# The q-Exponential Operator and the Bivariate Carlitz Polynomial with Numerical Applications

### Abstract

**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.**

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