## 1 Introduction
Furthermore, the ratio of the extensive variables yields a non-extensive variable, for example, the number density = (particle number/volume), which is the proportionality of the number of particles to the system's volume. [PERSON] proposed and [PERSON] [22] researched a suitable BGS entropy, statistically under equilibrium. The nonlinear and non-extensive situation of the entropies was continued by [PERSON]; the entropic (spectral) index \(q\) depicts the level of non-extensivity of the assumed framework. There are two major physical zones that can be identified: the \(-1<q\leq 1\) region, which has all velocities, and \(q\geq 1\), in which the plasma species distribution-function capacity indicates a thermal cut-off on the most extreme quality considered: the particles' velocity [23]. Negative ions' impact in space areas [24] and in the laboratory plasmas [25] was discovered in the earth's mesosphere, the ionosphere's D zone, and the sun's photosphere. The dynamics of IA dispersive waves have also been observed with negative ions [26]. [PERSON] et al. [27] looked at the parameters of IA waves in positive-/negative-ion plasma elements and found primary modes. They have demonstrated that in \(Xe^{+}-F^{-}-SF_{6}^{-}\) plasma, two modes exist considering the values of the electron temperature and ion temperature, and they also occurr in one mode in \(Ar^{+}-F^{-}-SF_{6}^{-}\) plasma, regardless of the multi-ions temperatures. [PERSON] et al. [28] studied the slow mode of IA waves with various features utilizing the \(Xe^{+}-F^{-}\) plasmas in two distinct DP devices. Later research looked at the IAWs in \(Xe^{+}-F^{-}-SF_{6}^{-}\) and \(Ar^{+}-F^{-}-SF_{6}^{-}\) plasma species, revealing that this work will help with the analysis of dusty plasmas with negative ions. By using \(Ar^{+}-Xe^{+}\), the heavy and light ion modes have been studied in lab plasma [29]. In addition to the negative ion plasma employed in the labs [30], sulfur hexafluoride \((SF_{6})\) gas has been employed to generate negative ions. It has been also discovered using mass spectrometry in \(Ar/SF_{6}\) plasma created in DP devices in which \(Ar^{+}\), \(F^{-}\), and \(SF_{6}\) were present for the sufficiently high negative ion to positive ion density ratios [27], and experiments were also conducted using such plasmas [27]. In the current paper, we will analyze the \(Xe^{+}-F^{-}-SF_{6}^{-}\) and \(Ar^{+}-F^{-}-SF_{6}^{-}\) dusty plasma and do numerical computations using the static negative dust particle. These negative ions are a form of plasma species that can arise naturally or be infused from an outside source in the space plasma and lab plasmas [31]. The control of an electrode in deposition and etching plasmas [32], the development of voids [33], and the extraction of negative ion electrode sources [34] are all dependent on negative ions. The DSP was previously [35] calculated using non-Maxwellian currents due to multi ions for dust particles that are negatively charged. We have studied the \(Xe^{+}-F^{-}-SF_{6}^{-}\) and the \(Ar^{+}-F^{-}-SF_{6}^{-}\) plasmas. To investigate the influence of plasma characteristics, we used numerical analysis on the negatively charged DSP. We noted the effects of the spectral indices \(\alpha\) and \(q\), positive ion streaming (\(U_{Ar^{+}}\) and \(U_{Xe^{+}}\)), and negatively ion streaming (\(U_{F^{-}}\) and \(U_{SF_{6}^{-}}\)) on DSP for both types of plasmas. The rest of this paper is divided into the following sections: we investigate the generalized Cairm-Tsallis \((q,\alpha)\)-distribution function in Section 2. In Section 3, we calculate the DSP for \(Xe^{+}-F^{-}-SF_{6}^{-}\) plasma. In Section 4, we have derived the DSP for \(Ar^{+}-F^{-}-SF_{6}^{-}\) plasma utilizing the same method of analysis. The Numerical results and discussion for the type of plasmas are studied in Section 5. Finally, in Section 6, a summary of the study's significant findings is presented.