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On \((i,q)\) Bernoulli and Euler numbers. (English) Zbl 1152.11008

Summary: The \(p\)-adic invariant \(q\)-integral on \(\mathbb Z_p\) was originally constructed by T. Kim [On a \(q\)-analogue of the \(p\)-adic log gamma function and related integrals, J. Number Theory 76, 320–329 (1999; Zbl 0941.11048)]. Recently, many authors have been studying the extended Bernoulli numbers or Euler numbers by using this \(p\)-adic \(q\)-integral in the fermionic or bosonic sense. Let \(i\in O_{\mathbb C_p}=\{x \in \mathbb C_p\mid |x|_p \leqslant 1 \}\). Then we consider new \((i,q)\)-Bernoulli and Euler numbers using \(p\)-adic \(q\)-integrals in this work.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Citations:

Zbl 0941.11048
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Full Text: DOI

References:

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