A generalized theory of the figure of the Earth: formulae

Abstract

Traditionally a laterally homogeneous and spherical base Earth model (e.g., the PREM model) is considered as input when computing the Earth’s equipotential surfaces, which are then resulted to be in symmetric shape. However, the Earth, known with a complex distribution of interior material and density, especially in the upper mantle and the crust, cannot be treated as a symmetric sphere. Recently, a CRUST1.0 model of crust layer is published and well accepted. But the effect caused by the asymmetric crust (and mantle) on equilibrium figures of the Earth cannot be analyzed by the traditional theories. A generalized theory of the figure of the Earth to third-order precision is firstly proposed in this paper, as well as the iterative calculation strategy to solve the complex equation system. In order to validate this generalized theory, the degeneration of this generalized theory with the PREM model is made and is compared with traditional theories, and it is shown that the result of this generalized theory, after degeneration, is consistent very well with traditional theory. Meanwhile, the effect (including both the direct and indirect effects) of the crust layer, from the CRUST1.0 model, on the figures of equipotential surfaces of the Earth’s interior, as well as their effects on the global dynamics flattening, will be presented as an application of this theory in accompanying paper.

Appendices

Appendix A: Derivation of ${r}^{k}$

The expression of $r^k$ can be obtained from Eq. (7):

$$\begin{aligned} r^k = s^k \left[ 1+ \sum _{j=1}^{k} C_k^j \left( \sum _{n=0}^{\infty } \sum _{m=-n}^{n} H_n^m Y_n^m\right) ^j\right] \end{aligned}$$

(A.1)

where $C_k^j$ is the binomial coefficients.

To simplify the derivation, we define the $Y_{n,m}^N$ as a generalized spherical harmonic function (Phinney and Burridge 1973; Huang and Liao 2003). However, we only need to use the function of $Y_{n,m}^0$ (i.e., $Y_n^m$) in this paper.

$$\begin{aligned} Y_{n,m}^N (\theta ,\lambda ) = P_{n,m}^N (\cos \theta ) \mathrm{e}^{i m \lambda } \end{aligned}$$

(A.2)

where

$$\begin{aligned} \begin{aligned} P_{n,m}^N (x)&= \frac{(-1)^{n-N}}{2^n (n-N)!} \left[ \frac{(n-N)! (n+m)!}{(n+N)! (n-m)!}\right] ^{\frac{1}{2}} \\ {}&\quad \cdot (1-x)^{-\frac{1}{2} (m-N)} (1+x)^{-\frac{1}{2} (m+N)} \\ {}&\quad \cdot \left( \frac{\mathrm {d}}{\mathrm {d} x}\right) ^{n-m} [(1-x)^{n-N} (1+x)^{n+N}] \end{aligned} \end{aligned}$$

(A.3)

Then we have

$$\begin{aligned}&Y_{l_1,m_1}^{N_1} Y_{l_2,m_2}^{N_2} \nonumber \\&\quad = \sum _{l=0}^{\infty } \sum _{m=-l}^{l} {\mathrm{func}} (l,m,N,l_1,m_1,N_1,l_2,m_2,N_2) Y_{l,m}^N\nonumber \\ \end{aligned}$$

(A.4)

where $N = N_1 + N_2$, and

$$\begin{aligned}&\mathrm{func} (l,m,N,l_1,m_1,N_1,l_2,m_2,N_2) \nonumber \\&\quad = (-1)^{N+m} \left[ \frac{(2l+1)(2l_1+1)(2l_2+1)}{4 \pi }\right] ^{1/2}\nonumber \\&\qquad \times \begin{pmatrix} l &{} l_1 &{} l_2 \\ -N &{} N_1 &{} N_2 \end{pmatrix} \begin{pmatrix} l &{} l_1 &{} l_2 \\ -m &{} m_1 &{} m_2 \end{pmatrix} \end{aligned}$$

(A.5)

both $\begin{pmatrix} l &{} l_1 &{} l_2 \\ -N &{} N_1 &{} N_2 \end{pmatrix}$ and $\begin{pmatrix} l &{} l_1 &{} l_2 \\ -m &{} m_1 &{} m_2 \end{pmatrix}$ are Wigner-3j symbol, their definition is:

$$\begin{aligned} \begin{aligned}&\begin{pmatrix} l &{} l_1 &{} l_2 \\ -m &{} m_1 &{} m_2 \end{pmatrix} \\&\quad = (-1)^{l_1+l_2+m} \times \frac{\sqrt{(l+l_1-l_2)!(l-l_1+l_2)!(-l+l_1+l_2)!}}{\sqrt{(l+l_1+l_2+1)!}}\\&\qquad \times \sqrt{(l+m)!(l-m)!(l_1+m_1)!(l_1-m_1)!(l_2+m_2)!(l_2-m_2)!} \\&\qquad \times \sum _k \frac{(-1)^k}{D_k} \end{aligned} \end{aligned}$$

(A.6)

where

$$\begin{aligned} \begin{aligned} D_k&= k!(l_1-m_1-k)!(l_2+m_2-k)! \\&\quad (-l+l_1+l_2-k)!(l-l_1-m_2+k)!(l-l_2+m_1+k)! \end{aligned} \end{aligned}$$

Combining Eqs. (A.1–A.6), the expression of $r^k$ can be simplified as follow:

$$\begin{aligned} r^k = s^k \sum _{n=0}^{\infty } \sum _{m=-n}^{n} L_{k,n,m} (s) Y_n^m (\theta , \lambda ), \quad k \ne 0 \end{aligned}$$

(A.7)

where $L_{k,n,m} (s)$ is function of $H_n^m (s)$.

Similarly, the Taylor expansion of $\ln r$ is:

$$\begin{aligned} \ln r = \ln s + \sum _{j=1}^{\infty } (-1)^{j+1}j^{-1} \left( \sum _{n=0}^{\infty }\sum _{m=-n}^{n} H_{n}^{m} Y_{n}^{m}\right) ^{j} \end{aligned}$$

(A.8)

Substituting Eq. (A.4) into above Eq. (A.8), one can get

$$\begin{aligned} \ln r= & {} \sum _{n=0}^{\infty } \sum _{m=-n}^{n} L_{0,n,m}(s) Y_{n}^{m}(\theta ,\lambda ) . \end{aligned}$$

(A.9)

Here two short terms ($L_{3,0,0}$ and $L_{2,2,0}$) in Eq. (A.7) and one short term ($L_{0,5,5}$) in Eq. (A.9) are listed for reference and check.

$$\begin{aligned} L_{3,0,0} (s) = k_1 (H_0^0)^3 + k_2 (H_0^0)^2 +k_3 H_0^0 + k_4 + 2 \sqrt{\pi }\nonumber \\ \end{aligned}$$

(A.10)

where $k_1, k_2, k_3$ and $k_4$ are given in “Appendix B”.

$$\begin{aligned} L_{2,2,0} (s)= & {} \frac{\sqrt{15} H_2^{-2} H_4^2}{7 \sqrt{\pi }} - \frac{\sqrt{30} H_2^{-1} H_4^1 }{7 \sqrt{\pi }} -\frac{6 H_2^0 H_4^0}{7 \sqrt{\pi }} \nonumber \\&- \frac{\sqrt{30} H_2^1 H_4^{-1}}{7 \sqrt{\pi }} + \frac{\sqrt{15} H_2^2 H_4^{-2}}{7 \sqrt{\pi }} + \frac{\sqrt{5} H_3^{-3} H_3^{3}}{3 \sqrt{\pi }} \nonumber \\&- \frac{\sqrt{5} H_3^{-1} H_3^1}{5 \sqrt{\pi }} - \frac{3 \sqrt{6} H_1^{-1} H_3^1}{\sqrt{105 \pi }} + \frac{2 \sqrt{5} (H_3^{0})^2}{15 \sqrt{\pi }} \nonumber \\&+ \frac{9 H_1^{0} H_3^0}{\sqrt{105 \pi }} - \frac{3 \sqrt{6} H_1^{1} H_3^{-1}}{\sqrt{105 \pi }} - \frac{2 \sqrt{5} H_2^{-2} H_2^2}{7 \sqrt{\pi }} \nonumber \\&- \frac{\sqrt{5} H_2^{-1} H_2^1}{\sqrt{7 \pi }} + \frac{\sqrt{5} (H_2^{0})^2}{7 \sqrt{\pi }} + \frac{H_0^{0} H_2^0}{\sqrt{\pi }} \nonumber \\&+ 2 H_2^{0} + \frac{\sqrt{5} H_1^{-1} H_1^1}{\sqrt{5 \pi }} + \frac{\sqrt{5} (H_1^{0})^2}{5 \sqrt{\pi }} \end{aligned}$$

(A.11)

$$\begin{aligned} L_{0,5,5} (s)= & {} \frac{(H_0^0)^2 H_5^5}{4 \pi } -\frac{H_0^0 H_5^5}{2 \sqrt{\pi }} + H_5^5 + \frac{\sqrt{5}\sqrt{33} H_0^0 H_1^1 H_4^4}{22 \pi } \nonumber \\&- \frac{\sqrt{5}\sqrt{33} H_1^1 H_4^4}{22 \sqrt{\pi }} + \frac{\sqrt{210} \sqrt{385} H_0^0 H_2^2 H_3^3}{462 \pi }\nonumber \\&- \frac{\sqrt{210} \sqrt{385} H_2^2 H_3^3}{462 \sqrt{\pi }} +\frac{\sqrt{5} \sqrt{21} \sqrt{33} (H_1^1)^2 H_3^3}{66 \sqrt{7 } \pi } \nonumber \\&+ \frac{5 \sqrt{5} \sqrt{33} H_1^1 (H_2^2)^2}{22 \sqrt{70} \pi } \end{aligned}$$

(A.12)

Appendix B: The coefficients of $k_1, k_2, k_3$ and $k_4$ in Eq. (A.10)

$$\begin{aligned} k_1= & {} \frac{1}{4 \pi } \end{aligned}$$

(B.1)

$$\begin{aligned} k_2= & {} \frac{3}{2 \sqrt{\pi }} \end{aligned}$$

(B.2)

$$\begin{aligned} k_3= & {} - \frac{3 H_{3}^{-3} H_{3}^{3}}{2 \pi } + \frac{3 H_{3}^{-2} H_{3}^{2}}{2 \pi } - \frac{3 H_{3}^{-1} H_{3}^{1}}{2 \pi } + \frac{3 (H_{3}^{0})^{2}}{2 \pi }+ \frac{3 H_{2}^{-2} H_{2}^{2}}{2 \pi } \nonumber \\&- \frac{3 H_{2}^{-1} H_{2}^{1}}{2 \pi }+ \frac{3 (H_{2}^{0})^{2} }{4 \pi }- \frac{3 H_{1}^{-1} H_{1}^{1} }{2 \pi }+ \frac{3 (H_{1}^{0})^{2}}{4 \pi } + 3 \end{aligned}$$

(B.3)

