A new q-Laplace transform with many examples

Thomas Ernst

Abstract


In the spirit of Hahn 1949, the purpose of this paper is to introduce a new \(q\)-Laplace transform for a Jackson \(q\)-integral \(\int_{0}^{a} f(t,q) \,d_q(t)\), with upper integration boundary \(\frac{1}{s(1-q)}\). For this purpose we redefine this \(q\)-integral with a \(\sigma\)-algebra and a discrete measure supported at the points \(x=aq^{n},\ n\in\mathbb{N}\). Then we prove \(q\)-analogues of many well-known Laplace transform formulas, including the formula for the transform of the delta distribution. The paper concludes with a list of \(q\)-Laplace transforms for (multiple) \(q\)-hypergeometric series, some with function arguments in the first \(q\)-real numbers \(\mathbb{R}_{\oplus_{q}}\). Elsewhere, other \(q\)-real numbers are defined in similar style as function arguments in formal power series.

Keywords


q-Laplace transform; q-hypergeometric series; Jackson q-integral; q-real numbers; Dirac distribution

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References


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DOI: http://dx.doi.org/10.17951/a.2025.79.1.25-51
Date of publication: 2025-07-31 20:53:33
Date of submission: 2025-07-24 19:01:45


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