Mizuno's rank three Nahm sums I: identities of index \((1,1,2)\)

Mizuno provided 19 examples of generalized rank three Nahm sums with symmetrizer \(\mathrm{diag}(1,1,2)\) which are conjecturally modular. We confirm their modularity by establishing Rogers--Ramanujan type identities of index \((1,1,2)\) for these examples. We first reduce these Nahm sums to some double sums or single sums, and then we use known results or apply the theory of Bailey pairs to prove the desired identities. Meanwhile, we generalize some triple sum identities to general multi-sum identities.