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Generalizations of Euler Numbers and Polynomials 1 GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS QIU-MING LUO AND FENG QI Abstract. In this paper, the concepts of Euler numbers and Euler polyno- mials are generalized, and some basic properties are investigated. 1. Introduction It is well-known that the Euler numbers and polynomials can be defined by the following definitions. Definition 1.1 ([1]). The Euler numbers Ek are defined by the following expansion t ∞ 2e X Ek = tk, |t| ≤ π. (1.1) e2t + 1 k! k=0 In [4, p. 5], the Euler numbers is defined by t/2 ∞ n 2n 2e t X (−1) En t = sech = , |t| ≤ π. (1.2) et + 1 2 (2n)! 2 n=0 Definition 1.2 ([1, 4]). The Euler polynomials Ek(x) for x ∈ R are defined by xt ∞ 2e X Ek(x) = tk, |t| ≤ π. (1.3) et + 1 k! k=0 It can also be shown that the polynomials Ei(t), i ∈ N, are uniquely determined by the following two properties 0 Ei(t) = iEi−1(t),E0(t) = 1; (1.4) i Ei(t + 1) + Ei(t) = 2t . (1.5) 2000 Mathematics Subject Classification. 11B68. Key words and phrases. Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. This paper was typeset using AMS-LATEX. 1 2 Q.-M. LUO AND F. QI Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4]. In this paper, we will give some generalizations of the concepts of Euler numbers and Euler polynomials, and research their basic properties. 2. Generalizations of Euler numbers and polynomials In this section, we give two definitions, the generalized Euler number and the generalized Euler polynomial, which generalize the concepts of Euler number and Euler polynomial. Definition 2.1. For positive numbers a, b, c, the generalized Euler numbers Ek(a, b, c) are defined by t ∞ 2c X Ek(a, b, c) = tk. (2.1) b2t + a2t k! k=0 Definition 2.2. For any given positive numbers a, b, c and x ∈ R, the generalized Euler polynomials Ek(x; a, b, c) are defined by xt ∞ 2c X Ek(x; a, b, c) = tk. (2.2) bt + at k! k=0 If taking a = 1, b = c = e, then Definition 1.1 and Definition 1.2 can be deduced from Definition 2.1 and Definition 2.2 respectively. Thus, the Definition 2.1 and Definition 2.2 generalize the concepts of Euler numbers and polynomials. 3. Some properties of the generalized Euler numbers In this section, we study some basic properties of the generalized Euler numbers defined in Definition 2.1. Theorem 3.1. For positive numbers a, b, c and real number x ∈ R, we have 1/2 x E0(a, b, c) = 1,Ek(1, e, e) = Ek,Ek(1, e , e ) = Ek(x), (3.1) ln c − 2 ln a E (a, b, c) = 2k(ln b − ln a)kE , (3.2) k k 2(ln b − ln a) k X k E (a, b, c) = (ln b − ln a)j(ln c − ln a − ln b)k−jE . (3.3) k j j j=0 Proof. The formulae in (3.1) follow from Definition 1.1, Definition 1.2, and Defini- tion 2.1 easily. GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS 3 By Definition 1.2 and Definition 2.1 and direct computation, we have ln c−2 ln a 2ct 2 exp · 2t(ln b − ln a) = 2(ln b−ln a) b2t − a2t exp2t(ln b − ln a) + 1 ∞ (3.4) X ln c − 2 ln a tk = 2k(ln b − ln a)kE . k 2(ln b − ln a) k! k=0 Then, the formula (3.2) follows. Pk −jk 1 k−j Substituting Ek(x) = j=0 2 j (x− 2 ) Ej into the formula (3.2) yields the formula (3.3). Theorem 3.2. For k ∈ N, we have k−1 1 X k E (a, b, c) = − (2 ln b − ln c)k−j + (2 ln a − ln c)k−j E (a, b, c), (3.5) k 2 j j j=0 Ek(a, b, c) = Ek(b, a, c), (3.6) α α α k Ek(a , b , c ) = α Ek(a, b, c). (3.7) Proof. By Definition 2.1, direct calculation yields " t t# ∞ 1 b2 a2 X tk 1 = + E (a, b, c) 2 c c k! k k=0 ∞ " k k# ∞ 1 X tk b2 a2 X tk = ln + ln E (a, b, c) 2 k! c c k! k (3.8) k=0 k=0 ∞ k " k−j k−j# 1 X X k b2 a2 tk = ln + ln E (a, b, c) . 