We use two different types of laddering equations which are realized by the associated Gegenbauer functions. We show that the Dirac oscillator equation with minimal lengths can be written to form associated Gegenbauer functions. This leads us to write the Dirac oscillator equation in terms of first order equations. By changing some variable in two types of equations, we have some u1,1 and u2 algebras. These algebras correspond to AdS2 and S2 . Finally, we conclude that all quantum states from the Dirac oscillator presence of minimal lengths correspond to the quantum states of free particle on AdS2 and S2 . © 2007 American Institute of Physics
