Gold 19

Classical orthogonal polynomials

Gold ID

19

Link

https://sigir21.wmflabs.org/wiki/Classical_orthogonal_polynomials#math.69.117

Formula

$T_{n}(x)={\frac {\Gamma (1/2){\sqrt {1-x^{2}}}}{(-2)^{n}\,\Gamma (n+1/2)}}\ {\frac {d^{n}}{dx^{n}}}\left([1-x^{2}]^{n-1/2}\right)$

TeX Source

T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Semantic LaTeX

Translation

\ChebyshevpolyT{n}@{x} = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\Gamma(n+1/2)} \deriv [n]{ }{x}([1 - x^2]^{n-1/2})

Expected (Gold Entry)

\ChebyshevpolyT{n}@{x} = \frac{\EulerGamma{1/2}\sqrt{1-x^2}}{(-2)^n\EulerGamma{n+1/2}} \deriv [n]{ }{x}([1 - x^2]^{n-1/2})

Mathematica

Translation

ChebyshevT[n, x] == Divide[\[CapitalGamma]*(1/2)*Sqrt[1 - (x)^(2)],(- 2)^(n)* \[CapitalGamma]*(n + 1/2)]*D[(1 - (x)^(2))^(n - 1/2), {x, n}]

Expected (Gold Entry)

ChebyshevT[n, x] == Divide[Gamma[1/2]*Sqrt[1 - (x)^(2)],(- 2)^(n)* Gamma[n + 1/2]]*D[(1 - (x)^(2))^(n - 1/2), {x, n}]

Maple

Translation

ChebyshevT(n, x) = (Gamma*(1/2)*sqrt(1 - (x)^(2)))/((- 2)^(n)* Gamma*(n + 1/2))*diff((1 - (x)^(2))^(n - 1/2), [x$(n)])

Expected (Gold Entry)

ChebyshevT(n, x) = (GAMMA(1/2)*sqrt(1 - (x)^(2)))/((- 2)^(n)* GAMMA(n + 1/2))*diff((1 - (x)^(2))^(n - 1/2), [x$\$$(n)])