LAUSR

Abstract In the spirit of our earlier articles [12] , [14] , [11] , and our work [13] , we define two dual q-Bernoulli polynomials, with corresponding vector and matrix forms. Following Aceto–Trigiante [1] , the q-L matrix, the indefinite q-integral of the q-Pascal matrix is the link between the q-Cauchy and the q-Bernoulli matrix. The q-analogue of the Bernoulli complementary argument theorem can be expressed in matrix form through the diagonal A n matrix. For the q-Euler polynomials corresponding results are obtained. The umbral calculus for generating functions of q-Appell polynomials is shown to be equivalent to a transform method, which maps polynomials to matrices, a true q-analogue of Arponen [6] . This is manifested by the Vein [21] matrix, which occurs as the transform of the q-difference operator. The Aceto–Trigiante shifted q-Bernoulli matrix has a simple connection to the q-Bernoulli Arponen matrix through the q-Pascal matrix. We reintroduce certain q-Stirling numbers ∈ Z ( q ) from [12] , which will be needed for the polynomial matrix definitions.