The q-exponential operator and the bivariate carlitz polynomial with numerical applications

Abstract EN

In this paper, we introduce two operator roles for the ?-exponential operator ?????, we represent the bivariate carlitz polynomial ?????? ?? ?? by the operator ????? to derive the generating function, symmetry property, mehler’s formula, rogers formula, linearization formula, the inverse linearization formula, another rogers-type formula.

also we give an extended generating function, extended mehler’s formula extended rogers formula, and another extended identities for the bivariate carlitz polynomial by using the roles of the ?-exponential operator ?????.

finally, we test the convergence conditions of all identities given in this paper and their effect numerically.