$$\begin{aligned} &(S_{a}f_{h})_{pi}=\sum _{\gamma\in\mathbb{Z}}a_{p\gamma}f_{i-\gamma} = a_{0}f_{i,h}+ \sum_{\gamma\leq-1}a_{p\gamma}f_{i-\gamma,h}+ \sum _{\gamma\geq1 }a_{p\gamma}f_{i-\gamma,h} \\ &\phantom{(S_{a}f_{h})_{pi}}= f_{i,h}-\biggl(\sum_{\tau\leq-1}a_{p\tau}+ \sum_{\tau\geq 1}a_{p\tau}\biggr)f_{i,h}+ [a_{-p}f_{i+1,h}+a_{{-2}p}f_{i+2,h}+\cdots+ a_{{-p}\gamma}f_{i+\gamma ,h}+\cdots] \\ &\phantom{(S_{a}f_{h})_{pi}=}{}+ [a_{p}f_{i-1,h}+a_{2p} f_{i-2,h}+ \cdots+a_{p\gamma}f_{i-\gamma ,h}+\cdots] \\ &\phantom{(S_{a}f_{h})_{pi}}= f_{i,h}-\biggl(\sum _{\tau\leq-1}a_{p\tau}+\sum_{\tau\geq1}a_{p\tau } \biggr)f_{i,h}+ \biggl[\sum_{\tau\leq-1}a_{p\tau}f_{i+1,h}- \sum_{\tau\leq-2}a_{p\tau} f_{i+1,h} \\ &\phantom{(S_{a}f_{h})_{pi}=}{}+\sum_{\tau\leq-2}a_{p\tau}f_{i+2,h}-\sum _{\tau\leq-3}a_{p\tau } f_{i+2,h}+\cdots+ \sum _{\tau\leq{-\gamma}}a_{p\tau}f_{i+\gamma,h}-\sum _{\tau\leq {-\gamma-1}}a_{p\tau} f_{i+\gamma,h}+\cdots\biggr] \\ &\phantom{(S_{a}f_{h})_{pi}=}{}+ \biggl[\sum_{\tau\geq1}a_{p\tau}f_{i-1,h}- \sum_{\tau\geq2}a_{p\tau } f_{i-1,h}+ \sum _{\tau\geq2}a_{p\tau}f_{i-2,h}-\sum _{\tau\geq3}a_{p\tau} f_{i-2,h}+\cdots \\ &\phantom{(S_{a}f_{h})_{pi}=}{} +\sum_{\tau\geq\gamma}a_{p\tau}f_{i-\gamma,h}-\sum _{\tau\geq {\gamma+1}}a_{p\tau} f_{i-\gamma,h}+\cdots\biggr] \\ &\phantom{(S_{a}f_{h})_{pi}}= f_{i,h}+ \sum_{\gamma\leq-1}\biggl(\sum _{\tau\leq\gamma}a_{p\tau }\biggr) (f_{i-\gamma,h}-f_{i-\gamma-1,h})+ \sum_{\gamma\geq1}\biggl(\sum_{\tau\geq\gamma}a_{p\tau} \biggr) (f_{i-\gamma ,h}-f_{i-\gamma+1,h}) \\ &\phantom{(S_{a}f_{h})_{pi}}= f_{i,h}+\sum_{\gamma\leq-1}\lambda_{0}( \gamma) (f_{i-\gamma ,h}-f_{i-\gamma-1,h})+ \sum_{\gamma\geq1} \lambda_{0}(\gamma) (f_{i-\gamma,h}-f_{i-\gamma +1,h}). \\ &(S_{a}f_{h})_{pi+1} =\sum _{\gamma\in\mathbb{Z}}a_{p\gamma+1}f_{i-\gamma} = a_{1}f_{i,h}+ \sum_{\gamma\leq-1}a_{p\gamma+1}f_{i-\gamma,h}+ \sum _{\gamma\geq1 }a_{p\gamma+1}f_{i-\gamma,h} \\ &\phantom{(S_{a}f_{h})_{pi+1}}= f_{i,h}-\biggl(\sum_{\tau\leq-1}a_{p\tau+1}+ \sum_{\tau\geq 1}a_{p\tau+1}\biggr)f_{i,h}+ [a_{-p+1}f_{i+1,h}+a_{{-2}p+1}f_{i+2,h} \\ &\phantom{(S_{a}f_{h})_{pi+1}=}{}+\cdots+ a_{{-p}\gamma+1}f_{i+\gamma,h}+\cdots]+ [a_{p+1}f_{i-1,h}+a_{2p+1} f_{i-2,h}+\cdots \\ &\phantom{(S_{a}f_{h})_{pi+1}=}{} +a_{p\gamma+1}f_{i-\gamma,h}+\cdots] \\ &\phantom{(S_{a}f_{h})_{pi+1}}= f_{i,h}-\biggl(\sum_{\tau\leq-1}a_{p\tau+1}+ \sum_{\tau\geq 1}a_{p\tau+1}\biggr)f_{i,h}+ \biggl[\sum_{\tau\leq-1}a_{p\tau+1}f_{i+1,h}- \sum_{\tau\leq-2}a_{p\tau +1} f_{i+1,h} \\ &\phantom{(S_{a}f_{h})_{pi+1}=}{} +\sum_{\tau\leq-2}a_{p\tau+1}f_{i+2,h}-\sum _{\tau\leq -3}a_{p\tau+1} f_{i+2,h}+\cdots+ \sum _{\tau\leq{-\gamma}}a_{p\tau+1}f_{i+\gamma,h}\\ &\phantom{(S_{a}f_{h})_{pi+1}=}{}-\sum _{\tau \leq{-\gamma-1}}a_{p\tau+1} f_{i+\gamma,h} +\cdots\biggr]\\ &\phantom{(S_{a}f_{h})_{pi+1}=}{}+ \biggl[\sum_{\tau\geq1}a_{p\tau+1}f_{i-1,h}- \sum_{\tau\geq2}a_{p\tau +1} f_{i-1,h}+ \sum _{\tau\geq2}a_{p\tau+1}f_{i-2,h}-\sum _{\tau\geq3}a_{p\tau +1} f_{i-2,h}+\cdots \\ &\phantom{(S_{a}f_{h})_{pi+1}=}{} +\sum_{\tau\geq\gamma}a_{p\tau+1}f_{i-\gamma,h}-\sum _{\tau \geq{\gamma+1}}a_{p\tau+1} f_{i-\gamma,h}+\cdots \biggr] \\ &\phantom{(S_{a}f_{h})_{pi+1}}= f_{i,h}+ \sum_{\gamma\leq-1}\biggl(\sum _{\tau\leq\gamma}a_{p\tau +1}\biggr) (f_{i-\gamma,h}-f_{i-\gamma-1,h})+ \sum_{\gamma\geq1}\biggl(\sum_{\tau\geq\gamma}a_{p\tau +1} \biggr) (f_{i-\gamma,h}-f_{i-\gamma+1,h}) \\ &\phantom{(S_{a}f_{h})_{pi+1}}= f_{i,h}+\sum_{\gamma\leq-1}\lambda_{1}( \gamma) (f_{i-\gamma ,h}-f_{i-\gamma-1,h})+ \sum_{\gamma\geq1} \lambda_{1}(\gamma) (f_{i-\gamma,h}-f_{i-\gamma +1,h}). \end{aligned}$$
$$\lambda_{l}(i)= \textstyle\begin{cases} \sum_{\tau\leq i}a_{p\tau+l},& i< 0,\\ 0,& i=0,\\ \sum_{\tau\geq i}a_{p\tau+l},& i>0, l=0,1,\ldots,p-1. \end{cases} $$