### Non-Archimedean Umbral Calculus

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Let K be a non-archimedean valued field which contains Q and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. V is the closure of the set {aq|n = 0,1,2,...} where a and q are two units of Z, q not a root of unity. C(V → K) is the Banach space of continuous functions from V to K, equipped with the supremum norm. Our aim is to find normal bases (r(x)) for C(V → K), where r(x) does not have to be a polynomial.

Let K be a non-Archimedean valued field which contains Qp, and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn | n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq --> K) (resp. C1(Vq --> K)) is the Banach space of continuous functions (resp. continuously differentiable functions) from Vq to K. Our aim is to find orthonormal bases for C(Vq --> K) and C1(Vq --> K).

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