New forms of the Cauchy operator and some of their applications
Tóm tắt
In this paper, we first construct the Cauchy q-shift operator T(a, b;D
xy
) and the Cauchy q-difference operator L(a, b; θ
xy
). We then apply these operators in order to represent and investigate some new families of q-polynomials which are defined in this paper. We derive some q-identities such as generating functions, symmetry properties and Rogers-type formulas for these q-polynomials. We also give an application for the q-exponential operator R(bD
q
).
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