Fungrim entry: 30b67b

\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right) \;\mathbin{\operatorname{and}}\; x \notin \left\{-1, 1\right\}

TeX:

\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right) \;\mathbin{\operatorname{and}}\; x \notin \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("30b67b"),
    Formula(Where(Equal(Add(Sub(Mul(Sub(1, Pow(x, 2)), ComplexDerivative(y(x), For(x, x, 2))), Mul(Mul(3, x), ComplexDerivative(y(x), For(x, x, 1)))), Mul(Mul(n, Add(n, 2)), y(x))), 0), Equal(y(x), Add(Mul(Subscript(c, 1), ChebyshevU(n, x)), Mul(Subscript(c, 2), Div(ChebyshevT(Add(n, 1), x), Sqrt(Sub(1, Pow(x, 2))))))))),
    Variables(n, x, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(n, ZZ), Element(x, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), Or(Equal(Subscript(c, 2), 0), NotElement(x, Union(OpenClosedInterval(Neg(Infinity), 1), ClosedOpenInterval(1, Infinity)))), NotElement(x, Set(-1, 1)))))

Topics using this entry

Chebyshev polynomials