left side of each identity can be written as the sum of two parts. Next, by using the identities in Lemmas \[lem-z-6\] and \[lem-z-7\], we obtain the identity.

For example, the left side of (i) can be written as $$\begin{array}{lll}
&&\displaystyle \sum_{t=0}^m\sum_{j=0}^{2k}{4k+1\choose 2j}{4m-4k-1\choose 4t-2j}\\
&=&\displaystyle \sum_{i=0}^{4k}{4k+1\choose 4i}\sum_{t=0}^m{4(m-k)-1\choose 4(t-i)}+\sum_{i=0}^{4k}{4k+1\choose 4i+2}\sum_{t=0}^m{4(m-k)-1\choose 4(t-i)-2}\\
&=&\displaystyle \sum_{i=0}^{4k}{4k+1\choose 4i}\sum_{t'=0}^{m-k-1}{4(m-k-1)+3\choose 4t'}+\sum_{i=0}^{4k}{4k+1\choose 4i+2}\sum_{t'=0}^{m-k-1}{4(m-k-1)+3\choose 4t'+2}.
\end{array}$$ By Lemma \[lem-z-7\] (a) (b) (f) and (g), we have $$\begin{array}{lll}
&&\displaystyle \sum_{i=0}^{4k}{4k+1\choose 4i}\sum_{t'=0}^{m-k-1}{4(m-k-1)+3\choose 4t'}+\sum_{i=0}^{4k}{4k+1\choose 4i+2}\sum_{t'=0}^{m-k-1}{4(m-k-1)+3\choose 4t'+2}\\
&=&(2^{4k-1}+(-1)^k2^{2k-1})(2^{4(m-k-1)+1}-(-1)^{m-k-1}2^{2(m-k-1)})\\
&&+(2^{4k-1}-(-1)^k2^{2k-1})(2^{4(m-k-1)+1}+(-1)^{m-k-1}2^{2(m-k-1)})\\
&=&2^{4m-3}+(-1)^m2^{2m-2}.\end{array}$$ It leads to (i).

By these same processes, one can verify the other identities one by one.

At last, we mention that (i), (ii) and (iii) of Theorem \[thm-z-3\] and the identity of Theorem \[thm-z-2\] give four distinct combination representations but achieve the same value $2^{4m-2}+(-1)^m2^{2m-1}$; the next four identities and the last four identities of Theorem \[thm-z-3\] also give four distinct combination representations but achieve the same values $2^{4m}+(-1)^m2^{2m}$ and $2^{4m}-(-1)^m2^{2m}$, respectively. Such a properties are important to verify strongly regular orbit Cayley graphs in the next section.

Strongly regular orbit Cayley graphs
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A connected $r$-regular graph on $n$ vertices is said to be *strongly regular* with parameter $(n,r,\lambda,\mu)$ if two vertices $x$ and $y$ have $\lambda$ common neighbors whenever they 