz}{a}\right)}}
\sum_{\ell=0}^{n}\frac{\qPochhammer{aq^{2}}{q^{2}}{\ell}\qPochhammer{a}{q^{2}}{n-\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q^{2}}{q^{2}}{n-\ell}}(q^{-1}z)^{\ell} = \frac{\qPochhammer{a}{q^{2}}{n}}{\qPochhammer{q^{2}}{q^{2}}{n}}\qgenhyperphi{2}{1}@@{aq^{2},q^{-2n}}{a^{-1}q^{2-2n}}{q^{2}}{\frac{qz}{a}}		 

Error

Sum[Divide[QPochhammer[a*(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[a, (q)^(2), n - \[ScriptL]],QPochhammer[(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[(q)^(2), (q)^(2), n - \[ScriptL]]]*((q)^(- 1)* z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[QPochhammer[a, (q)^(2), n],QPochhammer[(q)^(2), (q)^(2), n]]*QHypergeometricPFQ[{a*(q)^(2), (q)^(- 2*n)},{(a)^(- 1)* (q)^(2 - 2*n)},(q)^(2),Divide[q*z,a]]
Missing Macro Error Aborted Skip - symbolical successful subtest Skipped - Because timed out
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