## 2 A General Solution
### Solutions for Series Composed of Terms \(\left(j_{p,k}^{2}-j_{p+r,n}^{2}\right)^{-l}\)  
#### 2.1.1 Solutions for \(l=1\)  
The ratio (quotient) of Bessel functions in Equation (6) has been studied by mathematicians at least since the middle of the nineteenth century; as an example, please refer to the [PERSON] book entitled: \"Studien uber die Bessel'schen Functionen\" [20], second paragraph, p. 4. In the paper related to Bessel functions, [PERSON] derived an important relation between Bessel functions and special polynomials \(R_{a,b}(z)\)[21], p. 109:  
\[J_{m+\
u}(z)=J_{\
u}(z)\cdot R_{m,\
u}(z)-J_{\
u-1}(z)\cdot R_{m-1,\
u+1}(z) \tag{8}\]  
These polynomials are defined today as Lommel polynomials, and can be explicitly defined in at least three ways:  
(a) The explicit form (based on factorial and gamma functions);  
\[R_{m,\
u}(z)=\sum_{n=0}^{\lfloor m/2\rfloor}\left(-1\right)^{n}\frac{(m-n)! \Gamma(\
u+m-n)}{n!(m-2n)!\Gamma(\
u+n)}\left(\frac{z}{2}\right)^{2n-m} \tag{9}\]2. The hypergeometric representation [22]; \[R_{m,\
u}(z)=\frac{\Gamma(\
u+m)}{\Gamma(\
u)(z/2)^{m}}\ {}_{2}F_{3}\left(\frac{1}{2}(1-m),- \frac{1}{2}m;\
u,-m,1-\
u-m;\ z^{2}\right)\] (10) 3. The third possibility is expressing these polynomials with the help of the Bessel functions [21], p. 109: \[R_{m,\
u}(z)=\frac{\pi z}{2\text{sin}(\
u\pi)}\big{[}J_{\
u+m}(z)J_{-\
u+1}(z)+ (-1)^{m}J_{-\
u-m}(z)J_{\
u-1}(z)\big{]}\] (11)  
The Lommel polynomial function can be calculated using a recursive formula derived by [PERSON] [21], p. 114:  
\[\frac{2(m+\
u)}{z}R_{m,\
u}(z)=R_{m+1,\
u}(z)+R_{m-1,\
u}(z) \tag{12}\]  
If in Equation (8) one assumes \(\
u=p+1\), then:  
\[J_{m+p+1}(z)=J_{p+1}(z)\cdot R_{m,p+1}(z)-J_{p}(z)\cdot R_{m-1,p+2}(z) \tag{13}\]  
Rearranging this Equation gives the following:  
\[\frac{1}{R_{m,p+1}(z)}\frac{J_{m+p+1}(z)}{J_{p}(z)}+\frac{R_{m-1,p+2}(z)}{R_{m,p+1}(z)}=\frac{J_{p+1}(z)}{J_{p}(z)} \tag{14}\]  
If the above solution is used on the l.h.s of Equation (6), one obtains the following:  
\[\frac{1}{R_{m,p+1}(z)}\frac{J_{m+p+1}(z)}{J_{p}(z)}+\frac{R_{m-1,p+2}(z)}{R_{m,p+1}(z)}=2z\sum_{k=1}^{\infty}\frac{1}{j_{p,k}^{2}-z^{2}} \tag{15}\]  
The rational solutions presented in the left columns of Appendix A are valid for the first power of the difference of Bessel zeros in the denominator (A4, A10, A16, A22), which can be archived with the help of the derived Equation (15). Inserting \(z=j_{m+p+1,r}\) where \(r=1\), \(2\), \(\ldots\); \(m=0\), \(1\), \(2\), \(\ldots\) and \(j_{m+p+1,r}\) are the roots of the standard Bessel function of order \(\text{m}+\text{p}+1\)\(J_{m+p+1}\big{(}j_{m+p+1,r}\big{)}=0\) deduces the final Equation:  
\[\boxed{\frac{R_{m-1,p+2}\big{(}j_{m+p+1,r}\big{)}}{2j_{m+p+1,r}\cdot R_{m,p+1 }\big{(}j_{m+p+1,r}\big{)}}=\sum_{k=1}^{\infty}\frac{1}{j_{p,k}^{2}-j_{m+p+1,r }^{2}}} \tag{16}\]  
where \(R_{a,b}(z)\) are Lommel polynomials [19], p. 120.  
To receive the final simplified functional solution tabulated in Appendix A, one needs to insert proper Lommel polynomials in the above solution, Equation (16). The first few Lommel polynomials are presented in Appendix B. According to Equation (16) and in the Lommel polynomials shown in Appendix B, such substitutions as \(\
u=p+2\) and \(\
u=p+1\) are needed. Examples of calculations for the selected final results (Equations (A4), (A10), (A16) and (A22)) can be found in Appendix C.