On several q-special matrices, including the q-Bernoulli and q-Euler matrices

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 An 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.