## 2 A General Solution  
The Bessel functions (Figure 1) look at first glance to be strictly related to trigonometrical functions like \(\sin(z)\) and \(\cos(z)\). The main difference between them is that Bessel functions are a damped function, which means that the maximum and minimum values on the subsequent amplitudes of these functions for arguments \(z\geq 0\) always decrease. The zeros of these functions are the points when this function takes the zero value. From Figure 1, it is clear that for functions \(J_{0}(z)\) and \(J_{1}(z)\) the first zeros have values less than five (excluding situations where \(z=0\) and all standard Bessel functions with orders higher than 0 have the starting point), while for all others, \(J_{2}(z)\), \(J_{3}(z)\), \(J_{4}(z)\), \(\ldots\), this first zero has a value larger than five. The greater the value of the order of the Bessel function, the greater the value of the first zeros. Subsequent zeros never have the same values for different orders of the Bessel function. The \(p\)-th order Bessel function \(J_{p}(z)\) has the following approximate behavior for large real values of \(z\):  
\[J_{p}(z)\approx\sqrt{\frac{2}{\pi z}}\text{cos}\left(z-\frac{(2p+1)\pi}{4}\right) \tag{5}\]  
It follows that \(J_{p}(z)\) has infinitely many positive zeros whose values tend to infinity. The main interest in this paper is placed not on the Bessel function itself but on a function composed as the ratio of two Bessel functions when, in the numerator, there is a function of order that is one higher than the denominator \(J_{p+1}(z)/J_{p}(z)\). Such a ratio (quotient) has been analyzed by many mathematicians in the past who found that it can be presented in the form of a continued fraction and is useful in many practical applied science theoretical problems. This ratio, according to the oldest reference [19], p. 147, has an infinite series representation. \[\frac{J_{p+1}(z)}{J_{p}(z)}=2z\overset{\infty}{\underset{k=1}{\sum}}\frac{1}{j _{p,k}^{2}-z^{2}} \tag{6}\]  
Figure 1: Comparison of selected standard Bessel functions of the few first orders (\(p=0,1,2,3\)). As can be seen from Figure 2 that the ratio of these Bessel functions has similar geometric properties to the commonly known function, namely \(\tan(z)\) (here, it is worth noting that this ratio takes the form of the just mentioned \(\tan(z)\) for a special case when the half-integer orders are assumed \(J_{1/2}(z)/J_{-1/2}(z)\)). There are singularity points (places where the denominators reach zero values--the zeros of the corresponding Bessel functions); however, unlike the tangent function, the period of this function changes slightly. In the case of the function \(J_{1}(z)/J_{0}(z)\), this period gradually increases, while for all the others, on the contrary, it gradually decreases, as shown in Figure 3, which illustrates the differences from subsequent zeros of the Bessel function (\(j_{p,k+1}-j_{p,k}\)). Figure 3: Comparison of selected ratios of Bessel functions. Figure 2: Comparison of selected ratios of Bessel functions. This paper is devoted to the situation when the arguments \(z\) of this series defined by Equation (6) take a characteristic value defined by the zeros of Bessel functions of different \(p\) orders (excluding the occurrence of singularity as the effect of the zero value in the denominator--Calogero-Ahmed series):  
\[\boxed{\sum_{k=1}^{\infty}\frac{1}{\left(j_{p,k}^{2}-j_{p\pm(1+m),n}^{2}\right) ^{l}}\text{ where} \begin{cases}\;l=1,2,3,\ldots\\ m=0,1,2,\ldots\end{cases} } \tag{7}\]  
Such types of series, to the best of the author's knowledge in this work, have not been studied in detail yet. And because they can have important practical applications, which are demonstrated in Section 4, showing the example of fluid mechanic issues, they deserve much more attention.