2 j c c j k! k=0 j=0 Equating coefficients of tk in (3.8) gives us k " k−j k−j# X k b2 a2 ln + ln E (a, b, c) = 0. (3.9) j c c j j=0 Formula (3.5) follows. The rest formulae follow from Definition 2.1 and formula (3.2). Remark 3.1. For positive numbers a, b, and c, we have the following E0(a, b, c) = 1, E1(a, b, c) = ln c − ln a − ln b, (3.10) E2(a, b, c) = (ln c − 2 ln a)(ln c − 2 ln b), 2 2 E3(a, b, c) = [(ln c − ln a − ln b) − 3(ln b − ln a) ](ln c − ln a − ln b). 4 Q.-M. LUO AND F. QI 4. Some properties of the generalized Euler polynomials In this section, we will investigate properties of the generalized Euler polynomials defined by Definition 2.2. Theorem 4.1. For any given positive numbers a, b, c and x ∈ R, we have k k−j X k(ln c)k−j 1 E (x; a, b, c) = x − E (a, b, c), (4.1) k j 2j 2 j j=0 k j k−j X k b 1 ln c − 2 ln a E (x; a, b, c) = (ln c)k−j ln x − E , (4.2) k j a 2 j 2(ln b − ln a) j=0 k j ` k−j X X kj(ln c)k−j b h c ij−` 1 E (x; a, b, c) = ln ln x − E , (4.3) k j ` 2j a ab 2 ` j=0 `=0 1 E (a, b, c) = 2kE ; a, b, c , (4.4) k k 2 Ek(x) = Ek(x; 1, e, e). (4.5) Proof. By Definition 2.1 and Definition 2.2, we have ∞ 2c2xt X tk = 2kE (x; a, b, c) (4.6) b2t + a2t k k! k=0 and 2c2xt 2ct = · c(2x−1)t b2t + a2t b2t + a2t ∞ ! ∞ ! X tk X tk = E (a, b, c) (2x − 1)k(ln c)k k! k k! (4.7) k=0 k=0 ∞ k X X k tk = (ln c)k−j(2x − 1)k−jE (a, b, c) . j j k! k=0 j=0 tk Equating the coefficients of k! in (4.6) and (4.7) yields k X k 2kE (x; a, b, c) = (ln c)k−j(2x − 1)k−jE (a, b, c). k j j j=0 Formula (4.1) follows. The rest formulae follow directly from substituting formulae (3.2) and (3.3) into 1 (4.1) and taking x = 2 in (4.1), respectively. GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS 5 Theorem 4.2. For positive integer 1 ≤ p ≤ k, we have ∂p k! E (x; a, b, c) = (ln c)pE (x; a, b, c), (4.8) ∂xp k (k − p)! k−p Z x 1 Ek(t; a, b, c)dt = [Ek+1(x; a, b, c) − Ek+1(β; a, b, c)] . (4.9) β (k + 1) ln c Proof. Differentiating on both sides of formula (4.1) yields ∂ E (x; a, b, c) = k(ln c)E (x; a, b, c). (4.10) ∂x k k−1 Using formula (4.10) and by mathematical induction, the formulae (4.8) follows. Rearranging formula (4.10) produces 1 ∂ E (x; a, b, c) = E (x; a, b, c). (4.11) k (k + 1) ln c ∂x k+1 Formula (4.9) follows from integrating on both sides of formula (4.11). Theorem 4.3. For positive numbers a, b, c and x ∈ R, we have k X k E (x + 1; a, b, c) = (ln c)k−jE (x; a, b, c), (4.12) k j j j=0 k k Ek(x + 1; a, b, c) = 2x (ln c) k (4.13) X k + (ln c)k−j − (ln b)k−j − (ln a)k−j E (x; a, b, c), j j j=0 a b E (x + 1; a, b, c) = E x; , , c . (4.14) k k c c Proof. From Definition 2.2 and straightforward calculation, we have " ∞ #" ∞ # 2cxt X tk X tk · ct = E (x; a, b, c) (ln c)k bt + at k! k k! k=0 k=0 (4.15) ∞ k X X k tk = (ln c)k−jE (x; a, b, c) j j k! k=0 j=0 and ∞ 2cxt 2c(x+1)t X tk · ct = = E (x + 1; a, b, c). (4.16) bt + at bt + at k! k k=0 tk Therefore, from equating the coefficients of k! in (4.15) and (4.16), formula (4.12) follows.
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