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Fungrim entry: 807f3f

Kν(r) ⁣(z)=(1)r2rk=0r(rk)Kν+2kr ⁣(z)K^{(r)}_{\nu}\!\left(z\right) = \frac{{\left(-1\right)}^{r}}{{2}^{r}} \sum_{k=0}^{r} {r \choose k} K_{\nu + 2 k - r}\!\left(z\right)
Assumptions:νC  and  zC{0}  and  rZ0\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
K^{(r)}_{\nu}\!\left(z\right) = \frac{{\left(-1\right)}^{r}}{{2}^{r}} \sum_{k=0}^{r} {r \choose k} K_{\nu + 2 k - r}\!\left(z\right)

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
BesselKKν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(BesselK(nu, z, r), Mul(Div(Pow(-1, r), Pow(2, r)), Sum(Mul(Binomial(r, k), BesselK(Sub(Add(nu, Mul(2, k)), r), z)), For(k, 0, r))))),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

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2020-04-08 16:14:44.404316 UTC