$$\begin{aligned} k_4= & {} -\frac{3 H_{3}^{-3} H_{3}^{3}}{\sqrt{\pi }} - \frac{9 \sqrt{15} H_{1}^{-1} H_{2}^{-2} H_{3}^{3}}{2 \sqrt{105} \pi } + \frac{3 H_{3}^{-2} H_{3}^{2}}{\sqrt{\pi }} \nonumber \\&+ \frac{9 \sqrt{10} H_{1}^{-1} H_{2}^{-1} H_{3}^{2}}{2 \sqrt{105} \pi }+ \frac{9 \sqrt{5} H_{1}^{0} H_{2}^{-2} H_{3}^{2}}{2 \sqrt{105} \pi } -\frac{3 H_{3}^{-1} H_{3}^{1}}{ \sqrt{\pi } } \nonumber \\&- \frac{9 \sqrt{6} H_{1}^{-1} H_{2}^{0} H_{3}^{1}}{2 \sqrt{105} \pi } -\frac{9 \sqrt{2} H_{1}^{0} H_{2}^{-1} H_{3}^{1}}{\sqrt{105} \pi }-\frac{9 H_{1}^{1} H_{2}^{-2} H_{3}^{1}}{2 \sqrt{105} \pi }\nonumber \\&+\frac{3 (H_{3}^{0})^{2}}{2 \sqrt{\pi }} +\frac{9 \sqrt{3} H_{1}^{-1} H_{2}^{1} H_{3}^{0}}{2 \sqrt{105} \pi } +\frac{27 H_{1}^{0} H_{2}^{0} H_{3}^{0}}{2 \sqrt{105} \pi } \nonumber \\&+\frac{9 \sqrt{3} H_{1}^{1} H_{2}^{-1} H_{3}^{0}}{2 \sqrt{105} \pi } -\frac{9 H_{1}^{-1} H_{2}^{2} H_{3}^{-1}}{2 \sqrt{105} \pi } -\frac{9 \sqrt{2} H_{1}^{0} H_{2}^{1} H_{3}^{-1}}{\sqrt{105} \pi } \nonumber \\&-\frac{9 \sqrt{6} H_{1}^{1} H_{2}^{0} H_{3}^{-1}}{2 \sqrt{105} \pi } +\frac{9 \sqrt{5} H_{1}^{0} H_{2}^{2} H_{3}^{-2}}{2 \sqrt{105} \pi } +\frac{9 \sqrt{10} H_{1}^{1} H_{2}^{1} H_{3}^{-2}}{2 \sqrt{105} \pi } \nonumber \\&-\frac{9 \sqrt{15} H_{1}^{1} H_{2}^{2} H_{3}^{-3}}{2 \sqrt{105} \pi } -\frac{3 \sqrt{5} H_{2}^{-2} H_{2}^{0} H_{2}^{2}}{7 \pi } +\frac{3 \sqrt{5} \sqrt{6} (H_{2}^{-1})^{2} H_{2}^{2}}{28 \pi }\nonumber \\&+\frac{3 H_{2}^{-2} H_{2}^{2}}{\sqrt{\pi } } +\frac{3 \sqrt{5} \sqrt{6} (H_{1}^{-1})^{2} H_{2}^{2}}{20 \pi } +\frac{3 \sqrt{5} \sqrt{6} H_{2}^{-2} (H_{2}^{1})^{2}}{28 \pi }\nonumber \\&-\frac{3 \sqrt{5} H_{2}^{-1} H_{2}^{0} H_{2}^{1}}{14 \pi } -\frac{3 H_{2}^{-1} H_{2}^{1}}{\sqrt{\pi } }-\frac{3 \sqrt{3} \sqrt{5} H_{1}^{-1} H_{1}^{0} H_{2}^{1}}{10 \pi } \nonumber \\&+\frac{\sqrt{5} (H_{2}^{0})^{3}}{14 \pi } +\frac{3 (H_{2}^{0})^{2}}{2 \sqrt{\pi } } +\frac{3 \sqrt{5} H_{1}^{-1} H_{1}^{1} H_{2}^{0}}{10 \pi } +\frac{3 \sqrt{5} (H_{1}^{0})^{2} H_{2}^{0}}{10 \pi } \nonumber \\&-\frac{3 \sqrt{3} \sqrt{5} H_{1}^{0} H_{1}^{1} H_{2}^{-1}}{10 \pi } +\frac{3 \sqrt{5} \sqrt{6} (H_{1}^{1})^{2} H_{2}^{-2}}{20 \pi } -\frac{3 H_{1}^{-1} H_{1}^{1}}{\sqrt{\pi }} \nonumber \\&+\frac{3 (H_{1}^{0})^{2}}{2 \sqrt{\pi } } \end{aligned}$$

(B.4)

Appendix C: The derivation of $A _n^m$ and $B _n^m$

Let x = $s /s _0$, y = $s '/s _0$, where $s _0$ is the mean equi-volumetric radii of the equipotential surface crossing the point on uppermost boundary, so $s _0$ is 6371 km in Sect. 3.1 and 6296 km in Sect. 3.2. Dividing the Earth into L layers with the lowest and uppermost boundary ($x _1$ = 0, $x _{L+1}$ = 1) and $x$$\in$ [$x _k$, $x _{k+1}$], we can get the integral of $A _n^m$ and $B _n^m$:

$$\begin{aligned} A_{n}^{m}&= x^{-n-3} \int _{0}^{x} \frac{\mathrm{d} [y^{n+3}L_{n+3,n,m}]}{\mathrm{d} y} \cdot \delta \mathrm{d} y\nonumber \\&= x^{-n-3} \left[ \int _{x_k}^{x} \frac{\mathrm{d} y^{n+3} L_{n+3,n,m}}{\mathrm{d} y}\delta _{k}(y)\mathrm{d} y \right. \nonumber \\&\left. + \sum _{i=1}^{k-1} \int _{x_i}^{x_{i+1}}\frac{\mathrm{d} y^{n+3} L_{n+3,n,m}}{\mathrm{d} y}\delta _{i}(y) \mathrm{d} y\right] \nonumber \\&= x^{-n-3} \left[ y^{n+3} L_{n+3,n,m} \delta _{k}(y) |_{x_k}^{x} \right. - \int _{x_k}^{x} y^{n+3} L_{n+3,n,m}\frac{\mathrm{d} \delta _{k}(y)}{\mathrm{d} y}\mathrm{d} y\nonumber \\&\left. \quad + \sum _{i=1}^{k-1} y^{n+3} L_{n+3,n,m} \delta _{i}(y) |_{x_i}^{x_{i+1}}\right. \nonumber \\&\left. - \sum _{i=1}^{k-1} \int _{x_i}^{x_{i+1}} y^{n+3} L_{n+3,n,m}\frac{\mathrm{d} \delta _{i}(y)}{\mathrm{d} y}\mathrm{d} y\right] \nonumber \\&= - x^{-n-3} \int _{x_k}^{x} y^{n+3} L_{n+3,n,m}\frac{\mathrm{d} \delta _{k}(y)}{\mathrm{d} y}\mathrm{d} y - x^{-n-3} \sum _{i=1}^{k-1} \int _{x_i}^{x_{i+1}} y^{n+3} \nonumber \\&\quad \cdot L_{n+3,n,m}\frac{\mathrm{d} \delta _{i}(y)}{\mathrm{d} y}\mathrm{d} y + L_{n+3,n,m}(x) \delta _{k}(x) + x^{-n-3}\sum _{i=1}^{k-1} [ \delta _{i}(x_{i+1}) \nonumber \\&\quad -\, \delta _{i+1}(x_{i+1})]L_{n+3,n,m} (x_{i+1})x_{i+1}^{n+3} \end{aligned}$$

(C.1)

$$\begin{aligned} B_{n}^{m}&{\mathop {=}\limits ^{n\ne 2}} x^{n-2} \int _{x}^{1} \frac{\mathrm{d} [ y^{2-n}L_{2-n,n,m}]}{\mathrm{d} y} \cdot \delta y\nonumber \\&= x^{n-2} \left[ \int _{x}^{x_{k+1}} \frac{\mathrm{d} y^{2-n} L_{2-n,n,m}}{\mathrm{d} y}\delta _{k}(y) \mathrm{d} y \right. \nonumber \\&\left. \quad +\, \sum _{i=k+1}^{L} \int _{x_i}^{x_{i+1}}\frac{\mathrm{d} y^{2-n} L_{2-n,n,m}}{\mathrm{d} y}\delta _{i}(y) \mathrm{d} y\right] \nonumber \\&= x^{n-2} \left[ y^{2-n} L_{2-n,n,m} \delta _{k}(y) |_{x}^{x_{k+1}} \right. \left. - \int _{x}^{x_{k+1}} y^{2-n} L_{2-n,n,m}\frac{\mathrm{d} \delta _{k}(y)}{\mathrm{d} y} \mathrm{d} y \right. \nonumber \\&\left. \quad +\, \sum _{i=k+1}^{L} y^{2-n} L_{2-n,n,m} \delta _{i}(y)|_{x_i}^{x_{i+1}} \right. \nonumber \\&\quad \left. - \sum _{i=k+1}^{L} \int _{x_i}^{x_{i+1}} y^{2-n} L_{2-n,n,m}\frac{\mathrm{d} \delta _{i}(y)}{\mathrm{d} y} \mathrm{d} y\right] \nonumber \\&= -L_{2-n,n,m}(x) \delta _{k}(x) + x^{n-2} L_{2-n,n,m}(1)\delta _{L}(1) - x^{n-2} \int _{x}^{x_{k+1}} y^{2-n} \nonumber \\&\quad \cdot L_{2-n,n,m} \frac{\mathrm{d} \delta _{k}(y)}{\mathrm{d} y} \mathrm{d} y + x^{n-2}\sum _{i=k+1}^{L} [ \delta _{i-1}(x_{i}) - \delta _{i}(x_{i}) L_{2-n,n,m}(x_i)x_i^{2-n} \nonumber \\&\quad -\, x^{n-2} \sum _{i=k+1}^{L} \int _{x_i}^{x_{i+1}} y^{2-n} L_{2-n,n,m} \frac{\mathrm{d} \delta _{i}(y)}{\mathrm{d} y} \mathrm{d} y \end{aligned}$$

(C.2)

$$\begin{aligned} B_{n}^{m}&{\mathop {=}\limits ^{n = 2}} \int _{x}^{1} \frac{\mathrm{d} L_{0,2,m}}{\mathrm{d} y} \cdot \delta \mathrm{d} y\nonumber \\&= \sum _{i=k+1}^{L} \int _{x_i}^{x_{i+1}} \frac{\mathrm{d} L_{0,2,m}}{\mathrm{d} y} \cdot \delta _i (y) \mathrm{d} y\nonumber \\&= \sum _{i=k+1}^{L} L_{0,2,m} \delta _i (y) |_{x_i}^{x_{i+1}} - \sum _{i=k+1}^{L} \int _{x_i}^{x_{i+1}} L_{0,2,m} \frac{\mathrm{d} \delta _i (y)}{\mathrm{d} y} \mathrm{d} y \end{aligned}$$

(C.3)

where $\delta _i = \rho _i / \overline{\rho }, \rho _i$ is the density of the i-th layer and $\overline{\rho }$ is the mean density of the whole Earth for case (1) in Sect. 3.1, or the mean density of the dominant part of the Earth surrounded by the equipotential surface of equi-volume radii $s = 6296$ km for case (2) in Sect. 3.2.

Appendix D: The values of $C_n^m$ (n = 0, ..., 6; m = $-n$, ..., n)

The terms $C_n^m$ in Eq. (37) correspond to the gravitational contribution of the mass of crust layer (the depth up to 75 km, or $s_2$$\in$ [6296, 6371] km) in the CRUST1.0 model in this work. As the density distribution is given by an a prior (CRUST1.0 model here) and will not change with the variation of the equi- potential (or equi-density) surfaces interior, except that the bottom of the deepest 9th layer, i.e., the top of the inner dominant Earth; therefore, $C_n^m$ can be approximately regarded as constant for a given location. $C_n^m$ can thus be calculated at beginning for subsequent usage, by discrete integration with CRUST1.0, and do not need to take part in the iterations. Moreover, this table values are the final results after the uppermost equipotential surface $(S_{L+1})$ changes. All the 49 values of $C_n^m$ are normalized and listed in following.

The recording order is: $C_0^0, C_1^{-1}, C_1^0, C_1^1, C_2^{-2}, C_2^{-1}, C_2^0$, $C_2^1$, $C_2^2$, $C_3^{-3}$, $\ldots$, $C_6^5$, $C_6^6$ in the next lines with the real and imaginary parts of $C_n^m$ in the same bracket.

Appendix E: On the ellipticity (f)

If a symmetric Earth model like PREM is used as a starting Earth model, the equipotential surfaces are obtained to be rotational symmetry and equatorial symmetry with this generalized theory, which means that there are only coefficients $H_0^0$, $H_2^0$, $H_4^0$ and $H_6^0$ kept in Eq. (7). Then one can define an effective ellipticity (f) of this figure ($\mathbf{S_i} (s_i)$):

$$\begin{aligned} f(s_i)=\frac{a(s_i)-c(s_i)}{a(s_i)} \end{aligned}$$

(E.1)

where $a(s_i)$ and $c(s_i)$ are the equatorial radii and polar radii of the Earth.

$$\begin{aligned} a(s_i)= & {} r (s_i, \frac{\pi }{2}, 0) \nonumber \\ {}= & {} s_i[1+ H_0^0(s_i)Y_0^0(\frac{\pi }{2}, 0) + H_2^0(s_i)Y_2^0(\frac{\pi }{2}, 0) \nonumber \\ {}&+ H_4^0(s_i)Y_4^0(\frac{\pi }{2}, 0) + H_6^0(s_i)Y_6^0(\frac{\pi }{2}, 0)]\end{aligned}$$

(E.2)

$$\begin{aligned} c(s_i)= & {} r (s_i, 0, 0) \nonumber \\ {}= & {} s_i[1+ H_0^0(s_i)Y_0^0(0, 0) + H_2^0(s_i)Y_2^0(0, 0) \nonumber \\ {}&+ H_4^0(s_i)Y_4^0(0, 0) + H_6^0(s_i)Y_6^0(0, 0)] \end{aligned}$$

(E.3)

Appendix F: Some terms of $p_{n,m}, f_{k,n,m}, g_{k,n,m}, u_{k,n,m}$ and $DH_n^m$

In this appendix, we list some terms (e.g., $p_{n,m}$, $L_{k,n,m}$, $f_{k,n,m}$, $g_{k,n,m}$ and $u_{k,n,m}$) with $n = 2$ and $m = 0$ for reader’s reference and check:

$$\begin{aligned} p_{2,0} (s)= & {} - \frac{2 L_{2, 4, 0}}{7 \sqrt{5}} + \frac{5 L_{2,2,0}}{21} - \frac{L_{2,0,0}}{3 \sqrt{5}} \end{aligned}$$

(F.1)

$$\begin{aligned} f_{1,2,0}(s)= & {} -{{\sqrt{6}\sqrt{\pi }L_{-2 , 3 , -1} A_1^1} \over {2\sqrt{105}}}+{{ \sqrt{\pi }L_{-2 , 1 , -1} A_1^1}\over {6\sqrt{5}}} \nonumber \\&+{{3\sqrt{\pi }L_{-2 , 3 , 0} A_1^0} \over {2\sqrt{105}}}+{{ \sqrt{\pi }L_{-2 , 1 , 0} A_1^0}\over {3\sqrt{5}}}\nonumber \\&-{{\sqrt{6}\sqrt{\pi }L_{ -2 , 3 , 1} A_1^{-1}}\over {2\sqrt{105} }}+{{\sqrt{\pi }L_{-2 , 1 , 1} A_1^{-1} }\over {6\sqrt{5}}} \end{aligned}$$

