Painlevé Transcendents - 32.15 Orthogonal Polynomials

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
32.15.E1	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}\exp\left(-\tfrac{1}{4}\xi^{4}-z\xi^{2}\right)p_{m}(\xi)p_{n}(\xi)\mathrm{d}\xi=\delta_{m,n}}}$ \int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}		int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m](xi)* p[n](xi), xi = - infinity..infinity) = KroneckerDelta[m, n]	Integrate[Exp[-Divide[1,4]*\[Xi]^(4)- z*\[Xi]^(2)]*Subscript[p, m][\[Xi]]* Subscript[p, n][\[Xi]], {\[Xi], - Infinity, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n]	Failure	Failure	Failed [300 / 300] Result: -.5089051774+.3195154069*I Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1} Result: .4910948226+.3195154069*I Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.5089051767265081, 0.31951540648426185] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.49109482327349185, 0.31951540648426185] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
32.15.E2	${\displaystyle{\displaystyle a_{n+1}(z)p_{n+1}(\xi)=\xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)}}$ a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)		a[n + 1](z)* p[n + 1](xi) = xi*p[n](xi)- a[n](z)* p[n - 1](xi)	Subscript[a, n + 1][z]* Subscript[p, n + 1][\[Xi]] == \[Xi]*Subscript[p, n][\[Xi]]- Subscript[a, n][z]* Subscript[p, n - 1][\[Xi]]	Skipped - no semantic math	Skipped - no semantic math	-	-
32.15.E3	${\displaystyle{\displaystyle(u_{n+1}+u_{n}+u_{n-1})u_{n}=n-2zu_{n}}}$ (u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}		(u[n + 1]+ u[n]+ u[n - 1])*u[n] = n - 2*z*u[n]	(Subscript[u, n + 1]+ Subscript[u, n]+ Subscript[u, n - 1])*Subscript[u, n] == n - 2*z*Subscript[u, n]	Skipped - no semantic math	Skipped - no semantic math	-	-