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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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