# Fungrim entry: 807f3f

$K^{(r)}_{\nu}\!\left(z\right) = \frac{{\left(-1\right)}^{r}}{{2}^{r}} \sum_{k=0}^{r} {r \choose k} K_{\nu + 2 k - r}\!\left(z\right)$
Assumptions:$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
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}
Definitions:
Fungrim symbol Notation Short description
BesselK$K_{\nu}\!\left(z\right)$ Modified Bessel function of the second kind
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("807f3f"),
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)))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC