Q-analogues of Symbolic Operators

Here presented are 𝑞-extensions of several linear operators including a novel 𝑞-analogue of the derivative operator 𝐷. Some 𝑞-analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these 𝑞-substitution rules may be used to construct symbolic summation and series transformation formulas, including 𝑞-analogues of the classical Euler transformations for accelerating the convergence of alternating series.