Recurrence relations for Bessel functions

d56914

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d56914"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(z, Mul(2, nu)), Add(BesselJ(Sub(nu, 1), z), BesselJ(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

b6d600

Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b6d600"),
    Formula(Equal(BesselY(nu, z), Mul(Div(z, Mul(2, nu)), Add(BesselY(Sub(nu, 1), z), BesselY(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

4fb391

I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4fb391"),
    Formula(Equal(BesselI(nu, z), Mul(Div(z, Mul(2, nu)), Sub(BesselI(Sub(nu, 1), z), BesselI(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

9d98f8

K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d98f8"),
    Formula(Equal(BesselK(nu, z), Neg(Mul(Div(z, Mul(2, nu)), Sub(BesselK(Sub(nu, 1), z), BesselK(Add(nu, 1), z)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

f1afc0

J'_{0}\!\left(z\right) = -J_{1}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

J'_{0}\!\left(z\right) = -J_{1}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJDerivative`	$J^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the first kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f1afc0"),
    Formula(Equal(BesselJDerivative(0, z, 1), Neg(BesselJ(1, z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

8b6264

Y'_{0}\!\left(z\right) = -Y_{1}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y'_{0}\!\left(z\right) = -Y_{1}\!\left(z\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselYDerivative`	$Y^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the second kind
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8b6264"),
    Formula(Equal(BesselYDerivative(0, z, 1), Neg(BesselY(1, z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

c0247f

I'_{0}\!\left(z\right) = I_{1}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

I'_{0}\!\left(z\right) = I_{1}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselIDerivative`	$I^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the first kind
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c0247f"),
    Formula(Equal(BesselIDerivative(0, z, 1), BesselI(1, z))),
    Variables(z),
    Assumptions(Element(z, CC)))

81ffcd

K'_{0}\!\left(z\right) = -K_{1}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K'_{0}\!\left(z\right) = -K_{1}\!\left(z\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselKDerivative`	$K^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the second kind
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("81ffcd"),
    Formula(Equal(BesselKDerivative(0, z, 1), Neg(BesselK(1, z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

5aceb9

J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJDerivative`	$J^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the first kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5aceb9"),
    Formula(Equal(BesselJDerivative(nu, z, 1), Div(Sub(BesselJ(Sub(nu, 1), z), BesselJ(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

40aeb6

Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselYDerivative`	$Y^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the second kind
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("40aeb6"),
    Formula(Equal(BesselYDerivative(nu, z, 1), Div(Sub(BesselY(Sub(nu, 1), z), BesselY(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

58d91f

I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselIDerivative`	$I^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the first kind
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("58d91f"),
    Formula(Equal(BesselIDerivative(nu, z, 1), Div(Add(BesselI(Sub(nu, 1), z), BesselI(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

a0ff0b

K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselKDerivative`	$K^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the second kind
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a0ff0b"),
    Formula(Equal(BesselKDerivative(nu, z, 1), Neg(Div(Add(BesselK(Sub(nu, 1), z), BesselK(Add(nu, 1), z)), 2)))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

2488bb

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

J^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} J_{\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
`BesselJDerivative`	$J^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`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("2488bb"),
    Formula(Equal(BesselJDerivative(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Mul(Pow(-1, k), Binomial(r, k)), BesselJ(Sub(Add(nu, Mul(2, k)), r), z)), Tuple(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)))))

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)))))

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
`BesselIDerivative`	$I^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`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(BesselIDerivative(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Binomial(r, k), BesselI(Sub(Add(nu, Mul(2, k)), r), z)), Tuple(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)))))

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
`BesselKDerivative`	$K^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified 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("807f3f"),
    Formula(Equal(BesselKDerivative(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)), Tuple(k, 0, r))))),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

15ac84

\left({r}^{2} + 4 r - {n}^{2} + 4\right) J^{(r + 2)}_{n}\!\left(0\right) + \left(r + 1\right) \left(r + 2\right) J^{(r)}_{n}\!\left(0\right) = 0

