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In this paper, we study the desingularization problem in the first q -Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q -Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q -Weyl closure of a given q -difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator. [ABSTRACT FROM AUTHOR]

In this paper, we expand functions of specific q -exponential growth in terms of its even (odd) Askey-Wilson q -derivatives at 0 and η = (q 1 / 4 + q − 1 / 4) / 2. This expansion is a q -version of the celebrated Lidstone expansion theorem, where we expand the function in q -analogs of Lidstone polynomials, i.e., q-Bernoulli and q -Euler polynomials as in the classical case. We also raise and solve a q -extension of the problem of representing an entire function of the form f (z) = g (z + 1) − g (z) , where g (z) is also an entire function of the same order as f (z). [ABSTRACT FROM AUTHOR]