A Note on Eulerian Polynomials
We study Genocchi, Euler, and tangent numbers. From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers.
As is well known, the Eulerian polynomials, , are defined by generating function as follows: with the usual convention about replacing by (see [1–18]). From (1.1), we note that where is the Kronecker symbol (see ).
Thus, by (1.2), we get
In this paper, we give some identities on the Eulerian polynomials at associated with Genocchi, Euler, and tangent numbers.
2. Witt's Formula for Eulerian Polynomials
In this section, we assume that , , and will, respectively, denote the ring of -adic integers, the field of -adic numbers, and the completion of algebraic closure of . The -adic norm is normalized so that .
Let us take . Then, by (2.3), we get
Thus, from (2.4), we have
By Taylor expansion on the left-hand side of (2.5), we get
Theorem 2.1. For , one has where is an Eulerian polynomials.
Therefore, by (2.14), we obtain the following theorem.
Theorem 2.2. For , one has where is the th tangent numbers.
Corollary 2.3. For , one has
Corollary 2.4. For , one has
By (2.21), we get
Therefore, by (2.23), we obtain the following theorem.
Theorem 2.5. For , one has
Theorem 2.6. For , we have
3. Further Remark
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786. Also, the authors would like to thank the referees for their valuable comments and suggestions.
L. Euler, Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum, Methodus summanandi superior ulterius pro-mota, chapter VII, Academiae Imperialis ScientiarumPetropolitanae, St Petersbourg, Russia, 1755.
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