Fungrim entry: ff5e82

z \left({n}^{2} + 5 n + 6\right) a_{n + 3} + \left({n}^{2} + 5 n + 6\right) a_{n + 2} + z a_{n + 1} + a_{n} = 0\; \text{ where } a_{n} = \frac{{\operatorname{sinc}}^{(n)}(z)}{n !}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

TeX:

z \left({n}^{2} + 5 n + 6\right) a_{n + 3} + \left({n}^{2} + 5 n + 6\right) a_{n + 2} + z a_{n + 1} + a_{n} = 0\; \text{ where } a_{n} = \frac{{\operatorname{sinc}}^{(n)}(z)}{n !}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ff5e82"),
    Formula(Where(Equal(Add(Add(Add(Mul(Mul(z, Add(Add(Pow(n, 2), Mul(5, n)), 6)), a_(Add(n, 3))), Mul(Add(Add(Pow(n, 2), Mul(5, n)), 6), a_(Add(n, 2)))), Mul(z, a_(Add(n, 1)))), a_(n)), 0), Equal(a_(n), Div(ComplexDerivative(Sinc(z), For(z, z, n)), Factorial(n))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)))))

Topics using this entry

Sinc function