Fungrim entry: e284d7

I^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {r \choose k} I_{\nu + 2 k - r}\!\left(z\right)

Assumptions:

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

Alternative 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:

I^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {r \choose k} I_{\nu + 2 k - r}\!\left(z\right)

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

\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
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e284d7"),
    Formula(Equal(BesselI(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Binomial(r, k), BesselI(Sub(Add(nu, Mul(2, k)), r), z)), For(k, 0, r))))),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Recurrence relations for Bessel functions