On the q-extension of Euler and Genocchi numbers

Abstract

Carlitz has introduced an interesting q-analogue of Frobenius–Euler numbers in [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000; L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332–350]. He has indicated a corresponding Stadudt–Clausen theorem and also some interesting congruence properties of the q-Euler numbers. A recent author's study of more general q-Euler and Genocchi numbers can be found in previous publication [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]. In this paper we give a new construction of q-Euler numbers, which are different from Carlitz's q-extension and author's q-extension in previous publication (see [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]). By using our q-extension of Euler numbers, we can also consider a new q-extension of Genocchi numbers and obtain some interesting relations between q-extension of Euler numbers and q-extension of Genocchi numbers.