This study explores various properties of generalized Bernoulli numbers and polynomials using two distinct approaches: the generalized Fibonacci sequences of infinite order and the determinantal process. We establish recursive relations of infinite order for generating generalized Bernoulli and Euler numbers and polynomials. Several identities satisfied by Bernoulli, and Euler numbers and polynomials are derived, while others are recovered. Additionally, further properties of the generalized Bernoulli, Euler numbers and polynomials are developed using the determinantal approach. Finally, several application are provided. ...