An Extension of Generalized Euler Polynomials of the Second Kind

Abstract

Many mathematicians have studied various relations beweenEuler number En, Bernoulli number Bn and Genocchi number Gn (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

SYMMETRIC IDENTITIES FOR THE HIGHER-ORDER TWISTED $(h, q)$-EULER NUMBERS AND POLYNOMIALS

“…Many mathematicians have studied in the area of the q-extension of Euler numbers and polynomials(see [1,2,3,5,6,7,8,9,11,13]). Recently, Y.…”

“…For h ∈ Z, α, k ∈ N, and n ∈ Z + , we introduced the higher order twisted q-Euler polynomials with weight α as follows(see [7]):…”

Exploring the zeros of the generalized Euler polynomials of second kind

“…In [1], we introduced the the generalized Euler polynomials E (α) n (x) of the second kind. It is the aim of this paper to observe an interesting phenomenon of 'scattering' of the zeros of the generalized Euler polynomials E (α) n (x) of the second kind in complex plane.…”

“…Setting α = 1 in (2.1), we can obtain the corresponding definition for the second kind Euler polynomials(see [1,2,3,4 ]).…”

“…The following elementary properties of the generalized Euler polynomials E (α) n (x) of the second kind are readily derived form (2.1). We, therefore, choose to omit details involved(see [1]). Thus, we know that E…”