## 1 Introduction  
Rayleigh series [1] (also called Rayleigh sums or functions) has been widely known and used in engineering practice for years. Apart from mathematics, these solutions are also used in theoretical issues related to the following areas: solid mechanics, fluid mechanics, thermodynamics, electronics, physics and many, many more. In the late 1960s, a formula that made it possible to determine the final simplified solution, which is a rational function based on the order of the main Bessel functions for which zeros are used, was derived. Two types of Rayleigh series solutions can be distinguished: (1) those explicitly presented by [PERSON] [2] and (2) recursive ones (which are shown below) derived by [PERSON] [3], [PERSON] [4] and [PERSON] [5]:  
\[\sigma_{p}^{(2l)}=\sum_{k=1}^{\infty}\frac{1}{j_{p,k}^{2l}}=\bigg{(}\frac{1}{ p+l}\bigg{)}\sum_{m=1}^{l-1}\sigma_{p}^{(2m)}\sigma_{p}^{(2l-2m)}\text{ for }l\geq 2 \tag{1}\]  
where \(p\)--the order of the Bessel function, \(j_{p,k}\)--consecutive zeros of the Bessel function of \(p\) order \(j_{p}\Big{(}j_{p,k}\Big{)}=0\) and the Rayleigh series results for \(l=1\) are as follows: \(\sigma_{p}^{(2)}=\sum_{k=1}^{\infty}\frac{1}{j_{p,k}^{2l}}=\frac{1}{4(p+1)}\). A more complex series, whose successive terms contain the difference between selected values of zeros of the Bessel function in the denominators, was dealt with by [PERSON] [6, 7]. After developing the basic solutions (recalled in this work) in the following work [8], [PERSON], supported by [PERSON], developed knowledge about these series while also analyzing solutions for their modified forms. As shown in [9], the modified Calogero-Ahmed series can always be written in a form depending on the standard Rayleigh series and the standard Calogero series. Standard Calogero series is characterized by the fact that they are based on differences computed between zeros of Bessel functions of the same order as follows:  
\[\chi_{p}^{(l)}=\sum_{ \begin{subarray}{c}k=1\\ k\
eq n\end{subarray} }^{\infty}\frac{1}{\bigg{(}j_{p,n}^{2}-j_{p,k}^{2} \bigg{)}^{l}} \tag{2}\]A similar problem was analyzed by [PERSON] [10], although this researcher was the first to analyze the difference between the zeros of the Bessel function of the order \(p\) and \(p+1\), \(p\) and \(p-1\) in the denominator:  
\[\sum_{k}^{\infty}=1\,\frac{1}{\frac{1}{j_{p,k}^{2}-j_{p+1,n}^{2}}}\text{ and }\sum_{k}^{\infty}=1\,\frac{1}{j_{p,k}^{2}-j_{p-1,n}^{2}} \tag{3}\]  
Due to the fact that the values of zeros of the Bessel functions of different orders never coincide, there is no possibility of a singularity (a zero value in the denominator). Similar to [PERSON]'s consideration of the magnetic perturbations theory, it naturally led [PERSON] [11] to results involving two consecutive orders of Bessel function zeros: \(j_{p,n}\) and \(j_{p+1,k}\). This is the consequence of considering transitions between the corresponding quantum states. [PERSON] noticed the connection with the Thomas-Reiche-Kuhn sum rule, which helped him to write three identities for the static polarizability:  
\[\left\{ \begin{array}{c}\sum_{k}^{\infty}=1\,\frac{j_{p,n}^{2}j_{p+1,k}^{2}}{ \left(j_{p+1,k}^{2}-j_{p,n}^{2}\right)^{3}}=\frac{p+1}{8}\\ \sum_{k}^{\infty}=1\,\frac{j_{p,n}^{2}j_{p+1,k}^{2}}{\left(j_{p+1,k}^{2}-j_{p,n}^{2}\right)^{3}}=\frac{2(p-2)(p^{2}-1)+(4p+1)j_{p,n}^{2}}{192j_{p,n}^{2}}\\ \sum_{k}^{\infty}=1\,\frac{j_{p,n}^{2}j_{p+1,k}^{2}}{\left(j_{p+1,k}^{2}-j_{p,n}^{2}\right)^{7}}=\frac{4(p-4)(p-3)(p-2)(p^{2}-1)+2\left(16+p-49p^{2}+26p^{3} \right)j_{p,n}^{2}+(34p-1)j_{p,n}^{4}}{11520j_{p,n}^{8}}\end{array} \right. \tag{4}\]  
Unfortunately, the general equation to other such dependencies for \(l=9,\ 11,\ 13,\) etc., is not available. The other interesting identities found in the [PERSON] paper [11] are related to relativistic quantum mechanics models that enter low-energy physics through the realization that electrons in graphene behave as massless Dirac fermions. They are then mathematically described by the Dirac rather than the Schrodinger equation.