(F.2)

$$\begin{aligned} f_{3,2,0}(s)= & {} {{\sqrt{5}\sqrt{\pi }L_{-4 , 3 , -3 } A_3^3}\over {63}}-{{\sqrt{\pi } L_{-4 , 3 , -1} A_3^1 }\over {21\sqrt{5}}} \nonumber \\&-{{\sqrt{6}\sqrt{\pi }L_{-4 , 1 , -1}A_3^1 }\over {7\sqrt{105}}}+{{4 \sqrt{\pi }L_{-4 , 3 , 0} A_3^0}\over {63\sqrt{5}}} \nonumber \\&+{{3\sqrt{\pi }L_{-4 , 1 , 0} A_3^0}\over {7\sqrt{105}}}-{{ \sqrt{\pi }L_{-4 , 3 , 1} A_3^{ -1 }}\over {21\sqrt{5}}} \nonumber \\&-{{\sqrt{6}\sqrt{\pi }L _{-4 , 1 , 1} A_3^{ -1 }}\over {7 \sqrt{105}}}+{{\sqrt{5}\sqrt{\pi }L_{-4 , 3 , 3 } A_3^{ -3 }}\over {63}} \nonumber \\\end{aligned}$$

(F.3)

$$\begin{aligned} g_{1,2,0}(s)= & {} -{{2\sqrt{6}\sqrt{\pi }B_1^{ -1 } L_{1 , 3 , 1}}\over {\sqrt{105}}}+{{6 \sqrt{\pi } B_1^{ 0} L_{1 , 3 , 0 }}\over {\sqrt{105}}} \nonumber \\&-{{2\sqrt{6}\sqrt{\pi }B_1^1 L_{1 , 3 , -1}}\over {\sqrt{105 }}}+{{2\sqrt{\pi } B_1^{ -1 } L_{1 , 1 , 1}}\over {3\sqrt{5}}} \nonumber \\&+{{4\sqrt{\pi } B_1^{ 0} L_{1 , 1 , 0}}\over {3\sqrt{5}}}+{{2 \sqrt{\pi } B_1^{1} L_{1 , 1 , -1 }}\over {3\sqrt{5}}} \end{aligned}$$

(F.4)

$$\begin{aligned} g_{3,2,0}(s)= & {} -{{2\sqrt{5}\sqrt{\pi } B_3^{-3} L_{3 , 3 , 3}}\over {21}}+{{2\sqrt{\pi } B_3^{ -1} L_{3 , 3 , 1} }\over {7\sqrt{5}}} \nonumber \\&-{{8\sqrt{\pi }B_3^{ 0} L_{3 , 3 , 0}}\over {21\sqrt{5}}}+{{2\sqrt{\pi } B_3^{1} L_{3 , 3 , -1} }\over {7\sqrt{5}}}\nonumber \\&-{{2\sqrt{5}\sqrt{\pi } B_3^{ 3} L_{3 , 3 , -3}}\over {21}}+{{6\sqrt{6} \sqrt{\pi } B_3^{-1} L_{3 , 1 , 1 }}\over {7\sqrt{105}}} \nonumber \\&-{{18\sqrt{\pi } B_3^{ 0} L_{3 , 1 , 0}}\over {7\sqrt{105}}}+{{6 \sqrt{6}\sqrt{\pi } B_3^{1} L _{3 , 1 , -1}}\over {7\sqrt{105}}} \end{aligned}$$

(F.5)

$$\begin{aligned} u_{1,2,0}(s)= & {} -{{2\sqrt{6}\sqrt{\pi }C_1^{-1 } L_{1 , 3 , 1 }}\over {\sqrt{105}}}+{{6 \sqrt{\pi } C_1^{0} L_{1 , 3 , 0 }}\over {\sqrt{105}}} \nonumber \\&-{{2\sqrt{6}\sqrt{\pi } C_1^{1 } L_{1 , 3 , -1 }}\over { \sqrt{105}}}+{{2\sqrt{\pi } C_1^{-1 } L_{1 , 1 , 1 }}\over {3\sqrt{5}}} \nonumber \\&+{{4\sqrt{ \pi } C_1^{0 } L_{1 , 1 , 0 , I }}\over {3\sqrt{5}}}+{{2\sqrt{\pi } C_1^{1 }L_{1 , 1 , -1 }}\over {3\sqrt{5}}} \end{aligned}$$

(F.6)

$$\begin{aligned} u_{3,2,0}(s)= & {} {{2\sqrt{5}\sqrt{\pi } C_3^{-3} L_{3 , 3 , 3 }}\over {21}}-{{2\sqrt{ \pi }C_3^{-1 } L_{3 , 3 , 1 , I }}\over {7\sqrt{5}}}\nonumber \\&+{{8\sqrt{\pi } C_3^{0} L_{3 , 3 , 0 }}\over {21\sqrt{5}}}-{{2\sqrt{\pi } C_3^{1} L_{3 , 3 , -1 }}\over {7\sqrt{5}}} \nonumber \\&+{{2\sqrt{5}\sqrt{\pi } C_3^{3} L_{3 , 3 , -3 } }\over {21}}-{{6\sqrt{6}\sqrt{\pi } C_3^{-1 } L_{3 , 1 , 1 }}\over {7\sqrt{105}}}\nonumber \\&+{{18 \sqrt{\pi } C_3^{0} L_{3 , 1 , 0 }}\over {7\sqrt{105}}} -{{6\sqrt{6}\sqrt{\pi } C_3^{1} L_{3 , 1 , -1 } }\over {7\sqrt{105}}} \end{aligned}$$

(F.7)

$$\begin{aligned}&DH_{2}^{0}\\&= {{15\sqrt{15} (H_0^0)^4 H_2^{-2} H_4^2} \over {112\pi ^{{{5}\over {2}}}}} - {{5\sqrt{15} (H_0^0)^3 H_2^{-2} H_4^2 }\over {28\pi ^2}} \\&+{{3\sqrt{15} (H_0^0)^2 H_2^{-2} H_4^2 }\over {14\pi ^{{{3 }\over {2}}}}}-{{3\sqrt{15} H_0^0 H_2^{-2} H_4^2 }\over {14\pi }} \\&+{{\sqrt{15} H_2^{-2} H_4^2 }\over {7\sqrt{\pi }}}+{{15\sqrt{6 }\sqrt{15}\sqrt{21} (H_0^0)^3 (H_1^{-1})^2 H_4^2 }\over {56\sqrt{105}\pi ^{{{5}\over {2 }}}}} \\&-{{15\sqrt{6}\sqrt{15}\sqrt{21}(H_0^0)^2 (H_1^{-1})^2 H_4^2 }\over {56\sqrt{105}\pi ^2}}+{{3\sqrt{6}\sqrt{15}\sqrt{21} H_0^0 (H_1^{-1})^2 H_4^2 }\over {14\sqrt{105}\pi ^{ {{3}\over {2}}}}} \\&-{{3\sqrt{6}\sqrt{15}\sqrt{21} (H_1^{-1})^2 H_4^2 }\over {28\sqrt{105}\pi }}-{{15 \sqrt{30} (H_0^0)^4 H_2^{-1} H_4^1 }\over {112\pi ^{{{5}\over {2}}}}} \\&+{{5\sqrt{30} (H_0^0)^3 H_2^{-1} H_4^1 }\over {28\pi ^2}}-{{3\sqrt{30}(H_0^0)^2 H_2^{-1} H_4^1 }\over {14\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{30}H_0^0 H_2^{-1} H_4^1 }\over {14 \pi }}-{{\sqrt{30} H_2^{-1} H_4^1 }\over {7\sqrt{\pi }}}\\&-{{15\sqrt{6}\sqrt{15}\sqrt{21} (H_0^0)^3 H_1^{-1} H_1^0 H_4^1 }\over {56\sqrt{105}\pi ^{{{5}\over {2}}}}}-{{45 \sqrt{10}\sqrt{21} (H_0^0)^3 H_1^{-1} H_1^0 H_4^1 }\over {56\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{15\sqrt{6}\sqrt{15}\sqrt{21} (H_0^0)^2 H_1^{-1} H_1^0 H_4^1 }\over {56\sqrt{105}\pi ^2}}+{{45\sqrt{10}\sqrt{21} (H_0^0)^2 H_1^{-1} H_1^0 H_4^1 }\over {56\sqrt{105}\pi ^2}} \\&-{{3\sqrt{6} \sqrt{15}\sqrt{21}H_0^0 H_1^{-1} H_1^0 H_4^1 }\over {14\sqrt{105}\pi ^{ {{3}\over {2}}}}}-{{9\sqrt{10}\sqrt{21}H_0^0 H_1^{-1} H_1^0 H_4^1 }\over {14 \sqrt{105}\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{6}\sqrt{15}\sqrt{21 } H_1^{-1} H_1^0 H_4^1 }\over {28\sqrt{105}\pi }}+{{9\sqrt{10}\sqrt{21} H_1^{-1} H_1^0 H_4^1 }\over {28\sqrt{105 }\pi }} \\&+{{45 (H_0^0)^4 H_2^0 H_4^0 }\over {56\pi ^{{{5}\over {2}}}}}-{{15 (H_0^0)^3 H_2^0 H_4^0 }\over {14\pi ^2}} +{{9(H_0^0)^2 H_2^0 H_4^0 }\over {7\pi ^{{{3}\over {2}}}}} \\&-{{9H_0^0 H_2^0 H_4^0 }\over {7\pi }} +{{6H_2^0 H_4^0 }\over {7\sqrt{\pi }}}+{{45\sqrt{21} (H_0^0)^3 H_1^{-1} H_1^1 H_4^0 }\over {14\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{45 \sqrt{21}(H_0^0)^2 H_1^{-1} H_1^1 H_4^0 }\over {14\sqrt{105}\pi ^2}}+{{18 \sqrt{21}H_0^0 H_1^{-1} H_1^1 H_4^0 }\over {7\sqrt{105}\pi ^{{{3}\over {2}} }}} \\&-{{9\sqrt{21} H_1^{-1} H_1^1 H_4^0 }\over {7\sqrt{105}\pi }}+{{45\sqrt{21} (H_0^0)^3 (H_1^0)^2 H_4^0 }\over {14\sqrt{105}\pi ^{{{5}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&-{{45\sqrt{21} (H_0^0)^2 (H_1^0)^2 H_4^0 }\over {14\sqrt{105}\pi ^2}}+{{18\sqrt{21}H_0^0 (H_1^0)^2 H_4^0 }\over {7\sqrt{105} \pi ^{{{3}\over {2}}}}} \\&-{{9\sqrt{21} (H_1^0)^2 H_4^0 }\over {7\sqrt{105}\pi }}-{{15\sqrt{30} (H_0^0)^4 H_2^1 H_4^{-1} }\over {112 \pi ^{{{5}\over {2}}}}} \\&+{{5\sqrt{30} (H_0^0)^3 H_2^1 H_4^{-1} }\over {28\pi ^2}}-{{3\sqrt{30} (H_0^0)^2 H_2^1 H_4^{-1} }\over {14\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{30}H_0^0 H_2^1 H_4^{-1} }\over {14\pi }}-{{\sqrt{30 } H_2^1 H_4^{-1} }\over {7\sqrt{\pi }}} \\&-{{15\sqrt{6}\sqrt{15}\sqrt{21} (H_0^0)^3 H_1^0 H_1^1 H_4^{-1} }\over {56 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{45\sqrt{10}\sqrt{21} (H_0^0)^3 H_1^0 H_1^1 H_4^{-1} }\over {56\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{15 \sqrt{6}\sqrt{15}\sqrt{21}(H_0^0)^2 H_1^0 H_1^1 H_4^{-1} }\over {56\sqrt{105 }\pi ^2}}+{{45\sqrt{10}\sqrt{21}(H_0^0)^2 H_1^0 H_1^1 H_4^{-1} }\over {56 \sqrt{105}\pi ^2}} \\&-{{3\sqrt{6}\sqrt{15}\sqrt{21} H_0^0 H_1^0 H_1^1 H_4^{-1} }\over {14\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9\sqrt{10} \sqrt{21}H_0^0 H_1^0 H_1^1 H_4^{-1} }\over {14\sqrt{105}\pi ^{{{3}\over {2 }}}}} \\&+{{3\sqrt{6}\sqrt{15}\sqrt{21} H_1^0 H_1^1 H_4^{-1} }\over {28\sqrt{105}\pi }}+{{9\sqrt{10}\sqrt{21} H_1^0 H_1^1 H_4^{-1} }\over {28\sqrt{105}\pi }} \\&+{{15\sqrt{15} (H_0^0)^4 H_2^2 H_4^{-2} }\over {112\pi ^{{{5}\over {2}}}}}-{{5\sqrt{15} (H_0^0)^3 H_2^2 H_4^{-2} }\over {28\pi ^2}} \\&+{{3\sqrt{15}(H_0^0)^2 H_2^2 H_4^{-2} }\over {14\pi ^{{{3}\over {2}}}}}-{{3\sqrt{15} H_0^0 H_2^2 H_4^{-2} }\over {14\pi }} \\&+{{\sqrt{15} H_2^2 H_4^{-2} }\over {7 \sqrt{\pi }}}+{{15\sqrt{6}\sqrt{15}\sqrt{21} (H_0^0)^3 (H_1^1)^2 H_4^{-2} }\over {56 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{15\sqrt{6}\sqrt{15}\sqrt{21 }(H_0^0)^2 (H_1^1)^2 H_4^{-2} }\over {56\sqrt{105}\pi ^2}}+{{3\sqrt{6}\sqrt{15} \sqrt{21}H_0^0 (H_1^1)^2 H_4^{-2} }\over {14\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{3\sqrt{6} \sqrt{15}\sqrt{21} (H_1^1)^2 H_4^{-2} }\over {28\sqrt{105}\pi }}+{{5\sqrt{5} (H_0^0)^4 H_3^{-3} H_3^3 }\over {16\pi ^{{{5}\over {2 }}}}} \\&-{{5\sqrt{5} (H_0^0)^3 H_3^{-3} H_3^3 }\over {12\pi ^2}}+{{\sqrt{5} (H_0^0)^2 H_3^{-3} H_3^3 }\over {2\pi ^{{{3 }\over {2}}}}} \\&-{{\sqrt{5}H_0^0 H_3^{-3} H_3^3 }\over {2\pi }}+{{\sqrt{5} H_3^{-3} H_3^3 }\over {3\sqrt{\pi }}} \\&+{{9\sqrt{5} \sqrt{15} (H_0^0)^3 H_1^{-1} H_2^{-2} H_3^3 }\over {4\sqrt{105}\pi ^{{{5}\over {2}} }}}+{{3\sqrt{5}\sqrt{6}\sqrt{10} (H_0^0)^3 H_1^{-1} H_2^{-2} H_3^3 }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&-{{9\sqrt{5}\sqrt{15} (H_0^0)^2 H_1^{-1} H_2^{-2} H_3^3 }\over {4\sqrt{105}\pi ^2}}-{{3\sqrt{5} \sqrt{6}\sqrt{10}(H_0^0)^2 H_1^{-1} H_2^{-2} H_3^3 }\over {4\sqrt{105}\pi ^2 }} \\&+{{9\sqrt{5}\sqrt{15}H_0^0 H_1^{-1} H_2^{-2} H_3^3 }\over {5\sqrt{105 }\pi ^{{{3}\over {2}}}}}+{{3\sqrt{5}\sqrt{6}\sqrt{10} H_0^0 H_1^{-1} H_2^{-2} H_3^3 }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9\sqrt{5} \sqrt{15} H_1^{-1} H_2^{-2} H_3^3 }\over {10\sqrt{105}\pi }}-{{3\sqrt{5}\sqrt{6}\sqrt{10 } H_1^{-1} H_2^{-2} H_3^3 }\over {10\sqrt{105}\pi }} \\&+{{15\sqrt{6}\sqrt{15} (H_0^0)^2 (H_1^{-1})^3 H_3^3 }\over {8 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{5\sqrt{6}\sqrt{15} H_0^0 (H_1^{-1})^3 H_3^3 }\over {4 \sqrt{105}\pi ^2}} \\&+{{\sqrt{6}\sqrt{15} (H_1^{-1})^3 H_3^3 }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{3 \sqrt{5}\sqrt{6}\sqrt{15} (H_0^0)^3 H_1^{-1} H_2^{-1} H_3^2 }\over {4\sqrt{105 }\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{10} (H_0^0)^3 H_1^{-1} H_2^{-1} H_3^2 }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}}-{{3\sqrt{5} \sqrt{6}\sqrt{15}(H_0^0)^2 H_1^{-1} H_2^{-1} H_3^2 }\over {4\sqrt{105}\pi ^2 }} \\&+{{9\sqrt{5}\sqrt{10}(H_0^0)^2 H_1^{-1} H_2^{-1} H_3^2 }\over {4\sqrt{105 }\pi ^2}}+{{3\sqrt{5}\sqrt{6}\sqrt{15}H_0^0 H_1^{-1} H_2^{-1} H_3^2 }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{10} H_0^0 H_1^{-1} H_2^{-1} H_3^2 }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{3 \sqrt{5}\sqrt{6}\sqrt{15} H_1^{-1} H_2^{-1} H_3^2 }\over {10\sqrt{105}\pi }} \\&+{{9\sqrt{5 }\sqrt{10} H_1^{-1} H_2^{-1} H_3^2 }\over {10\sqrt{105}\pi }}+{{15\sqrt{3}\sqrt{6} \sqrt{15}(H_0^0)^2 (H_1^{-1})^2 H_1^0 H_3^2 }\over {56\sqrt{105}\pi ^{{{5 }\over {2}}}}} \\&+{{27\sqrt{2}\sqrt{6}\sqrt{10} (H_0^0)^2 (H_1^{-1})^2 H_1^0 H_3^2 }\over {28\sqrt{105}\pi ^{{{5}\over {2}}}}}-{{9\sqrt{3} \sqrt{10}(H_0^0)^2 (H_1^{-1})^2 H_1^0 H_3^2 }\over {8\sqrt{105}\pi ^{{{5 }\over {2}}}}} \\&-{{45\sqrt{5}\sqrt{6}(H_0^0)^2 (H_1^{-1})^2 H_1^0 H_3^2 }\over {28 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{5\sqrt{3}\sqrt{6}\sqrt{15 }H_0^0 (H_1^{-1})^2 H_1^0 H_3^2 }\over {28\sqrt{105}\pi ^2}} \\&-{{9\sqrt{2} \sqrt{6}\sqrt{10}H_0^0 (H_1^{-1})^2 H_1^0 H_3^2 }\over {14\sqrt{105}\pi ^2 }}+{{3\sqrt{3}\sqrt{10}H_0^0 (H_1^{-1})^2 H_1^0 H_3^2 }\over {4\sqrt{105 }\pi ^2}} \end{aligned}$$

