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Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 1872–1876 Research Article On the Appell type λ-Changhee polynomials Dongkyu Lima,∗, Feng Qib,c a School of Mathematical Sciences, Nankai University, Tianjin Ciy, 300071, China. b Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China. c Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China. Communicated by C. Park Abstract In the paper, by virtue of the p-adic fermionic integral on Zp , the authors consider a λ-analogue of the c Changhee polynomials and present some properties and identities of these polynomials. 2016 All rights reserved. Keywords: Identity, property, Appell type λ-Changhee polynomial, Changhee polynomial, degenerate Changhee polynomial. 2010 MSC: 05A30, 05A40, 11B68, 11S80. 1. Introduction Let p be a fixed odd prime number and let Zp , Qp , and Cp denote respectively the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp . The p-adic norm is normally defined as |p| = p1 . Recently, degenerate Changhee polynomials Chn,λ (x) are defined [6, p. 296] by ∞ 2λ ln(1 + λt) x X tn 1+ = Chn,λ (x) . 2λ + ln(1 + λt) λ n! n=0 When x = 0, we call Chn,λ = Chn,λ (0) the degenerate Changhee numbers. It is common knowledge that the Euler polynomials En (x) are given by ∞ X 2 tn xt e = E (x) n et + 1 n! n=0 ∗ Corresponding author Email addresses: [email protected] (Dongkyu Lim), [email protected], [email protected] (Feng Qi) Received 2015-08-18 D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876 1873 and that, when x = 0, we call En = 2n En 12 the Euler numbers. Let C(Zp ) be the space of all continuous functions on Zp . For f ∈ C(Zp ), the fermionic p-adic integral on Zp was defined [3, p. 134] by Z f (x) d µ−1 (x) = lim I−1 (f ) = N →∞ Zp N −1 pX N f (x)µ−1 x + p Zp = lim N →∞ x=0 N −1 pX f (x)(−1)x . x=0 From the above definition, we can derive I−1 (f1 ) + I−1 (f ) = 2f (0), where f1 (x) = f (x + 1). Consequently, it follows from [2, p. 1256], [4, p. 994], and [5, p. 366] that Z ∞ X tn x+y (1 + t) d µ−1 (y) = Chn (x) , n! Zp n=0 where Chn (x) are called the Changhee polynomials. We note that the Euler polynomials En (x) may also be represented by Z ∞ X 2 tn (x+y)t xt e d µ−1 (y) = t e = En (x) . e +1 n! Zp n=0 The purpose of this paper is to construct a new type of polynomials, the Appell type λ-Changhee polynomials, and to investigate some properties and identities of these polynomials. 2. Appell type λ-Changhee polynomials Assume that λ, t ∈ Cp such that | λt |p < p−1/(p−1) and (1 + λt)x/λ = ex ln(1+λt)/λ . Now we define the Appell type λ-Changhee polynomials Chn (x|λ) by Z ∞ X 2 tn xt ey ln(1+λt)/λ+xt d µ−1 (y) = Ch (x|λ) . e = n n! (1 + λt)1/λ + 1 Zp n=0 When x = 0, we call Chn (λ) = Chn (0|λ) the λ-Changhee numbers. Note that Chn (1) = Chn for n ≥ 0. Theorem 2.1. For n ≥ 0, we have n X n Chn (x|λ) = Chm (λ)xn−m . m m=0 Proof. From (2.1), we can derive ∞ X n=0 "∞ # ∞ ! " n # ∞ X X X xl X tn n tn Chn (x|λ) = Chn (λ) tl = . Chm (λ)xn−m n! l! m n! n=0 l=0 n=0 m=0 Equating coefficients on the very ends of the above identity arrives at the required result. Theorem 2.2. For n ≥ 0, we have n X k X n k−m Chn (x|λ) = λ Em (0)S1 (k, m)xn−k , k k=0 m=0 where S1 (n, m) is the Stirling number of the first kind. (2.1) D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876 1874 Proof. From Theorem 2.1, it follows that n n X X d n n−1 Chn (x|λ) = Chm (λ)(n − m)xn−m−1 = n Chm (λ)xn−m−1 dx m m−1 m=1 m=1 n−1 X n − 1 =n Chn−m−1 (λ)xm = nChn−1 (x|λ). m m=0 This means that Chn (x|λ) is an Appel