  This work provides the general framework for obtaining strong Szeg\H{o} limit
theorems for multi-bordered, semi-framed, framed, and multi-framed Toeplitz
determinants, extending the results of Basor et al. (2022) beyond the (single)
bordered Toeplitz case. For the two-bordered and also the semi-framed Toeplitz
determinants, we compute the strong Szeg\H{o} limit theorems associated with
certain classes of symbols, and for the $k$-bordered (${k \geq 3}$), framed,
and multi-framed Toeplitz determinants we demonstrate the recursive fashion
offered by the Dodgson condensation identities via which strong Szeg\H{o} limit
theorems can be obtained. One instance of appearance of semi-framed Toeplitz
determinants is in calculations related to the entanglement entropy for
disjoint subsystems in the XX spin chain (Brightmore et al. (2020) and
Jin-Korepin (2011)). In addition, in the recent work Gharakhloo and Liechty
(2024) and in an unpublished work of Professor Nicholas Witte, such
determinants have found relevance respectively in the study of ensembles of
nonintersecting paths and in the study of off-diagonal correlations of the
anisotropic square-lattice Ising model. Besides the intrinsic mathematical
interest in these structured determinants, the aforementioned applications have
further motivated the study of the present work.
