Fungrim entry: 68cc2f

Y^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} Y_{\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:

Y^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} Y_{\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
`BesselYDerivative`	$Y^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("68cc2f"),
    Formula(Equal(BesselYDerivative(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Mul(Pow(-1, k), Binomial(r, k)), BesselY(Sub(Add(nu, Mul(2, k)), r), z)), Tuple(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

Recurrence relations for Bessel functions