$$\begin{aligned}&+{{15\sqrt{5}\sqrt{6}H_0^0 (H_1^{-1})^2 H_1^0 H_3^2 }\over {14 \sqrt{105}\pi ^2}}+{{\sqrt{3}\sqrt{6}\sqrt{15} (H_1^{-1})^2 H_1^0 H_3^2 }\over {14\sqrt{105 }\pi ^{{{3}\over {2}}}}} \\&+{{9\sqrt{2}\sqrt{6}\sqrt{10} (H_1^{-1})^2 H_1^0 H_3^2 }\over {35 \sqrt{105}\pi ^{{{3}\over {2}}}}}-{{3\sqrt{3}\sqrt{10} (H_1^{-1})^2 H_1^0 H_3^2 }\over {10 \sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{3\sqrt{5}\sqrt{6} (H_1^{-1})^2 H_1^0 H_3^2 }\over {7 \sqrt{105}\pi ^{{{3}\over {2}}}}}-{{3\sqrt{5} (H_0^0)^4 H_3^{-1} H_3^1 }\over {16\pi ^{{{5 }\over {2}}}}} \\&+{{\sqrt{5} (H_0^0)^3 H_3^{-1} H_3^1 }\over {4\pi ^2}}-{{3\sqrt{5} (H_0^0)^2 H_3^{-1} H_3^1 }\over {10 \pi ^{{{3}\over {2}}}}}\\&+{{3\sqrt{5}H_0^0 H_3^{-1} H_3^1 }\over {10\pi }}-{{\sqrt{5} H_3^{-1} H_3^1 }\over {5\sqrt{\pi }}} \\&-{{45 \sqrt{2}\sqrt{5} (H_0^0)^3 H_1^0 H_2^{-1} H_3^1 }\over {4\sqrt{105}\pi ^{ {{5}\over {2}}}}}+{{45\sqrt{2}\sqrt{5}(H_0^0)^2 H_1^0 H_2^{-1} H_3^1 }\over {4 \sqrt{105}\pi ^2}} \\&-{{9\sqrt{2}\sqrt{5}H_0^0 H_1^0 H_2^{-1} H_3^1 }\over {\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{9\sqrt{2}\sqrt{5} H_1^0 H_2^{-1} H_3^1 }\over {2 \sqrt{105}\pi }} \\&+{{45\sqrt{5} (H_0^0)^3 H_1^1 H_2^{-2} H_3^1 }\over {4 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{45\sqrt{5} (H_0^0)^2 H_1^1 H_2^{-2} H_3^1 }\over {4\sqrt{105}\pi ^2}} \\&+{{9\sqrt{5} H_0^0 H_1^1 H_2^{-2} H_3^1 }\over {\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9\sqrt{5} H_1^1 H_2^{-2} H_3^1 }\over {2 \sqrt{105}\pi }} \\&+{{135\sqrt{6}(H_0^0)^2 (H_1^{-1})^2 H_1^1 H_3^1 }\over {8 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{45\sqrt{6} H_0^0 (H_1^{-1})^2 H_1^1 H_3^1 }\over {4\sqrt{105}\pi ^2}} \\&+{{9\sqrt{6} (H_1^{-1})^2 H_1^1 H_3^1 }\over {2\sqrt{105 }\pi ^{{{3}\over {2}}}}}-{{45\sqrt{10}\sqrt{15} (H_0^0)^2 H_1^{-1} (H_1^0)^2 H_3^1 }\over {56\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{6} (H_0^0)^2 H_1^{-1} (H_1^0)^2 H_3^1 }\over {2\sqrt{105}\pi ^{{{5}\over {2}}}}}+{{9 \sqrt{2}\sqrt{3}(H_0^0)^2 H_1^{-1} (H_1^0)^2 H_3^1 }\over {14\sqrt{105}\pi ^{ {{5}\over {2}}}}} \\&+{{15\sqrt{10}\sqrt{15}H_0^0 H_1^{-1} (H_1^0)^2 H_3^1 }\over {28\sqrt{105}\pi ^2}}+{{9\sqrt{6}H_0^0 H_1^{-1} (H_1^0)^2 H_3^1 }\over {\sqrt{105}\pi ^2}} \\&-{{3\sqrt{2}\sqrt{3} H_0^0 H_1^{-1} (H_1^0)^2 H_3^1 }\over {7\sqrt{105}\pi ^2}}-{{3\sqrt{10}\sqrt{15} H_1^{-1} (H_1^0)^2 H_3^1 }\over {14\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{18\sqrt{6} H_1^{-1} (H_1^0)^2 H_3^1 }\over {5 \sqrt{105}\pi ^{{{3}\over {2}}}}}+{{6\sqrt{2}\sqrt{3} H_1^{-1} (H_1^0)^2 H_3^1 }\over {35 \sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{45\sqrt{6} (H_0^0)^4 H_1^{-1} H_3^1 }\over {16\sqrt{105 }\pi ^{{{5}\over {2}}}}}+{{15\sqrt{6} (H_0^0)^3 H_1^{-1} H_3^1 }\over {4\sqrt{105}\pi ^2 }} \end{aligned}$$