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

TeX:

\left({r}^{2} + 4 r - {n}^{2} + 4\right) J^{(r + 2)}_{n}\!\left(0\right) + \left(r + 1\right) \left(r + 2\right) J^{(r)}_{n}\!\left(0\right) = 0

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselJDerivative`	$J^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("15ac84"),
    Formula(Equal(Add(Mul(Add(Sub(Add(Pow(r, 2), Mul(4, r)), Pow(n, 2)), 4), BesselJDerivative(n, 0, Add(r, 2))), Mul(Mul(Add(r, 1), Add(r, 2)), BesselJDerivative(n, 0, r))), 0)),
    Variables(nu, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

f303c9

\left({r}^{2} + 4 r - {n}^{2} + 4\right) I^{(r + 2)}_{n}\!\left(0\right) - \left(r + 1\right) \left(r + 2\right) I^{(r)}_{n}\!\left(0\right) = 0

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

TeX:

\left({r}^{2} + 4 r - {n}^{2} + 4\right) I^{(r + 2)}_{n}\!\left(0\right) - \left(r + 1\right) \left(r + 2\right) I^{(r)}_{n}\!\left(0\right) = 0

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselIDerivative`	$I^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f303c9"),
    Formula(Equal(Sub(Mul(Add(Sub(Add(Pow(r, 2), Mul(4, r)), Pow(n, 2)), 4), BesselIDerivative(n, 0, Add(r, 2))), Mul(Mul(Add(r, 1), Add(r, 2)), BesselIDerivative(n, 0, r))), 0)),
    Variables(nu, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

9b2f38

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{J^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{J^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) + {z}^{2} - {\nu}^{2} + 4\right) \frac{J^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} + 2 z \frac{J^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} + \frac{J^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

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:

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{J^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{J^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) + {z}^{2} - {\nu}^{2} + 4\right) \frac{J^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} + 2 z \frac{J^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} + \frac{J^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

\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
`Pow`	${a}^{b}$	Power
`BesselJDerivative`	$J^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the first kind
`Factorial`	$n !$	Factorial
`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("9b2f38"),
    Formula(Equal(Add(Add(Add(Add(Mul(Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12)), Div(BesselJDerivative(nu, z, Add(r, 4)), Factorial(Add(r, 4)))), Mul(Mul(z, Add(Add(Mul(2, Pow(r, 2)), Mul(11, r)), 15)), Div(BesselJDerivative(nu, z, Add(r, 3)), Factorial(Add(r, 3))))), Mul(Add(Sub(Add(Mul(r, Add(r, 4)), Pow(z, 2)), Pow(nu, 2)), 4), Div(BesselJDerivative(nu, z, Add(r, 2)), Factorial(Add(r, 2))))), Mul(Mul(2, z), Div(BesselJDerivative(nu, z, Add(r, 1)), Factorial(Add(r, 1))))), Div(BesselJDerivative(nu, z, r), Factorial(r))), 0)),
    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)))))

e85dee

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{Y^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{Y^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) + {z}^{2} - {\nu}^{2} + 4\right) \frac{Y^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} + 2 z \frac{Y^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} + \frac{Y^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

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:

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{Y^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{Y^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) + {z}^{2} - {\nu}^{2} + 4\right) \frac{Y^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} + 2 z \frac{Y^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} + \frac{Y^{(r)}_{\nu}\!\left(z\right)}{r !} = 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
`Pow`	${a}^{b}$	Power
`BesselYDerivative`	$Y^{(r)}_{\nu}\!\left(z\right)$	Differentiated Bessel function of the second kind
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e85dee"),
    Formula(Equal(Add(Add(Add(Add(Mul(Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12)), Div(BesselYDerivative(nu, z, Add(r, 4)), Factorial(Add(r, 4)))), Mul(Mul(z, Add(Add(Mul(2, Pow(r, 2)), Mul(11, r)), 15)), Div(BesselYDerivative(nu, z, Add(r, 3)), Factorial(Add(r, 3))))), Mul(Add(Sub(Add(Mul(r, Add(r, 4)), Pow(z, 2)), Pow(nu, 2)), 4), Div(BesselYDerivative(nu, z, Add(r, 2)), Factorial(Add(r, 2))))), Mul(Mul(2, z), Div(BesselYDerivative(nu, z, Add(r, 1)), Factorial(Add(r, 1))))), Div(BesselYDerivative(nu, z, r), Factorial(r))), 0)),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

e233b0

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{I^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{I^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) - {z}^{2} - {\nu}^{2} + 4\right) \frac{I^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} - 2 z \frac{I^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} - \frac{I^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

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:

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{I^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{I^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) - {z}^{2} - {\nu}^{2} + 4\right) \frac{I^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} - 2 z \frac{I^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} - \frac{I^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

\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
`Pow`	${a}^{b}$	Power
`BesselIDerivative`	$I^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the first kind
`Factorial`	$n !$	Factorial
`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("e233b0"),
    Formula(Equal(Sub(Sub(Add(Add(Mul(Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12)), Div(BesselIDerivative(nu, z, Add(r, 4)), Factorial(Add(r, 4)))), Mul(Mul(z, Add(Add(Mul(2, Pow(r, 2)), Mul(11, r)), 15)), Div(BesselIDerivative(nu, z, Add(r, 3)), Factorial(Add(r, 3))))), Mul(Add(Sub(Sub(Mul(r, Add(r, 4)), Pow(z, 2)), Pow(nu, 2)), 4), Div(BesselIDerivative(nu, z, Add(r, 2)), Factorial(Add(r, 2))))), Mul(Mul(2, z), Div(BesselIDerivative(nu, z, Add(r, 1)), Factorial(Add(r, 1))))), Div(BesselIDerivative(nu, z, r), Factorial(r))), 0)),
    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)))))

7377c8

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{K^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{K^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) - {z}^{2} - {\nu}^{2} + 4\right) \frac{K^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} - 2 z \frac{K^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} - \frac{K^{(r)}_{\nu}\!\left(z\right)}{r !} = 0

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:

{z}^{2} \left({r}^{2} + 7 r + 12\right) \frac{K^{(r + 4)}_{\nu}\!\left(z\right)}{\left(r + 4\right)!} + z \left(2 {r}^{2} + 11 r + 15\right) \frac{K^{(r + 3)}_{\nu}\!\left(z\right)}{\left(r + 3\right)!} + \left(r \left(r + 4\right) - {z}^{2} - {\nu}^{2} + 4\right) \frac{K^{(r + 2)}_{\nu}\!\left(z\right)}{\left(r + 2\right)!} - 2 z \frac{K^{(r + 1)}_{\nu}\!\left(z\right)}{\left(r + 1\right)!} - \frac{K^{(r)}_{\nu}\!\left(z\right)}{r !} = 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
`Pow`	${a}^{b}$	Power
`BesselKDerivative`	$K^{(r)}_{\nu}\!\left(z\right)$	Differentiated modified Bessel function of the second kind
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7377c8"),
    Formula(Equal(Sub(Sub(Add(Add(Mul(Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12)), Div(BesselKDerivative(nu, z, Add(r, 4)), Factorial(Add(r, 4)))), Mul(Mul(z, Add(Add(Mul(2, Pow(r, 2)), Mul(11, r)), 15)), Div(BesselKDerivative(nu, z, Add(r, 3)), Factorial(Add(r, 3))))), Mul(Add(Sub(Sub(Mul(r, Add(r, 4)), Pow(z, 2)), Pow(nu, 2)), 4), Div(BesselKDerivative(nu, z, Add(r, 2)), Factorial(Add(r, 2))))), Mul(Mul(2, z), Div(BesselKDerivative(nu, z, Add(r, 1)), Factorial(Add(r, 1))))), Div(BesselKDerivative(nu, z, r), Factorial(r))), 0)),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

Consecutive orders

Derivatives in terms of the direct functions

Consecutive derivatives