In this paper, we introduced and studied the $(t,s)$-Oresme and $(t,s)$-Oresme-Lucas sequences, which were generalizations of the classical Oresme and Oresme-Lucas numbers. We presented the recurrence relations, generating functions, and Binet formulas for these sequences. Furthermore, we derived various algebraic identities, including those of Tagiuri, d'Ocagne, Catalan, Cassini, Ruggles, and Honsberger. A matrix representation for these sequences was also established, and we defined the associated $(t,s)$-Oresme matrix sequence, exploring its fundamental properties and relations. Additionally, we provided the summation formulas for the first $n$ terms of these sequences. This work extended the existing theory of Oresme numbers and offered a unified framework for their further investigation.

In this paper, we introduce a new generalization of the Frank matrix, which is a lower Hessenberg matrix called the generalized max $r$-Frank matrix. We obtain a recurrence relation provided by the characteristic polynomial, inverse, determinant, and norm properties of this matrix. We also present an example to illustrate the results obtained.