$$\begin{aligned}&-{{9\sqrt{6}(H_0^0)^2 H_1^{-1} H_3^1 }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{9 \sqrt{6}H_0^0 H_1^{-1} H_3^1 }\over {2\sqrt{105}\pi }} \\&-{{3\sqrt{6} H_1^{-1} H_3^1 }\over {\sqrt{105}\sqrt{\pi }}}+{{\sqrt{5 } (H_0^0)^4 (H_3^0)^2 }\over {8\pi ^{{{5 }\over {2}}}}} -{{\sqrt{5} (H_0^0)^3 (H_3^0)^2 }\over {6\pi ^2}}\\&+{{\sqrt{5}(H_0^0)^2 (H_3^0)^2 }\over {5\pi ^{{{3}\over {2}}}}}-{{\sqrt{5}H_0^0 (H_3^0)^2 }\over {5\pi }}+{{2\sqrt{5 } (H_3^0)^2 }\over {15\sqrt{\pi }}}\\&-{{3\sqrt{2} \sqrt{5}\sqrt{6} (H_0^0)^3 H_1^{-1} H_2^1 H_3^0 }\over {4\sqrt{105}\pi ^{{{5 }\over {2}}}}}-{{9\sqrt{3}\sqrt{5} (H_0^0)^3 H_1^{-1} H_2^1 H_3^0 }\over {4 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{3\sqrt{2}\sqrt{5}\sqrt{6 }(H_0^0)^2 H_1^{-1} H_2^1 H_3^0 }\over {4\sqrt{105}\pi ^2}}+{{9\sqrt{3} \sqrt{5}(H_0^0)^2 H_1^{-1} H_2^1 H_3^0 }\over {4\sqrt{105}\pi ^2}} \\&-{{3 \sqrt{2}\sqrt{5}\sqrt{6}H_0^0 H_1^{-1} H_2^1 H_3^0 }\over {5\sqrt{105 }\pi ^{{{3}\over {2}}}}}-{{9\sqrt{3}\sqrt{5} H_0^0 H_1^{-1} H_2^1 H_3^0 }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{2} \sqrt{5}\sqrt{6} H_1^{-1} H_2^1 H_3^0 }\over {10\sqrt{105}\pi }}+{{9\sqrt{3}\sqrt{5 } H_1^{-1} H_2^1 H_3^0 }\over {10\sqrt{105}\pi }} \\&+{{45\sqrt{5} (H_0^0)^3 H_1^0 H_2^0 H_3^0 }\over {2 \sqrt{105}\pi ^{{{5}\over {2}}}}}-{{45\sqrt{5} (H_0^0)^2 H_1^0 H_2^0 H_3^0 }\over {2\sqrt{105}\pi ^2}} \\&+{{18\sqrt{5} H_0^0 H_1^0 H_2^0 H_3^0 }\over {\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9\sqrt{5} H_1^0 H_2^0 H_3^0 }\over { \sqrt{105}\pi }} \\&-{{3\sqrt{2}\sqrt{5}\sqrt{6} (H_0^0)^3 H_1^1 H_2^{-1} H_3^0 }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}}-{{9\sqrt{3} \sqrt{5} (H_0^0)^3 H_1^1 H_2^{-1} H_3^0 }\over {4\sqrt{105}\pi ^{{{5}\over {2}} }}} \\&+{{3\sqrt{2}\sqrt{5}\sqrt{6}(H_0^0)^2 H_1^1 H_2^{-1} H_3^0 }\over {4 \sqrt{105}\pi ^2}}+{{9\sqrt{3}\sqrt{5}(H_0^0)^2 H_1^1 H_2^{-1} H_3^0 }\over {4\sqrt{105}\pi ^2}} \\&-{{3\sqrt{2}\sqrt{5}\sqrt{6}H_0^0 H_1^1 H_2^{-1} H_3^0 }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9 \sqrt{3}\sqrt{5}H_0^0 H_1^1 H_2^{-1} H_3^0 }\over {5\sqrt{105}\pi ^{ {{3}\over {2}}}}} \\&+{{3\sqrt{2}\sqrt{5}\sqrt{6} H_1^1 H_2^{-1} H_3^0 }\over {10\sqrt{105 }\pi }}+{{9\sqrt{3}\sqrt{5} H_1^1 H_2^{-1} H_3^0 }\over {10\sqrt{105}\pi }} \\&-{{15 \sqrt{6}\sqrt{10}\sqrt{15}(H_0^0)^2 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {28\sqrt{105}\pi ^{{{5}\over {2}}}}}\\&+{{27\sqrt{2} \sqrt{3}\sqrt{6}(H_0^0)^2 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {14 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{117(H_0^0)^2 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}}+{{5 \sqrt{6}\sqrt{10}\sqrt{15}H_0^0 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {14\sqrt{105}\pi ^2}} \\&-{{9\sqrt{2}\sqrt{3} \sqrt{6}H_0^0 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {7\sqrt{105 }\pi ^2}}+{{39H_0^0 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {2 \sqrt{105}\pi ^2}} \\&-{{\sqrt{6}\sqrt{10}\sqrt{15} H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {7\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{18\sqrt{2} \sqrt{3}\sqrt{6} H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {35\sqrt{105}\pi ^{ {{3}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&-{{39 H_1^{-1} H_1^0 H_1^1 H_3^0 }\over {5\sqrt{105}\pi ^{{{3 }\over {2}}}}}+{{315(H_0^0)^2 (H_1^0)^3 H_3^0 }\over {8\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{105 H_0^0 (H_1^0)^3 H_3^0 }\over {4\sqrt{105}\pi ^2}}+{{21 (H_1^0)^3 H_3^0 }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{135 (H_0^0)^4 H_1^0 H_3^0 }\over {16 \sqrt{105}\pi ^{{{5}\over {2}}}}}\\&-{{45 (H_0^0)^3 H_1^0 H_3^0 }\over {4\sqrt{105}\pi ^2}} +{{27(H_0^0)^2 H_1^0 H_3^0 }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{27 H_0^0 H_1^0 H_3^0 }\over {2\sqrt{105 }\pi }} \\&+{{9 H_1^0 H_3^0 }\over {\sqrt{105 }\sqrt{\pi }}}+{{45\sqrt{5} (H_0^0)^3 H_1^{-1} H_2^2 H_3^{-1} }\over {4\sqrt{105 }\pi ^{{{5}\over {2}}}}} \\&-{{45\sqrt{5}(H_0^0)^2 H_1^{-1} H_2^2 H_3^{-1} }\over {4\sqrt{105}\pi ^2}}+{{9\sqrt{5}H_0^0 H_1^{-1} H_2^2 H_3^{-1} }\over {\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9\sqrt{5} H_1^{-1} H_2^2 H_3^{-1} }\over {2\sqrt{105 }\pi }}-{{45\sqrt{2}\sqrt{5} (H_0^0)^3 H_1^0 H_2^1 H_3^{-1} }\over {4 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{45\sqrt{2}\sqrt{5} (H_0^0)^2 H_1^0 H_2^1 H_3^{-1} }\over {4\sqrt{105}\pi ^2}}-{{9\sqrt{2} \sqrt{5}H_0^0 H_1^0 H_2^1 H_3^{-1} }\over {\sqrt{105}\pi ^{{{3}\over {2}}} }} \\&+{{9\sqrt{2}\sqrt{5} H_1^0 H_2^1 H_3^{-1} }\over {2\sqrt{105}\pi }}+{{135\sqrt{6} (H_0^0)^2 H_1^{-1} (H_1^1)^2 H_3^{-1} }\over {8\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{45 \sqrt{6}H_0^0 H_1^{-1} (H_1^1)^2 H_3^{-1} }\over {4\sqrt{105}\pi ^2}}+{{9 \sqrt{6} H_1^{-1} (H_1^1)^2 H_3^{-1} }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{45\sqrt{10} \sqrt{15}(H_0^0)^2 (H_1^0)^2 H_1^1 H_3^{-1} }\over {56\sqrt{105}\pi ^{{{5}\over {2 }}}}}-{{27\sqrt{6}(H_0^0)^2 (H_1^0)^2 H_1^1 H_3^{-1} }\over {2\sqrt{105}\pi ^{ {{5}\over {2}}}}} \\&+{{9\sqrt{2}\sqrt{3}(H_0^0)^2 (H_1^0)^2 H_1^1 H_3^{-1} }\over {14 \sqrt{105}\pi ^{{{5}\over {2}}}}}+{{15\sqrt{10}\sqrt{15}H_0^0 (H_1^0)^2 H_1^1 H_3^{-1} }\over {28\sqrt{105}\pi ^2}} \\&+{{9\sqrt{6}H_0^0 (H_1^0)^2 H_1^1 H_3^{-1} }\over {\sqrt{105}\pi ^2}}-{{3\sqrt{2}\sqrt{3 }H_0^0 (H_1^0)^2 H_1^1 H_3^{-1} }\over {7\sqrt{105}\pi ^2}} \\&-{{3\sqrt{10} \sqrt{15} (H_1^0)^2 H_1^1 H_3^{-1} }\over {14\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{18\sqrt{6} (H_1^0)^2 H_1^1 H_3^{-1} }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&+{{6\sqrt{2}\sqrt{3} (H_1^0)^2 H_1^1 H_3^{-1} }\over {35\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{45\sqrt{6} (H_0^0)^4 H_1^1 H_3^{-1} }\over {16 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{15\sqrt{6} (H_0^0)^3 H_1^1 H_3^{-1} }\over {4\sqrt{105 }\pi ^2}}-{{9\sqrt{6}(H_0^0)^2 H_1^1 H_3^{-1} }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&+{{9 \sqrt{6}H_0^0 H_1^1 H_3^{-1} }\over {2\sqrt{105}\pi }}-{{3\sqrt{6} H_1^1 H_3^{-1} }\over {\sqrt{105}\sqrt{\pi }}} \\&+{{3 \sqrt{5}\sqrt{6}\sqrt{15} (H_0^0)^3 H_1^1 H_2^1 H_3^{-2} }\over {4\sqrt{105 }\pi ^{{{5}\over {2}}}}}-{{9\sqrt{5}\sqrt{10} (H_0^0)^3 H_1^1 H_2^1 H_3^{-2} }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}}\\&-{{3\sqrt{5} \sqrt{6}\sqrt{15}(H_0^0)^2 H_1^1 H_2^1 H_3^{-2} }\over {4\sqrt{105}\pi ^2 }}+{{9\sqrt{5}\sqrt{10}(H_0^0)^2 H_1^1 H_2^1 H_3^{-2} }\over {4\sqrt{105 }\pi ^2}} \\&+{{3\sqrt{5}\sqrt{6}\sqrt{15}H_0^0 H_1^1 H_2^1 H_3^{-2} }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9\sqrt{5}\sqrt{10} H_0^0 H_1^1 H_2^1 H_3^{-2} }\over {5\sqrt{105}\pi ^{{{3}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&-{{3 \sqrt{5}\sqrt{6}\sqrt{15} H_1^1 H_2^1 H_3^{-2} }\over {10\sqrt{105}\pi }}+{{9 \sqrt{5}\sqrt{10} H_1^1 H_2^1 H_3^{-2} }\over {10\sqrt{105}\pi }} \\&+{{15\sqrt{3} \sqrt{6}\sqrt{15}(H_0^0)^2 H_1^0 (H_1^1)^2 H_3^{-2} }\over {56\sqrt{105}\pi ^{ {{5}\over {2}}}}}\\&+{{27\sqrt{2}\sqrt{6}\sqrt{10} (H_0^0)^2 H_1^0 (H_1^1)^2 H_3^{-2} }\over {28\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{3} \sqrt{10}(H_0^0)^2 H_1^0 (H_1^1)^2 H_3^{-2} }\over {8\sqrt{105}\pi ^{{{5}\over {2 }}}}}-{{45\sqrt{5}\sqrt{6}(H_0^0)^2 H_1^0 (H_1^1)^2 H_3^{-2} }\over {28 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{5\sqrt{3}\sqrt{6}\sqrt{15} H_0^0 H_1^0 (H_1^1)^2 H_3^{-2} }\over {28\sqrt{105}\pi ^2}}\\&-{{9\sqrt{2} \sqrt{6}\sqrt{10}H_0^0 H_1^0 (H_1^1)^2 H_3^{-2} }\over {14\sqrt{105}\pi ^2 }} \\&+{{3\sqrt{3}\sqrt{10}H_0^0 H_1^0 (H_1^1)^2 H_3^{-2} }\over {4\sqrt{105 }\pi ^2}}+{{15\sqrt{5}\sqrt{6}H_0^0 H_1^0 (H_1^1)^2 H_3^{-2} }\over {14 \sqrt{105}\pi ^2}} \\&+{{\sqrt{3}\sqrt{6}\sqrt{15} H_1^0 (H_1^1)^2 H_3^{-2} }\over {14 \sqrt{105}\pi ^{{{3}\over {2}}}}}+{{9\sqrt{2}\sqrt{6}\sqrt{10} H_1^0 (H_1^1)^2 H_3^{-2} }\over {35\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{3\sqrt{3}\sqrt{10 } H_1^0 (H_1^1)^2 H_3^{-2} }\over {10\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{3\sqrt{5}\sqrt{6} H_1^0 (H_1^1)^2 H_3^{-2} }\over {7\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&+{{9\sqrt{5}\sqrt{15} (H_0^0)^3 H_1^1 H_2^2 H_3^{-3} }\over {4\sqrt{105}\pi ^{{{5}\over {2}}}}}+{{3 \sqrt{5}\sqrt{6}\sqrt{10} (H_0^0)^3 H_1^1 H_2^2 H_3^{-3} }\over {4\sqrt{105 }\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{15} (H_0^0)^2 H_1^1 H_2^2 H_3^{-3} }\over {4\sqrt{105}\pi ^2}}-{{3\sqrt{5}\sqrt{6}\sqrt{10 }(H_0^0)^2 H_1^1 H_2^2 H_3^{-3} }\over {4\sqrt{105}\pi ^2}} \\&+{{9\sqrt{5} \sqrt{15}H_0^0 H_1^1 H_2^2 H_3^{-3} }\over {5\sqrt{105}\pi ^{{{3}\over {2 }}}}}+{{3\sqrt{5}\sqrt{6}\sqrt{10}H_0^0 H_1^1 H_2^2 H_3^{-3} }\over {5 \sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{15} H_1^1 H_2^2 H_3^{-3} }\over {10 \sqrt{105}\pi }}-{{3\sqrt{5}\sqrt{6}\sqrt{10} H_1^1 H_2^2 H_3^{-3} }\over {10\sqrt{105 }\pi }} \\&+{{15\sqrt{6}\sqrt{15}(H_0^0)^2 (H_1^1)^3 H_3^{-3} }\over {8\sqrt{105}\pi ^{{{5 }\over {2}}}}}-{{5\sqrt{6}\sqrt{15}H_0^0 (H_1^1)^3 H_3^{-3} }\over {4\sqrt{105}\pi ^2 }} \end{aligned}$$

