Fungrim entry: eca10b

$\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = \frac{-1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f\!\left(z\right) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)$
Assumptions:$r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, {c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = \frac{-1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f\!\left(z\right) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, {c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
Factorial$n !$ Factorial
Pow${a}^{b}$ Power
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
CoulombG$G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ConstI$i$ Imaginary unit
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("eca10b"),
Assumptions(And(Element(r, ZZGreaterEqual(0)), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), Element(ell, CC), Element(eta, CC), And(NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0))), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))