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Fungrim entry: ff5e82

z(n2+5n+6)an+3+(n2+5n+6)an+2+zan+1+an=0   where an=sinc(n)(z)n!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:zC  and  nZ0z \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
Powab{a}^{b} Power
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Factorialn!n ! Factorial
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\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)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC