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Factorial-type recurrence relations and $p$-adic incomplete gamma functions

Volume 212 / 2024

Paul Buckingham Acta Arithmetica 212 (2024), 1-29 MSC: Primary 11S80 DOI: 10.4064/aa221031-19-9 Published online: 12 December 2023


We introduce an automorphism $\mathcal S$ of the space $C(\mathbb Z_p,\mathbb C_p)$ of continuous functions $\mathbb Z_p \to \mathbb C_p$ and show that it can be used to give an alternative construction of the $p$-adic incomplete $\Gamma $-functions recently introduced by O’Desky and Richman. We then describe various properties of $\mathcal S$, showing in particular that it is self-adjoint with respect to a certain non-degenerate symmetric bilinear form defined in terms of $p$-adic integration, and introducing a $p$-adic integral transform to which $\mathcal S$ is related. We also derive an integral-transform formula for the $p$-adic incomplete $\Gamma $-functions.


  • Paul BuckinghamDepartment of Mathematical and Statistical Sciences
    University of Alberta
    Edmonton, Alberta, T6G 2G1, Canada

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