$$\begin{aligned}&+{{\sqrt{6}\sqrt{15} (H_1^1)^3 H_3^{-3} }\over {2\sqrt{105}\pi ^{{{3}\over {2}}}}}+{{75 (H_0^0)^3 H_2^{-2} H_2^0 H_2^2 }\over {28\pi ^{{{5}\over {2}}}}} \\&-{{75(H_0^0)^2 H_2^{-2} H_2^0 H_2^2 }\over {28\pi ^2}}+{{15H_0^0 H_2^{-2} H_2^0 H_2^2 }\over {7\pi ^{{{3}\over {2}} }}}\\&-{{15 H_2^{-2} H_2^0 H_2^2 }\over {14\pi }}+{{675\sqrt{6}\sqrt{21} (H_0^0)^2 (H_1^{-1})^2 H_2^0 H_2^2 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{45\sqrt{5} \sqrt{6}(H_0^0)^2 (H_1^{-1})^2 H_2^0 H_2^2 }\over {98\pi ^{{{5}\over {2}}}}}-{{225 \sqrt{6}\sqrt{21}H_0^0 (H_1^{-1})^2 H_2^0 H_2^2 }\over {196\sqrt{105}\pi ^2 }} \\&-{{15\sqrt{5}\sqrt{6}H_0^0 (H_1^{-1})^2 H_2^0 H_2^2 }\over {49\pi ^2}}+{{45\sqrt{6}\sqrt{21} (H_1^{-1})^2 H_2^0 H_2^2 }\over {98\sqrt{105}\pi ^{{{3}\over {2 }}}}} \\&+{{6\sqrt{5}\sqrt{6} (H_1^{-1})^2 H_2^0 H_2^2 }\over {49\pi ^{{{3}\over {2}}}}}-{{5 \sqrt{5}\sqrt{30} (H_0^0)^3 (H_2^{-1})^2 H_2^2 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{5\sqrt{10} \sqrt{15} (H_0^0)^3 (H_2^{-1})^2 H_2^2 }\over {196\pi ^{{{5}\over {2}}}}}+{{5\sqrt{5}\sqrt{30 }(H_0^0)^2 (H_2^{-1})^2 H_2^2 }\over {196\pi ^2}} \\&-{{5\sqrt{10}\sqrt{15} (H_0^0)^2 (H_2^{-1})^2 H_2^2 }\over {196\pi ^2 }}-{{\sqrt{5}\sqrt{30}H_0^0 (H_2^{-1})^2 H_2^2 }\over {49\pi ^{{{3}\over {2}}}}} \\&+{{ \sqrt{10}\sqrt{15}H_0^0 (H_2^{-1})^2 H_2^2 }\over {49\pi ^{{{3}\over {2}}}}}+{{\sqrt{5} \sqrt{30} (H_2^{-1})^2 H_2^2 }\over {98\pi }} \\&-{{\sqrt{10}\sqrt{15} (H_2^{-1})^2 H_2^2 }\over {98\pi }}-{{45\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21 }(H_0^0)^2 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {392\sqrt{105} \pi ^{{{5}\over {2}}}}} \\&-{{135\sqrt{5}\sqrt{10}\sqrt{21} (H_0^0)^2 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}}\\&-{{9\sqrt{3}\sqrt{5}\sqrt{6} (H_0^0)^2 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{81 \sqrt{2}\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {98 \pi ^{{{5}\over {2}}}}}\\&+{{15\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21 }H_0^0 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {196\sqrt{105}\pi ^2 }} \\&+{{45\sqrt{5}\sqrt{10}\sqrt{21}H_0^0 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {196\sqrt{105}\pi ^2}}\\&+{{3\sqrt{3} \sqrt{5}\sqrt{6}H_0^0 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {98 \pi ^2}} \\&-{{27\sqrt{2}\sqrt{5}H_0^0 H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {49\pi ^2}}\\&-{{3\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21 } H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {98\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9 \sqrt{5}\sqrt{10}\sqrt{21} H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {98\sqrt{105 }\pi ^{{{3}\over {2}}}}}-{{3\sqrt{3}\sqrt{5}\sqrt{6} H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {245\pi ^{{{3}\over {2}}}}} \\&+{{54\sqrt{2}\sqrt{5} H_1^{-1} H_1^0 H_2^{-1} H_2^2 }\over {245\pi ^{{{3}\over {2}}}}}+{{135\sqrt{21} (H_0^0)^2 H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {98\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{891\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {196\pi ^{{{5}\over {2}}}}}-{{45\sqrt{21} H_0^0 H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {49\sqrt{105}\pi ^2}} \\&-{{297 \sqrt{5}H_0^0 H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {98\pi ^2}}+{{18\sqrt{21} H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {49\sqrt{105}\pi ^{{{3 }\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&+{{297\sqrt{5} H_1^{-1} H_1^1 H_2^{-2} H_2^2 }\over {245\pi ^{ {{3}\over {2}}}}}+{{135\sqrt{21}(H_0^0)^2 (H_1^0)^2 H_2^{-2} H_2^2 }\over {98 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{5} (H_0^0)^2 (H_1^0)^2 H_2^{-2} H_2^2 }\over {98\pi ^{{{5}\over {2}}}}}-{{45\sqrt{21} H_0^0 (H_1^0)^2 H_2^{-2} H_2^2 }\over {49\sqrt{105}\pi ^2}} \\&+{{9\sqrt{5} H_0^0 (H_1^0)^2 H_2^{-2} H_2^2 }\over {49\pi ^2}}+{{18\sqrt{21} (H_1^0)^2 H_2^{-2} H_2^2 }\over {49\sqrt{105}\pi ^{ {{3}\over {2}}}}} \\&-{{18\sqrt{5} (H_1^0)^2 H_2^{-2} H_2^2 }\over {245\pi ^{{{3}\over {2}}}}}-{{15 \sqrt{5} (H_0^0)^4 H_2^{-2} H_2^2 }\over {56\pi ^{{{5}\over {2}}}}} \\&+{{5\sqrt{5} (H_0^0)^3 H_2^{-2} H_2^2 }\over {14\pi ^2}}-{{3\sqrt{5}(H_0^0)^2 H_2^{-2} H_2^2 }\over {7\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{5} H_0^0 H_2^{-2} H_2^2 }\over {7\pi }}-{{2\sqrt{5} H_2^{-2} H_2^2 }\over {7 \sqrt{\pi }}} \\&+{{45\sqrt{6}H_0^0 (H_1^{-1})^3 H_1^1 H_2^2 }\over {28\pi ^{{{5 }\over {2}}}}}-{{15\sqrt{6} (H_1^{-1})^3 H_1^1 H_2^2 }\over {28\pi ^2}} \\&-{{9\sqrt{10} \sqrt{15}H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {784\pi ^{{{5}\over {2}}}}}\\&+{{3 \sqrt{3}\sqrt{5}\sqrt{6}\sqrt{15}H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {784\pi ^{{{5}\over {2}}}}} \\&+{{27\sqrt{2}\sqrt{5}\sqrt{6} \sqrt{10}H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {1960\pi ^{{{5}\over {2}}}}}\\&-{{9\sqrt{3}\sqrt{5}\sqrt{10}H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {560 \pi ^{{{5}\over {2}}}}} \\&-{{1017\sqrt{6}H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {3920\pi ^{{{5}\over {2}}}}}+{{99\sqrt{2}\sqrt{3} H_0^0 (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {490\pi ^{{{5}\over {2}}}}} \\&+{{3\sqrt{10}\sqrt{15 } (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {784\pi ^2}}-{{\sqrt{3}\sqrt{5}\sqrt{6}\sqrt{15} (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {784\pi ^2}} \\&-{{9\sqrt{2}\sqrt{5}\sqrt{6}\sqrt{10} (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {1960\pi ^2}}+{{3\sqrt{3}\sqrt{5}\sqrt{10} (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {560 \pi ^2}} \\&+{{339\sqrt{6} (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {3920\pi ^2}}-{{33\sqrt{2} \sqrt{3} (H_1^{-1})^2 (H_1^0)^2 H_2^2 }\over {490\pi ^2}} \\&-{{15\sqrt{6} (H_0^0)^3 (H_1^{-1})^2 H_2^2 }\over {28\pi ^{{{5}\over {2 }}}}}+{{15\sqrt{6}(H_0^0)^2 (H_1^{-1})^2 H_2^2 }\over {28\pi ^2}} \end{aligned}$$

$$\begin{aligned}&-{{3\sqrt{6} H_0^0 (H_1^{-1})^2 H_2^2 }\over {7\pi ^{ {{3}\over {2}}}}}+{{3\sqrt{6} (H_1^{-1})^2 H_2^2 }\over {14\pi }} \\&-{{5\sqrt{5}\sqrt{30} (H_0^0)^3 H_2^{-2} (H_2^1)^2 }\over {196\pi ^{{{5}\over {2}}}}}+{{5\sqrt{10}\sqrt{15} (H_0^0)^3 H_2^{-2} (H_2^1)^2 }\over {196\pi ^{{{5 }\over {2}}}}}\\&+{{5\sqrt{5}\sqrt{30}(H_0^0)^2 H_2^{-2} (H_2^1)^2 }\over {196\pi ^2}}-{{5 \sqrt{10}\sqrt{15}(H_0^0)^2 H_2^{-2} (H_2^1)^2 }\over {196\pi ^2}} \\&-{{\sqrt{5}\sqrt{30}H_0^0 H_2^{-2} (H_2^1)^2 }\over {49\pi ^{{{3}\over {2}}}}}+{{\sqrt{10}\sqrt{15} H_0^0 H_2^{-2} (H_2^1)^2 }\over {49\pi ^{ {{3}\over {2}}}}} \\&+{{\sqrt{5}\sqrt{30} H_2^{-2} (H_2^1)^2 }\over {98\pi }}-{{\sqrt{10}\sqrt{15} H_2^{-2} (H_2^1)^2 }\over {98\pi }} \\&+{{45\sqrt{6} \sqrt{10}\sqrt{15}\sqrt{21}(H_0^0)^2 (H_1^{-1})^2 (H_2^1)^2 }\over {392\sqrt{105}\pi ^{{{5 }\over {2}}}}}\\&-{{135\sqrt{5}(H_0^0)^2 (H_1^{-1})^2 (H_2^1)^2 }\over {196\pi ^{{{5}\over {2}}}}} \\&-{{15 \sqrt{6}\sqrt{10}\sqrt{15}\sqrt{21}H_0^0 (H_1^{-1})^2 (H_2^1)^2 }\over {196\sqrt{105} \pi ^2}}+{{45\sqrt{5}H_0^0 (H_1^{-1})^2 (H_2^1)^2 }\over {98\pi ^2}} \\&+{{3\sqrt{6}\sqrt{10} \sqrt{15}\sqrt{21} (H_1^{-1})^2 (H_2^1)^2 }\over {98\sqrt{105}\pi ^{{{3}\over {2}}}}}-{{9\sqrt{5} (H_1^{-1})^2 (H_2^1)^2 }\over {49\pi ^{{{3}\over {2 }}}}} \\&-{{75 (H_0^0)^3 H_2^{-1} H_2^0 H_2^1 }\over {28\pi ^{{{5}\over {2}}}}}+{{75 (H_0^0)^2 H_2^{-1} H_2^0 H_2^1 }\over {28\pi ^2}} \\&-{{15H_0^0 H_2^{-1} H_2^0 H_2^1 }\over {7 \pi ^{{{3}\over {2}}}}}+{{15 H_2^{-1} H_2^0 H_2^1 }\over {14\pi }} \\&-{{45\sqrt{6}\sqrt{15 }\sqrt{21}\sqrt{30}(H_0^0)^2 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}}\\&-{{135\sqrt{10} \sqrt{21}\sqrt{30}(H_0^0)^2 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {392 \sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{2}\sqrt{5}\sqrt{6 }(H_0^0)^2 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {196\pi ^{{{5 }\over {2}}}}}\\&+{{9\sqrt{3}\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{15\sqrt{6} \sqrt{15}\sqrt{21}\sqrt{30}H_0^0 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {196\sqrt{105}\pi ^2}}\\&+{{45\sqrt{10}\sqrt{21} \sqrt{30}H_0^0 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {196\sqrt{105 }\pi ^2}} \\&+{{9\sqrt{2}\sqrt{5}\sqrt{6}H_0^0 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {98\pi ^2}}-{{3\sqrt{3}\sqrt{5}H_0^0 H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {98\pi ^2}} \\&-{{3 \sqrt{6}\sqrt{15}\sqrt{21}\sqrt{30} H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {98 \sqrt{105}\pi ^{{{3}\over {2}}}}}\\&-{{9\sqrt{10}\sqrt{21} \sqrt{30} H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {98\sqrt{105}\pi ^{{{3}\over {2 }}}}} \\&-{{9\sqrt{2}\sqrt{5}\sqrt{6} H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {245 \pi ^{{{3}\over {2}}}}}+{{3\sqrt{3}\sqrt{5} H_1^{-1} H_1^0 H_2^0 H_2^1 }\over {245\pi ^{{{3}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&+{{270\sqrt{21} (H_0^0)^2 H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {49\sqrt{105}\pi ^{{{5}\over {2 }}}}}-{{54\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {49 \pi ^{{{5}\over {2}}}}} \\&-{{180\sqrt{21}H_0^0 H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {49\sqrt{105}\pi ^2}}+{{36\sqrt{5}H_0^0 H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {49\pi ^2}}\\&+{{72 \sqrt{21} H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {49\sqrt{105}\pi ^{{{3}\over {2 }}}}}-{{72\sqrt{5} H_1^{-1} H_1^1 H_2^{-1} H_2^1 }\over {245\pi ^{{{3}\over {2 }}}}} \\&+{{270\sqrt{21}(H_0^0)^2 (H_1^0)^2 H_2^{-1} H_2^1 }\over {49\sqrt{105} \pi ^{{{5}\over {2}}}}}-{{1377\sqrt{5}(H_0^0)^2 (H_1^0)^2 H_2^{-1} H_2^1 }\over {392 \pi ^{{{5}\over {2}}}}} \\&-{{180\sqrt{21}H_0^0 (H_1^0)^2 H_2^{-1} H_2^1 }\over {49 \sqrt{105}\pi ^2}}+{{459\sqrt{5}H_0^0 (H_1^0)^2 H_2^{-1} H_2^1 }\over {196 \pi ^2}} \\&+{{72\sqrt{21} (H_1^0)^2 H_2^{-1} H_2^1 }\over {49\sqrt{105}\pi ^{{{3}\over {2 }}}}}-{{459\sqrt{5} (H_1^0)^2 H_2^{-1} H_2^1 }\over {490\pi ^{{{3}\over {2}}}}} \\&-{{15\sqrt{5} (H_0^0)^4 H_2^{-1} H_2^1 }\over {112\pi ^{{{5}\over {2}}}}}+{{5\sqrt{5} (H_0^0)^3 H_2^{-1} H_2^1 }\over {28\pi ^2}} \\&-{{3\sqrt{5}(H_0^0)^2 H_2^{-1} H_2^1 }\over {14\pi ^{{{3}\over {2}}}}}+{{3\sqrt{5} H_0^0 H_2^{-1} H_2^1 }\over {14\pi }} \\&-{{\sqrt{5} H_2^{-1} H_2^1 }\over {7 \sqrt{\pi }}}-{{45\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21} (H_0^0)^2 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{135\sqrt{5}\sqrt{10}\sqrt{21} (H_0^0)^2 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {392\sqrt{105}\pi ^{{{5}\over {2 }}}}} \end{aligned}$$

