**Appendix C**

In Sections 2 and 3, the AFCs *ψ*5,2(*<sup>α</sup>*, *θ*) and *ψ*6,3(*<sup>α</sup>*, *θ*) were calculated with detailed derivations. In Section 4, the AFCs *ψ*7,3(*<sup>α</sup>*, *θ*) and *ψ*8,4(*<sup>α</sup>*, *θ*) were presented with a very brief derivations. The corresponding results were summarized in Appendices A and B. The current Appendix presents the AFCs *ψ*9,4 ≡ *ψ*9,4(*<sup>α</sup>*, *θ*) and *ψ*10,5 ≡ *ψ*10,5(*<sup>α</sup>*, *θ*) without derivations. To find the latter AFCs, the methods described in the main sections were used.

So, the first AFC under consideration can be represented as:

$$\psi\_{\mathfrak{P},4} = 2Z^4[2ZX\_1(\mathfrak{a},\theta) - X\_2(\mathfrak{a},\theta)],\tag{A18}$$

where

$$X\_1(\boldsymbol{\alpha}, \boldsymbol{\theta}) = \check{a}s\boldsymbol{\alpha}\check{f}\_1 + \check{a}s\boldsymbol{\alpha}\check{f}\_2 + \check{a}\_{84}\check{f}\_3 \tag{A19}$$

$$X\_2(u, \theta) = \frac{35}{\pi^{3/2}} \sqrt{\frac{2}{7}} \, d\_{84} f\_4 + \frac{(\pi - 2)(5\pi - 14)}{123451776000\pi^4} \left[ c\_5 f\_5 + c\_6 f\_6 + c\_7 f\_7 + 16(c\_8 f\_8 + c\_9 f\_9) \right]. \tag{A20}$$

Here, *<sup>a</sup>*ˇ8*l* = *<sup>Z</sup>*−4*a*8*l*, where the coefficients *<sup>a</sup>*8*l* are defined by Equations (157) and (158), whereas the other coefficients are:

$$c\varepsilon = \pi (29757524 - 4780401\pi) - 46286848, \quad c\_6 = \pi (9581100\pi - 59458928) + 92239360,$$

$$c\_7 = \pi (28060 + 10149\pi) - 167168, \quad c\_8 = 9\pi (134543\pi - 828732) + 11488128,$$

$$c\_9 = \pi (4804833\pi - 29773780) + 46119680. \tag{A21}$$

The functions ˇ *fi* ≡ ˇ *fi*(*<sup>α</sup>*, *θ*) are:

$$f\_1 = -\frac{(\rho + 1)(563\rho^8 + 1012\rho^7 - 8932\rho^6 - 3668\rho^5 + 23954\rho^4 - 3668\rho^3 - 8932\rho^2 + 1012\rho + 563)}{1260\pi^{3/2}(\rho^2 + 1)^{9/2}},\tag{A22}$$

$$f\_2 = -\left(\frac{8}{\pi^{3/2}}\sqrt{\frac{10}{7}}\right)\frac{\rho^2(1+\rho)(126+49\rho-424\rho^2+49\rho^3+126\rho^4)}{300(\rho^2+1)^{9/2}}P\_2(\cos\theta),\tag{A23}$$

$$f\_3 = -\frac{2}{5\pi^{3/2}}\sqrt{\frac{2(1+\sin a)}{7}}\sin^4 a \,\, P\_2(\cos \theta) ,\tag{A24}$$

$$\check{f}\_4 = -\frac{\check{\varsigma}(315 - 1680\zeta^2 + 2814\zeta^4 - 1854\zeta^6 + 419\zeta^8)}{1260},\tag{A25}$$

$$\check{f}\_5 = -\frac{\check{\xi}}{60} (2\check{\xi}^2 - 3)(2\check{\xi}^2 - 1)(4\check{\xi}^4 - 10\check{\xi}^2 + 5). \tag{A26}$$

The remaining ˇ *f* functions are represented by series:

$$\tilde{f}\_{\hat{j}} = \frac{(\rho^2 + 1)^{-9/2}}{k\_{\hat{j}}} \sum\_{l=0}^{\infty} \frac{\rho^{l} \tilde{\xi}\_{jl}(\rho)}{(2l - 1)(2l + 3)} P\_l(\cos \theta), \qquad (j = 6, 7, 8, 9) \tag{A27}$$

where

$$k\_6 = 6, \qquad k\_7 = 60, \qquad k\_8 = 24, \qquad k\_9 = 40,\tag{A28}$$

and the corresponding ˇ *ζ* functions are:

$$\begin{split} \check{\zeta}\_{\text{el}}(\rho) &= \frac{(2l-15)(2l-1)(l+1)\rho^{10}}{(2l+7)(2l+11)} + \frac{(22l^2-5l-12)\rho^8}{(2l+7)} + \frac{10(2l^2+11l+3)\rho^6}{2l+7} - \\ & - \frac{10(2l^2-7l-6)\rho^4}{2l-5} - \frac{(22l^2+49l+15)\rho^2}{2l-5} - \frac{l(2l+3)(2l+17)}{(2l-9)(2l-5)}, \end{split} \tag{A29}$$

$$\begin{split} \dot{\xi}\_{7l}(\rho) &= \frac{(2l-1)(4l^2+160l-189)\rho^{10}}{(2l+7)(2l+11)} + \frac{35(4l^2+40l-9)\rho^8}{(2l+7)} - \frac{350(4l^2+16l+3)\rho^6}{2l+7} + \\ &+ \frac{350(4l^2-8l-9)\rho^4}{2l-5} - \frac{35(4l^2-32l-45)\rho^2}{2l-5} - \frac{(2l+3)(4l^2-152l-345)}{(2l-9)(2l-5)},\end{split} \tag{A30}$$

$$
\begin{split}
\check{\zeta}\_{s}\mathrm{s}(\rho) &= -\frac{(2l-1)(56l^3 + 250l^2 + 338l + 171)\rho^{10}}{(2l+5)(2l+7)(2l+11)} - \frac{(136l^3 + 314l^2 - 110l - 153)\rho^8}{(2l+5)(2l+7)} - \\
& - \frac{2(80l^4 + 652l^3 + 566l^2 - 1824l - 873)\rho^6}{(2l-3)(2l+5)(2l+7)} + \frac{2(80l^4 - 332l^3 - 910l^2 + 1320l + 945)\rho^4}{(2l-5)(2l-3)(2l+5)} \\
& + \frac{(136l^3 + 94l^2 - 330l - 135)\rho^2}{(2l-5)(2l-3)} + \frac{(2l+3)(56l^3 - 82l^2 + 6l - 27)}{(2l-9)(2l-5)(2l-3)},\end{split}
\tag{A31}
$$

$$\begin{split} \mathcal{J}\_{9l}(\rho) &= \frac{(2l-1)(24l^3 - 150l^2 - 670l - 439)\rho^{10}}{(2l+5)(2l+7)(2l+11)} + \frac{5(72l^3 + 162l^2 - 70l - 103)\rho^8}{(2l+5)(2l+7)} + \\ &+ \frac{10(16l^4 + 22l0^3 + 222l^2 - 804l - 423)\rho^6}{(2l-3)(2l+5)(2l+7)} - \frac{10(16l^4 - 156l^3 - 342l^2 + 652l + 399)\rho^4}{(2l-5)(2l-3)(2l+5)} - \\ &- \frac{5(72l^3 + 54l^2 - 178l - 57)\rho^2}{(2l-5)(2l-3)} - \frac{(2l+3)(24l^3 + 222l^2 - 298l - 57)}{(2l-9)(2l-5)(2l-3)}, \end{split} \tag{A32}$$

It is important to emphasize that the representations (A27)–(A32) are valid only for 0 ≤ *ρ* ≤ 1. For values *ρ* > 1, one should replace *ρ* with 1/*ρ*, which is equivalent to simply redefining *ρ* as cot(*α*/2).

The second AFC under consideration is of the form:

$$\Psi\_{10,5} = -\frac{Z^5(\pi - 2)(5\pi - 14)}{\pi^{7/2}} [b\_{10,1} Y\_{10,1}(a,\theta) + b\_{10,3} Y\_{10,3}(a,\theta) + b\_{10,5} Y\_{10,5}(a,\theta)],\tag{A33}$$

where

$$b\_{10,1} = \frac{\pi [3\pi (6840010557\pi - 63828704998) + 595609133656] - 617517605744}{40102537860000\sqrt{105}},\tag{A34}$$

$$b\_{10,3} = \frac{\pi \left[ \pi \left( 9194460432\pi - 85833963053 \right) + 267084629592 \right] - 277009842768}{100256344650000\sqrt{30}},\tag{A35}$$

$$b\_{10,5} = \frac{\pi \left[ \pi (622341848670\pi - 5812646794643) + 18095537797140 \right] - 18776793358080}{1002563446500000\sqrt{42}},\tag{A36}$$

and *<sup>Y</sup>*10,*<sup>l</sup>*(*<sup>α</sup>*, *θ*) with *l* = 1, 3, 5 are the normalized HHs.