sequence. Furthermore, we observe that Z y ln(1+λt)/λ+xt e Zp ∞ X ! ∞ ! X xl 1 m l d µ−1 (y) = y µ−1 (y) (ln(1 + λt)) t λ m! l! Zp m=0 l=0 ! ∞ ! ∞ ∞ k tk X xl X X λ = tl λ−m Em (0) S1 (k, m) k! l! m=0 l=0 k=m ! ! ∞ ! ∞ k k X xl X X t = λk−m Em (0)S1 (k, m) tl k! l! l=0 k=0 m=0 ! ∞ n k n X XX n k−m n−k t . = λ Em (0)S1 (k, m)x n! k n=0 −m Z m k=0 m=0 Combining this with (2.1) yields the required identity. Theorem 2.3. For n ≥ 0, we have n X Chm (x|λ)λn−m S2 (n, m) = m=0 n X n m=0 m Bm x m λ En−m (0), λ where S2 (m, n) is the Stirling number of the second kind. Proof. By replacing t by eλt −1 λ in (2.1), we obtain ∞ X 2 1 eλt − 1 m x(eλt −1)/λ e = Chm (x|λ) et + 1 m! λ m=0 ∞ X ∞ 1 X λn tn S (n, m) 2 λm n=m n! m=0 " # ∞ X n X tn n−m = . Chm (x|λ)λ S2 (n, m) n! = Chm (x|λ) (2.2) n=0 m=0 Recall from [1, p. 265] that the Bell polynomials Bn (x) are generated by e x(et −1) = ∞ X Bn (x) n=0 tn . n! Therefore, we acquire that 2 λt ex(e −1)/λ = t e +1 ∞ X El (0) l=0 l! ! t l ∞ X m=0 ! m m! X ∞ n X x λ t n x m tn Bm = Bm λ En−m (0) . λ m! m λ n! Comparing this with (2.2) leads to the required identity. n=0 m=0 D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876 1875 (r) For r ∈ N, define the higher order λ-Changhee polynomials Chn (x) by Z Z ··· e (x1 +···+xr ) ln(1+λt)/λ+xt Zrp 2 d µ−1 (x1 ) · · · d µ−1 (xr ) = (1 + λt)1/λ + 1 (r) r e xt = ∞ X Ch(r) n (x|λ) n=0 (r) When x = 0, we call Chn (λ) = Chn (0|λ) the higher order λ-Changhee numbers. Theorem 2.4. For n ≥ 1, we have Ch(r) n (x|λ) = n X n m m=0 n−m Ch(r) m (λ)x d (r) Chn(r) (x|λ) = nChn−1 (x|λ). dx and Proof. This follows from the observation that ∞ X tn Ch(r) n (x|λ) 2 = n! (1 + λt)1/λ + 1 n=0 r xt e " n # ∞ X n X n (r) n−m t = Chm (λ)x . m n! n=0 m=0 (r) Recall from [8, p. 12] that the higher order Euler polynomials En (x) may be represented by Z Z (x1 +···+xr +x)t ··· e Zrp 2 d µ−1 (x1 ) · · · d µ−1 (xr ) t e +1 r ext = ∞ X En(r) (x) n=0 tn . n! (r) When x = 0, we call En (0) the higher order modified Euler numbers. Theorem 2.5. For n ≥ 0, we have Ch(r) n (x|λ) n X k X n k−m (r) = λ Em (0)S1 (k, m)xn−k . k k=0 m=0 Proof. We observe that Z Z ··· e(x1 +···+xr ) ln(1+λt)/λ+xt d µ−1 (x1 ) · · · d µ−1 (xr ) Zrp ∞ Z X [ln(1 + λt)]m = ··· (x1 +· · ·+ xr ) d µ−1 (x1 ) · · · d µ−1 (xr ) m!λm Zrp m=0 ! ! ∞ ∞ ∞ X X X xl l λk tk (r) = t λ−m Em (0) S1 (k, m) k! l! m=0 l=0 k=m ! ! ! ∞ ∞ k k X xl X X t (r) tl = λk−m Em (0)S1 (k, m) k! l! k=0 m=0 l=0 ! ∞ n k n X XX n k−m (r) n−k t = λ Em (0)S1 (k, m)x . k n! n=0 Z m k=0 m=0 Combination of this identity with (2.3) results in the required identity. Theorem 2.6. For n ≥ 0, we have n X m=0 n−m λ Ch(r) m (x|λ)S2 (n, m) = n X m=0 x m (r) Bm λ En−m (0). λ ! ∞ X xl l=0 l! tl tn . (2.3) n! D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876 Proof. Substituting 2 t e +1 r eλt −1 λ for t in (2.3) gives λt −1)/λ ex(e ∞ X = Ch(r) m (x|λ) m=0 = ∞ X m=0 1876 1 Ch(r) m (x|λ) m λ m 1 1 λt e −1 m λ m! ∞ X " n # ∞ λn tn X X n−m (r) tn S2 (n, m) = λ Chm (x|λ)S2 (n, m) . n! n! n=m n=0 m=0 On the other hand, 2 t e +1 r x(eλt −1)/λ e = "∞ X l=0 tl (r) El (0) l! #" ∞ X m=0 " n # m m# X ∞ X x λ t x m (r) tn Bm = Bm λ En−m (0) . λ m! λ n! n=0 m=0 The required result thus follows. Remark 2.7. This paper is a slightly modified version of the preprint [7]. References [1] E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258–277. 2 [2] D. V. Dolgy, T. Kim, S.-H. Rim, J.-J. Seo, A note on Changhee polynomials and numbers with q-parameter, Int. J. Math. Anal., 8 (2014), 1255–1264. 1 [3] T. 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