## 1 Introduction  
Re-configurable intelligent surface (RIS)-assisted communications (ACs) are envisioned for beyond 5G and 6G wireless systems [1, 2]. In a smart propagation environment, RIS blocks are capable of optimizing and improving system performances by controlling incident transmission waves in a directed and programmable way. RIS-ACs for smart radio environments with suitable RIS applications are considered in [3]. Path-loss models for RIS-ACs supported by experimental and simulation results are provided in [4]. Moreover, a comparison between relay-ACs and RIS-ACs is provided in [5]. In [6], mmWave RIS-ACs systems are considered, whereas visible light communication (VLC) RIS-ACs systems are considered in [7, 8]. Free-space optical (FSO) communication is a technology that, due to narrow beam widths, can be distinguished as highly secure, interference immune and energy efficient. Additionally, the FSO system is capable of providing a relatively large bandwidth and can be further distinguished as a license-free and cost effective transmission technology. In turn, the main cause of terrestrial FSO system performance degradation is atmospheric turbulence. The gamma-gamma (G-G) distribution is the most commonly used turbulence-induced (TI) fading model that is mathematically tractable and experimentally validated for moderate-to-strong TI fading conditions [9, 10]. Moreover, the log-normal TI fading model is suitable for weak TI fading conditions, but is mathematically less tractable and often can lead to complex analytical expressions [11], whereas the general Malaga \(\mathcal{M}\) TI fading model can be used to address weak-to-strong TI fading severity conditions, but is mathematically less tractable if compared with the G-G TI fading model [12]. The Fisher-Snedecor (F-S) \(\mathcal{F}\) distribution has recently been proposed as an accurate and experimentally verified FSO TI fading model [13]. The F-S \(\mathcal{F}\) TI fading channel for FSO communications with pointing errors for weak, moderate and strong TI fading conditions is addressed in [14], whereas the hybrid mmWave/FSO transmission over F-S \(\mathcal{F}\) TI fading channels is considered in [15]. Moreover, the secrecy performance of the FSO system over F-S \(\mathcal{F}\) TI fading propagation channels is reported in [16]. Initially, the F-S \(\mathcal{F}\) distribution was proposed in [17], whereas, in [18; 19; 20; 21], the authors considered wireless systems over the composite F-S \(\mathcal{F}\) fading channels to account for multi-path and shadowing fading conditions. The second-order (S-O) statistics of composite \(\mathcal{F}\) fading are given in [22], whereas the S-O metrics of bivariate F-S \(\mathcal{F}\) distribution are considered in [23]. An FSO RIS-ACs system in moderate-to-strong severity conditions modelled over the G-G TI fading channels is well investigated in [24], whereas a mixed RF-FSO, RIS-ACs over G-G TI fading channels is further considered in [25]. Moreover, FSO RIS-ACs in weak-to-strong TI fading conditions modelled with both log-normal (in order to account for weak turbulence conditions) and G-G (in order to account for moderate-to-strong turbulence conditions) distributions are considered in [26]. A unified performance analysis of FSO RIS-ACs over G-G, F-S \(\mathcal{F}\) and Malaga \(\mathcal{M}\) TI fading channels for the first-order (F-O) statistical measures that are expressed through Meijer G and Fox H functions is conducted in [27]. In addition to F-O statistics, the S-O statistics can broaden the understanding of FSO RIS-ACs systems' behaviour over rapidly time-variant TI fading channels. The S-O statistics for FSO systems over TI fading channels have been investigated in [28; 29; 30; 31; 32]. In [33], the authors analysed the S-O statistics of an FSO N-hop relay communications system over F-S \(\mathcal{F}\) TI fading channels. Moreover, the S-O statistics have been considered in a variety of 5G and beyond 5G communication systems, [34; 35; 36; 37].