## 3 Numerical Verification of Solutions
### Analysis of Corrected Afanasiev Solutions (Difference of One Order)  
The completed simulation tests in which the improved Afanasiev formulas (Equations (A4)-(A9)) were analyzed are shown in Figure 4. The tests were carried out for three assumed values of the orders of the Bessel function \(p=1\), \(p=3\) and \(p=5\). The research for different \(p\) aimed to capture the trends in the obtained results.  
From the analysis of the series described by Equation (A4) for \(l=1\), it follows that the solution to an infinite series of this type is always equal to zero, regardless of which Bessel root is subtracted (\(\dot{j}_{p+1,n}^{2}\)) and what order of the Bessel function \(p\) was taken into account, the result always remains identical. This situation changes when the difference from the zeros of the Bessel function is raised to the second power, i.e., when \(l=2\) Equation (A5). It can be seen here that when the first zero values of \(\dot{j}_{p+1,1}^{2}\) are subtracted, the results obtained take maximum values, with the maximum value recorded for the lowest \(p\) values (case when \(p=0\), not shown graphically). The larger the value of \(p\), the lower the initial values of this analyzed function and the greater the flattening of the course toward the asymptotic value, i.e., \(0\). When \(l=3\) in Equation (A6), it can be seen that this function starts with negative values and gradually increases with the subsequent subtraction of zeros (as \(n\) increases) until it reaches an asymptotic zero value (as in the previously analyzed case of Equation (A5)). This case also shows that for \(l=3\), a few steps (\(n\approx 10\)) are enough for this function to reach values that are close to asymptotic. The above means that significant results changes are visible only for low values of \(n\), while higher values of \(n\) results are very similar.  
An analysis of the solutions collected in the right column of Table A1 (Equations (A7)-(A9)), i.e., those in which the differences are in the denominator of the series, and the zeros of the Bessel function of one order lower (\(j_{p-1,n}^{2}\)) are subtracted, showing that the following:  
* When \(l=1\), the maximum values for \(n=1\) are obtained at the lowest \(p\) value; in this case, \(p=1\) (in the analysis it was not possible to assume a lower value, i.e., \(p=0\), because then the zeros of the function \(j_{-1,n}^{2}\) would be subtracted, and this  
Figure 4: Comparison of selected results of Afanasiev series (serie) with the new polynomial solution (newSol)—difference of one order.  
is what we wanted to avoid). The obtained results show that for lower values of \(p\), the dynamics of the decrease in the function value with increasing values of \(n\) were greater, as evidenced by the fact that the results for \(n=2,\ 3,\ 4\ldots\) were lower when the difference from the zeros of the higher orders of the Bessel function were taken into account;
* Situations with changing dynamics were not noticed when \(l=2\) and Equation (13) were analyzed. Here, the shape of the function course and the values remained at a level very similar to those obtained from Equation (14);
* An interesting behavior was shown by the analysis of Equation (15) obtained with power \(l=3\). It can be seen from the graph that for \(p=1\), the values of this function for \(n=2,\ 3,\ 4\ldots\) were negative, while a similar behavior was not observed for larger values of \(p\geq 3\) (values only positive). The zoom of Figure 4 for \(n\geq 10\) is presented in Appendix D.  
To sum up, it should be noted that from all graphical comparisons presented (Figure 4), one can clearly see the complete perfect agreement of the proposed polynomial solution, which describes the infinite series based on these differences from the zeros of the Bessel function.