$$\begin{aligned}&-{{9\sqrt{3}\sqrt{5}\sqrt{6}(H_0^0)^2 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{81\sqrt{2} \sqrt{5}(H_0^0)^2 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {98\pi ^{{{5 }\over {2}}}}}\\&+{{15\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21}H_0^0 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {196\sqrt{105}\pi ^2 }} \\&+{{45\sqrt{5}\sqrt{10}\sqrt{21}H_0^0 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {196\sqrt{105}\pi ^2}}\\&+{{3\sqrt{3} \sqrt{5}\sqrt{6}H_0^0 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {98 \pi ^2}} \\&-{{27\sqrt{2}\sqrt{5}H_0^0 H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {49\pi ^2}}-{{3\sqrt{5}\sqrt{6}\sqrt{15}\sqrt{21 } H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {98\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{9 \sqrt{5}\sqrt{10}\sqrt{21} H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {98\sqrt{105 }\pi ^{{{3}\over {2}}}}}-{{3\sqrt{3}\sqrt{5}\sqrt{6} H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {245\pi ^{{{3}\over {2}}}}} \\&+{{54\sqrt{2}\sqrt{5} H_1^0 H_1^1 H_2^{-2} H_2^1 }\over {245\pi ^{{{3}\over {2}}}}}-{{3\sqrt{3} \sqrt{6}\sqrt{10}\sqrt{15}H_0^0 (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {784\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{6} \sqrt{10}H_0^0 (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {392\pi ^{{{5 }\over {2}}}}}+{{351\sqrt{2}\sqrt{6}H_0^0 (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {392\pi ^{{{5}\over {2}}}}} \\&-{{81\sqrt{3} H_0^0 (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {56\pi ^{{{5}\over {2}} }}}+{{\sqrt{3}\sqrt{6}\sqrt{10}\sqrt{15} (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {784\pi ^2}} \\&+{{3\sqrt{5}\sqrt{6}\sqrt{10} (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {392\pi ^2}}-{{117\sqrt{2}\sqrt{6} (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {392\pi ^2}}\\&+{{27\sqrt{3} (H_1^{-1})^2 H_1^0 H_1^1 H_2^1 }\over {56\pi ^2}}-{{9\sqrt{2}\sqrt{10}\sqrt{15}H_0^0 H_1^{-1} (H_1^0)^3 H_2^1 }\over {392\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{2} \sqrt{6}H_0^0 H_1^{-1} (H_1^0)^3 H_2^1 }\over {56\pi ^{{{5}\over {2}}}}}-{{81 \sqrt{3}H_0^0 H_1^{-1} (H_1^0)^3 H_2^1 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{3 \sqrt{2}\sqrt{10}\sqrt{15} H_1^{-1} (H_1^0)^3 H_2^1 }\over {392\pi ^2}}+{{9\sqrt{2} \sqrt{6} H_1^{-1} (H_1^0)^3 H_2^1 }\over {56\pi ^2}} \\&+{{27\sqrt{3} H_1^{-1} (H_1^0)^3 H_2^1 }\over {196\pi ^2}}-{{9 \sqrt{2}\sqrt{6} (H_0^0)^3 H_1^{-1} H_1^0 H_2^1 }\over {28\pi ^{{{5}\over {2}} }}} \\&+{{3\sqrt{3} (H_0^0)^3 H_1^{-1} H_1^0 H_2^1 }\over {28\pi ^{{{5}\over {2}} }}}+{{9\sqrt{2}\sqrt{6}(H_0^0)^2 H_1^{-1} H_1^0 H_2^1 }\over {28\pi ^2}} \\&-{{3\sqrt{3}(H_0^0)^2 H_1^{-1} H_1^0 H_2^1 }\over {28\pi ^2}}-{{9\sqrt{2} \sqrt{6}H_0^0 H_1^{-1} H_1^0 H_2^1 }\over {35\pi ^{{{3}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&+{{3 \sqrt{3}H_0^0 H_1^{-1} H_1^0 H_2^1 }\over {35\pi ^{{{3}\over {2}}}}}+{{9 \sqrt{2}\sqrt{6} H_1^{-1} H_1^0 H_2^1 }\over {70\pi }} \\&-{{3\sqrt{3} H_1^{-1} H_1^0 H_2^1 }\over {70\pi }}+{{75 (H_0^0)^3 (H_3^0)^3 }\over {56\pi ^{{{5 }\over {2}}}}} \\&-{{75(H_0^0)^2 (H_3^0)^3 }\over {56\pi ^2}}+{{15H_0^0 (H_3^0)^3 }\over {14\pi ^{{{3}\over {2}}}}} \\&-{{15 (H_3^0)^3 }\over {28 \pi }}+{{405\sqrt{21}(H_0^0)^2 H_1^{-1} H_1^1 (H_2^0)^2 }\over {98\sqrt{105 }\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^1 (H_2^0)^2 }\over {392\pi ^{{{5}\over {2}}}}}-{{135\sqrt{21} H_0^0 H_1^{-1} H_1^1 (H_2^0)^2 }\over {49\sqrt{105}\pi ^2}} \\&+{{3\sqrt{5} H_0^0 H_1^{-1} H_1^1 (H_2^0)^2 }\over {196\pi ^2}}+{{54\sqrt{21} H_1^{-1} H_1^1 (H_2^0)^2 }\over {49\sqrt{105}\pi ^{{{3}\over {2}}}}} \\&-{{3\sqrt{5} H_1^{-1} H_1^1 (H_2^0)^2 }\over {490\pi ^{{{3}\over {2}}}}}+{{405 \sqrt{21}(H_0^0)^2 (H_1^0)^2 (H_2^0)^2 }\over {98\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&+{{117 \sqrt{5}(H_0^0)^2 (H_1^0)^2 (H_2^0)^2 }\over {49\pi ^{{{5}\over {2}}}}}-{{135\sqrt{21}H_0^0 (H_1^0)^2 (H_2^0)^2 }\over {49\sqrt{105}\pi ^2}} \\&-{{78\sqrt{5}H_0^0 (H_1^0)^2 (H_2^0)^2 }\over {49\pi ^2}}+{{54 \sqrt{21} (H_1^0)^2 (H_2^0)^2 }\over {49 \sqrt{105}\pi ^{{{3}\over {2}}}}} +{{156\sqrt{5} (H_1^0)^2 (H_2^0)^2 }\over {245\pi ^{{{3}\over {2}}}}}\\&+{{15 \sqrt{5} (H_0^0)^4 (H_2^0)^2 }\over {112 \pi ^{{{5}\over {2}}}}} -{{5\sqrt{5} (H_0^0)^3 (H_2^0)^2 }\over {28\pi ^2}}+{{3\sqrt{5} (H_0^0)^2 (H_2^0)^2 }\over {14\pi ^{{{3}\over {2}}}}} \\&-{{3 \sqrt{5}H_0^0 (H_2^0)^2 }\over {14\pi }}-{{45 \sqrt{6}\sqrt{15}\sqrt{21}\sqrt{30}(H_0^0)^2 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}}\\&+{{\sqrt{5} (H_2^0)^2 }\over {7\sqrt{\pi }}}-{{135 \sqrt{10}\sqrt{21}\sqrt{30}(H_0^0)^2 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{2} \sqrt{5}\sqrt{6}(H_0^0)^2 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {196 \pi ^{{{5}\over {2}}}}} \end{aligned}$$

$$\begin{aligned}&+{{9\sqrt{3}\sqrt{5} (H_0^0)^2 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{15 \sqrt{6}\sqrt{15}\sqrt{21}\sqrt{30}H_0^0 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {196\sqrt{105}\pi ^2}}\\&+{{45\sqrt{10} \sqrt{21}\sqrt{30}H_0^0 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {196 \sqrt{105}\pi ^2}} \\&+{{9\sqrt{2}\sqrt{5}\sqrt{6} H_0^0 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {98\pi ^2}}-{{3\sqrt{3}\sqrt{5 }H_0^0 H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {98\pi ^2}} \\&-{{3 \sqrt{6}\sqrt{15}\sqrt{21}\sqrt{30} H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {98 \sqrt{105}\pi ^{{{3}\over {2}}}}}\\&-{{9\sqrt{10}\sqrt{21} \sqrt{30} H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {98\sqrt{105}\pi ^{{{3}\over {2 }}}}} \\&-{{9\sqrt{2}\sqrt{5}\sqrt{6} H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {245 \pi ^{{{3}\over {2}}}}}+{{3\sqrt{3}\sqrt{5} H_1^0 H_1^1 H_2^{-1} H_2^0 }\over {245\pi ^{{{3}\over {2}}}}} \\&+{{675\sqrt{6}\sqrt{21} (H_0^0)^2 (H_1^1)^2 H_2^{-2} H_2^0 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}}+{{45 \sqrt{5}\sqrt{6}(H_0^0)^2 (H_1^1)^2 H_2^{-2} H_2^0 }\over {98\pi ^{{{5}\over {2 }}}}} \\&-{{225\sqrt{6}\sqrt{21}H_0^0 (H_1^1)^2 H_2^{-2} H_2^0 }\over {196\sqrt{105 }\pi ^2}}-{{15\sqrt{5}\sqrt{6}H_0^0 (H_1^1)^2 H_2^{-2} H_2^0 }\over {49\pi ^2}} \\&+{{45\sqrt{6}\sqrt{21} (H_1^1)^2 H_2^{-2} H_2^0 }\over {98\sqrt{105}\pi ^{{{3}\over {2 }}}}}+{{6\sqrt{5}\sqrt{6} (H_1^1)^2 H_2^{-2} H_2^0 }\over {49\pi ^{{{3}\over {2}}}}} \\&+{{135 H_0^0 (H_1^{-1})^2 (H_1^1)^2 H_2^0 }\over {28\pi ^{{{5}\over {2}}}}}-{{45 (H_1^{-1})^2 (H_1^1)^2 H_2^0 }\over {28 \pi ^2}} \\&-{{9\sqrt{6}\sqrt{10}\sqrt{15}H_0^0 H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {196\pi ^{{{5}\over {2}}}}}\\&+{{9\sqrt{2} \sqrt{3}\sqrt{6}H_0^0 H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {245 \pi ^{{{5}\over {2}}}}} \\&-{{513H_0^0 H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {140\pi ^{{{5}\over {2}}}}}+{{3\sqrt{6}\sqrt{10} \sqrt{15} H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {196\pi ^2}} \\&-{{3\sqrt{2} \sqrt{3}\sqrt{6} H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {245\pi ^2}}+{{171 H_1^{-1} (H_1^0)^2 H_1^1 H_2^0 }\over {140\pi ^2}} \\&-{{75 (H_0^0)^3 H_1^{-1} H_1^1 H_2^0 }\over {28 \pi ^{{{5}\over {2}}}}}+{{75(H_0^0)^2 H_1^{-1} H_1^1 H_2^0 }\over {28\pi ^2}}\\&-{{15H_0^0 H_1^{-1} H_1^1 H_2^0 }\over {7\pi ^{{{3}\over {2}}}}}+{{15 H_1^{-1} H_1^1 H_2^0 }\over {14\pi }} +{{405H_0^0 (H_1^0)^4 H_2^0 }\over {112\pi ^{{{5}\over {2}}}}}\\&-{{135 (H_1^0)^4 H_2^0 }\over {112\pi ^2}}+{{165 (H_0^0)^3 (H_1^0)^2 H_2^0 }\over {56\pi ^{ {{5}\over {2}}}}}-{{165(H_0^0)^2 (H_1^0)^2 H_2^0 }\over {56\pi ^2}} \end{aligned}$$

