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An invariant $$p$$-adic integral associated with Daehee numbers. (English) Zbl 1016.11008
Let $$z,q,u\in \mathbb C_p$$, $$|1-q|_p<p^{-1/(p-1)}$$, $$|1-u|_p\geq 1$$. The author introduces, using $$p$$-adic integration, a sequence $$D_n(z:q)$$, $$n=1,2,\ldots$$, of so-called Daehee numbers, in such a way that $$D_n(q:q)$$ coincide with the $$p$$-adic $$q$$-Bernoulli numbers, while $$D_n(u:q)=H_n(u^{-1}:q)$$ where $$H_n$$ are the $$p$$-adic $$q$$-Euler numbers. Explicit expressions for the Daehee numbers and related Daehee polynomials are found.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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