## 2 A General Solution
### Solutions for Series Composed of Terms \(\left(j_{p,k}^{2}-j_{p+r,n}^{2}\right)^{-l}\)
#### 2.2.2 Solutions for \(l\geq 2\)  
The derivative of l.h.s. of Equation (30) provides the following:  
\[\frac{\partial}{\partial z}\bigg{(}\frac{p}{z^{2}}-\frac{J_{p-1}(z)}{2 zJ_{p}( z)}\bigg{)}=-\frac{2p}{z^{3}}+\frac{1}{2z}\bigg{(}\frac{J_{p-1}(z)}{J_{p}(z)} \bigg{)}^{2}-\frac{(p-1)}{z^{2}}\bigg{(}\frac{J_{p-1}(z)}{J_{p}(z)}\bigg{)}+ \frac{1}{2z} \tag{36}\]  
Knowing that the derivative of the infinite series on the right side of Equation (30) and Equation (6) gives the result presented by Formula (20), the following final formula is obtained after ordering:  
\[{\sum_{k=1}^{\infty}}\frac{1}{\big{(}j_{p,k}^{2}-z^{2}\big{)}^{2}}=-\frac{p}{ z^{4}}+\frac{1}{4z^{2}}\bigg{(}\frac{J_{p-1}(z)}{J_{p}(z)}\bigg{)}^{2}-\frac{(p-1)}{2 z^{3}}\bigg{(}\frac{J_{p-1}(z)}{J_{p}(z)}\bigg{)}+\frac{1}{4z^{2}} \tag{37}\]  
Upon comparing the solution presented in Equation (37) with the one previously derived and described in Equation (21), a profound similarity becomes apparent. Apart from the first term from the right, which occurs in this solution, the others are analogous to the solution (21) if \(p+1\) is replaced by \(p-1\).  
With the help of Equation (33), the solution (37) can be written in the following form:  
\[\sum_{k=1}^{\infty}\frac{1}{\big{(}j_{p,k}^{2}-j_{p-m-1}^{2}\big{)}^{2}}=- \frac{p}{z^{4}}+\frac{1}{4z^{2}}\Big{(}\big{(}-1\big{)}^{m}\frac{I_{p-m-1}(z)} {I_{p}(z)}\frac{1}{R_{m,\ 1-p}(z)}-\frac{R_{m-1,\ 2-p}(z)}{R_{m,1-p}(z)}\Big{)}^{2}-\frac{(p-1)}{2 z^{3}}\Big{(}\big{(}-1\big{)}^{m}\frac{I_{p-m-1}(z)}{I_{p}(z)}\frac{1}{R_{m,\ 1-p}(z)}-\frac{R_{m-1,\ 2-p}(z)}{R_{m,1-p}(z)}\Big{)}+\frac{1}{4z^{2}} \tag{38}\]  
Substituting \(z=j_{p-m-1,r}\), i.e., the roots (zeros) of the function \(J_{p-m-1}\big{(}j_{p-m-1,r}\big{)}\), it reduces Equation (38) to the following final form:  
\[\boxed{\frac{\sum_{k=1}^{\infty}}{\sum_{\left(j_{p,k}^{2}-j_{p-m-1,r}^{2} \right)^{2}}}{2j_{p-m-1,r}^{2}}=-\frac{p}{j_{p-m-1,r}^{2}}+\frac{1}{4j_{p-m-1,r }^{2}}\bigg{(}-\frac{R_{m-1,\ 2-p}\big{(}j_{p-m-1,r}\big{)}}{R_{m,1-p}\big{(}j_{p-m-1,r}\big{)}} \bigg{)}^{2}+} \tag{39}\]  
Using the above Equation (39), the solutions of Equations (A8), (A14), (A20), (A25), etc., can be found.  
The next derivative of the function on the r.h.s. in Equation (37) can be written after ordering as follows:\[\frac{\frac{3}{24}}\left(-\frac{p}{2}+\frac{1}{4^{2}}\left(\frac{\left(p-1(z)\right) }{\left(p(z)\right)}\right)^{2}-\frac{\left(p-1\right)}{2z^{3}}\left(\frac{ \left(p-1(z)\right)}{\left(p(z)\right)}\right)+\frac{1}{4^{2}}\right)=\frac{4p }{5}+\frac{\left(p-2\right)}{2z^{3}}-\frac{\left[z^{2}+2\left(p-1\right)\left(p -2\right)\right]}{2z^{4}}\frac{\left(p-1(z)\right)}{\left(p(z)\right)}+\frac{3 \left(p-1\right)}{2z^{3}}\left(\frac{\left(p-1(z)\right)}{\left(p(z)\right)} \right)^{2}-\frac{1}{2z^{3}}\left(\frac{\left(p-1(z)\right)}{\left(p(z)\right) }\right)^{3} \tag{40}\]  
Then:  
\[\sum_{k=1}^{\infty}\frac{1}{\left(\dot{p}_{p,k}^{2}-z^{2}\right)^{3}}=-\frac{1 }{8z^{4}}\left(\frac{\left(J_{p-1}(z)\right)}{\left(J_{p}(z)\right)}^{3}+ \frac{3(p-1)}{8z^{4}}\left(\frac{\left(J_{p-1}(z)\right)}{\left(J_{p}(z) \right)}\right)^{2}-\frac{\left[z^{2}+2\left(p-1\right)\left(p-2\right)\right] }{8z^{5}}\frac{\left(J_{p-1}(z)\right)}{\left(J_{p}(z)\right)}+\frac{\left(p-2 \right)}{8z^{4}}+\frac{p}{z^{6}} \tag{41}\]  
and the final form based on the Lommel polynomials (when \(z=j_{p-m-1,r}\)) is as follows:  
\[\boxed{\sum_{k=1}^{\infty}\frac{1}{\left(\dot{p}_{p,k}^{2}-\dot{p}_{p-m-1,r}^{ 2}\right)^{3}}=\frac{1}{8\dot{p}_{p-m-1,r}^{4}}\left(\frac{\left(R_{m-1,\,2-p }\left(j_{p-m-1,r}\right)}{R_{m,1-p}\left(j_{p-m-1,r}\right)}\right)^{3}+ \frac{3(p-1)}{8\dot{p}_{p-m-1,r}^{4}}\left(-\frac{R_{m-1,\,2-p}\left(j_{p-m-1, r}\right)}{R_{m,1-p}\left(j_{p-m-1,r}\right)}\right)^{2}+} \tag{42}\]  
With the help of Equation (42), the solutions defined by Equations (A9), (A15), (A21), etc., can be found.  
By determining successive derivatives (from Equation (41)), solutions for higher powers of \(l\) (higher than \(l=2\) and \(l=3\)) can be obtained in a similar way.