$$\begin{aligned}&+{{33H_0^0 (H_1^0)^2 H_2^0 }\over {14\pi ^{{{3}\over {2 }}}}}-{{33 (H_1^0)^2 H_2^0 }\over {28\pi }} +{{3 (H_0^0)^5 H_2^0 }\over {16\pi ^{{{5 }\over {2}}}}}\\&-{{5 (H_0^0)^4 H_2^0 }\over {16 \pi ^2}}+{{ (H_0^0)^3 H_2^0 }\over {2\pi ^{{{3}\over {2}}}}}-{{3(H_0^0)^2 H_2^0 }\over {4\pi }} \\&+{{H_0^0 H_2^0 }\over { \sqrt{\pi }}}+{{45\sqrt{6}\sqrt{10}\sqrt{15}\sqrt{21} (H_0^0)^2 (H_1^1)^2 (H_2^{-1})^2 }\over {392\sqrt{105}\pi ^{{{5}\over {2}}}}} \\&-{{135\sqrt{5} (H_0^0)^2 (H_1^1)^2 (H_2^{-1})^2 }\over {196\pi ^{{{5}\over {2}}}}}-{{15\sqrt{6}\sqrt{10}\sqrt{15 }\sqrt{21}H_0^0 (H_1^1)^2 (H_2^{-1})^2 }\over {196\sqrt{105}\pi ^2}} \\&+{{45\sqrt{5} H_0^0 (H_1^1)^2 (H_2^{-1})^2 }\over {98 \pi ^2}}+{{3\sqrt{6}\sqrt{10}\sqrt{15}\sqrt{21} (H_1^1)^2 (H_2^{-1})^2 }\over {98\sqrt{105}\pi ^{{{3 }\over {2}}}}} \\&-{{9\sqrt{5} (H_1^1)^2 (H_2^{-1})^2 }\over {49\pi ^{{{3}\over {2}}}}}-{{3\sqrt{3}\sqrt{6} \sqrt{10}\sqrt{15}H_0^0 H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {784\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{5}\sqrt{6}\sqrt{10} H_0^0 H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {392\pi ^{{{5}\over {2 }}}}}\\&+{{351\sqrt{2}\sqrt{6}H_0^0 H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {392\pi ^{{{5}\over {2}}}}} \\&-{{81\sqrt{3} H_0^0 H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {56\pi ^{{{5}\over {2}}}}}+{{ \sqrt{3}\sqrt{6}\sqrt{10}\sqrt{15} H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {784\pi ^2}} \\&+{{3\sqrt{5}\sqrt{6}\sqrt{10} H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {392\pi ^2}}-{{117\sqrt{2}\sqrt{6} H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {392\pi ^2}} \\&+{{27\sqrt{3} H_1^{-1} H_1^0 (H_1^1)^2 H_2^{-1} }\over {56\pi ^2}}-{{9\sqrt{2}\sqrt{10}\sqrt{15} H_0^0 (H_1^0)^3 H_1^1 H_2^{-1} }\over {392\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{2}\sqrt{6}H_0^0 (H_1^0)^3 H_1^1 H_2^{-1} }\over {56\pi ^{{{5}\over {2}}}}}-{{81\sqrt{3} H_0^0 (H_1^0)^3 H_1^1 H_2^{-1} }\over {196\pi ^{{{5}\over {2}}}}} \\&+{{3\sqrt{2} \sqrt{10}\sqrt{15} (H_1^0)^3 H_1^1 H_2^{-1} }\over {392\pi ^2}}+{{9\sqrt{2}\sqrt{6} (H_1^0)^3 H_1^1 H_2^{-1} }\over {56 \pi ^2}}\\&+{{27\sqrt{3} (H_1^0)^3 H_1^1 H_2^{-1} }\over {196\pi ^2}}-{{9\sqrt{2}\sqrt{6} (H_0^0)^3 H_1^0 H_1^1 H_2^{-1} }\over {28\pi ^{{{5}\over {2}}}}} \\&+{{3\sqrt{3} (H_0^0)^3 H_1^0 H_1^1 H_2^{-1} }\over {28\pi ^{{{5}\over {2}}}}}+{{9\sqrt{2} \sqrt{6}(H_0^0)^2 H_1^0 H_1^1 H_2^{-1} }\over {28\pi ^2}} \\&-{{3\sqrt{3} (H_0^0)^2 H_1^0 H_1^1 H_2^{-1} }\over {28\pi ^2}}-{{9\sqrt{2}\sqrt{6}H_0^0 H_1^0 H_1^1 H_2^{-1} }\over {35\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{3}H_0^0 H_1^0 H_1^1 H_2^{-1} }\over {35\pi ^{{{3}\over {2}}}}}+{{9\sqrt{2} \sqrt{6} H_1^0 H_1^1 H_2^{-1} }\over {70\pi }} \end{aligned}$$

$$\begin{aligned}&-{{3\sqrt{3} H_1^0 H_1^1 H_2^{-1} }\over {70\pi }}+{{45\sqrt{6}H_0^0 H_1^{-1} (H_1^1)^3 H_2^{-2} }\over {28\pi ^{{{5}\over {2}}}}} \\&-{{15\sqrt{6} H_1^{-1} (H_1^1)^3 H_2^{-2} }\over {28\pi ^2}}-{{9\sqrt{10}\sqrt{15}H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {784\pi ^{{{5}\over {2}}}}} \\&+{{3\sqrt{3}\sqrt{5}\sqrt{6} \sqrt{15}H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {784\pi ^{{{5}\over {2}}}}}\\&+{{27\sqrt{2}\sqrt{5}\sqrt{6}\sqrt{10}H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {1960\pi ^{{{5}\over {2}}}}} \\&-{{9\sqrt{3}\sqrt{5}\sqrt{10 }H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {560\pi ^{{{5}\over {2}}}}}-{{1017 \sqrt{6}H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {3920\pi ^{{{5}\over {2}}}}} \\&+{{99 \sqrt{2}\sqrt{3}H_0^0 (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {490\pi ^{{{5}\over {2 }}}}}+{{3\sqrt{10}\sqrt{15} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {784\pi ^2}} \\&-{{\sqrt{3} \sqrt{5}\sqrt{6}\sqrt{15} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {784\pi ^2}}-{{9\sqrt{2} \sqrt{5}\sqrt{6}\sqrt{10} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {1960\pi ^2}} \\&+{{3\sqrt{3} \sqrt{5}\sqrt{10} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {560\pi ^2}}+{{339\sqrt{6} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {3920 \pi ^2}} \\&-{{33\sqrt{2}\sqrt{3} (H_1^0)^2 (H_1^1)^2 H_2^{-2} }\over {490\pi ^2}}-{{15\sqrt{6} (H_0^0)^3 (H_1^1)^2 H_2^{-2} }\over {28\pi ^{{{5}\over {2}}}}} \\&+{{15\sqrt{6} (H_0^0)^2 (H_1^1)^2 H_2^{-2} }\over {28\pi ^2 }}-{{3\sqrt{6}H_0^0 (H_1^1)^2 H_2^{-2} }\over {7\pi ^{{{3}\over {2}}}}} \\&+{{3\sqrt{6} (H_1^1)^2 H_2^{-2} }\over {14\pi }}+{{9\sqrt{5 } (H_1^{-1})^3 (H_1^1)^3 }\over {28\pi ^{{{5 }\over {2}}}}} \\&-{{3\sqrt{5}\sqrt{6}\sqrt{10}\sqrt{15} (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {2800\pi ^{{{5}\over {2}}}}}\\&+{{3\sqrt{3}\sqrt{15} (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {1960\pi ^{{{5}\over {2}}}}} \\&-{{81\sqrt{3}\sqrt{6}\sqrt{10 } (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {19600\pi ^{{{5}\over {2}}}}}\\&+{{27\sqrt{2}\sqrt{10} (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {4900\pi ^{{{5}\over {2}}}}}\\&+{{873\sqrt{2}\sqrt{3} \sqrt{5}\sqrt{6} (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {24500\pi ^{{{5}\over {2}}}}}\\&-{{8451 \sqrt{5} (H_1^{-1})^2 (H_1^0)^2 (H_1^1)^2 }\over {49000\pi ^{{{5}\over {2}}}}} \\&-{{27\sqrt{5} (H_0^0)^2 (H_1^{-1})^2 (H_1^1)^2 }\over {28\pi ^{{{5}\over {2}}}}}+{{9\sqrt{5}H_0^0 (H_1^{-1})^2 (H_1^1)^2 }\over {14\pi ^2}} \end{aligned}$$

$$\begin{aligned}&-{{9 \sqrt{5} (H_1^{-1})^2 (H_1^1)^2 }\over {35 \pi ^{{{3}\over {2}}}}}-{{3\sqrt{5}\sqrt{6}\sqrt{10}\sqrt{15} H_1^{-1} (H_1^0)^4 H_1^1 }\over {4900\pi ^{{{5}\over {2}}}}} \nonumber \\&-{{3\sqrt{2}\sqrt{3}\sqrt{5} \sqrt{10}\sqrt{15} H_1^{-1} (H_1^0)^4 H_1^1 }\over {4900\pi ^{{{5}\over {2}}}}}\nonumber \\&-{{102 \sqrt{2}\sqrt{3}\sqrt{5}\sqrt{6} H_1^{-1} (H_1^0)^4 H_1^1 }\over {6125\pi ^{{{5}\over {2 }}}}} \nonumber \\&-{{10233\sqrt{5} H_1^{-1} (H_1^0)^4 H_1^1 }\over {98000\pi ^{{{5}\over {2}}}}}-{{9\sqrt{2 }\sqrt{3}\sqrt{5}\sqrt{6}(H_0^0)^2 H_1^{-1} (H_1^0)^2 H_1^1 }\over {140\pi ^{ {{5}\over {2}}}}} \nonumber \\&-{{27\sqrt{5}(H_0^0)^2 H_1^{-1} (H_1^0)^2 H_1^1 }\over {280\pi ^{ {{5}\over {2}}}}}+{{3\sqrt{2}\sqrt{3}\sqrt{5}\sqrt{6}H_0^0 H_1^{-1} (H_1^0)^2 H_1^1 }\over {70\pi ^2}} \nonumber \\&+{{9\sqrt{5} H_0^0 H_1^{-1} (H_1^0)^2 H_1^1 }\over {140\pi ^2}}-{{3\sqrt{2}\sqrt{3}\sqrt{5}\sqrt{6 } H_1^{-1} (H_1^0)^2 H_1^1 }\over {175\pi ^{{{3}\over {2}}}}} \nonumber \\&-{{9\sqrt{5} H_1^{-1} (H_1^0)^2 H_1^1 }\over {350\pi ^{ {{3}\over {2}}}}}+{{3\sqrt{5} (H_0^0)^4 H_1^{-1} H_1^1 }\over {16\pi ^{{{5}\over {2}}}}} -{{ \sqrt{5} (H_0^0)^3 H_1^{-1} H_1^1 }\over {4\pi ^2}}\nonumber \\&+{{3\sqrt{5}(H_0^0)^2 H_1^{-1} H_1^1 }\over {10\pi ^{{{3}\over {2 }}}}} -{{3\sqrt{5}H_0^0 H_1^{-1} H_1^1 }\over {10\pi }}+{{\sqrt{5} H_1^{-1} H_1^1 }\over {5\sqrt{\pi }}} \nonumber \\&+{{9\sqrt{5} (H_1^0)^6 }\over {112\pi ^{{{5}\over {2}}}}}+{{27\sqrt{5} (H_0^0)^2 (H_1^0)^4 }\over {56\pi ^{{{5}\over {2 }}}}} -{{9\sqrt{5}H_0^0 (H_1^0)^4 }\over {28\pi ^2}}+{{9\sqrt{5} (H_1^0)^4 }\over {70 \pi ^{{{3}\over {2}}}}} \nonumber \\&+{{3\sqrt{5} (H_0^0)^4 (H_1^0)^2 }\over {16\pi ^{{{5}\over {2}}}}}-{{\sqrt{5} (H_0^0)^3 (H_1^0)^2 }\over {4\pi ^2}} +{{3 \sqrt{5}(H_0^0)^2 (H_1^0)^2 }\over {10\pi ^{{{3}\over {2}}}}}\nonumber \\&-{{3\sqrt{5}H_0^0 (H_1^0)^2 }\over {10\pi }} +{{\sqrt{5} (H_1^0)^2 }\over {5 \sqrt{\pi }}} \end{aligned}$$

(F.8)