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Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 4 (2016), pp. 2819–2827 © Research India Publications http://www.ripublication.com/gjpam.htm Symmetric properties for the degenerate q-tangent polynomials associated with p-adic integral on Zp C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract In [5], we studied the degenerate q-tangent numbers and polynomials associated with p-adic integral on Zp . In this paper, by using the symmetry of p-adic integral on Zp , we give recurrence identities the degenerate q-tangent polynomials and the generalized factorial sums. AMS subject classification: Keywords: Degenerate tangent numbers and polynomials, degenerate q-tangent numbers and polynomials, generalized factorial sums. 1. Introduction L. Carlitz introduced the degenerate Bernoulli polynomials(see [1]). Feng Qi et al. [2] studied the partially degenerate Bernoull polynomials of the first kind in p-adic field. T. Kim studied the Barnes’ type multiple degenerate Bernoulli and Euler polynomials (see [3]), Recently, Ryoo introduced the degenerate q-tangent numbers Tn,q (λ) and polynomials Tn,q (x, λ) (see [5]). In this paper, by using these numbers and polynomials, we give some interesting relations between the generalized factorial sums and the degenerate q-tangent polynomials. Let p be a fixed odd prime number. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of rational numbers, N denotes the set of natural numbers, C denotes the complex number field, Cp denotes the completion of algebraic closure of Qp , N denotes the set of natural numbers and Z+ = N ∪ {0}, and C denotes the set of complex numbers. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assumes that 2820 C. S. Ryoo |q| < 1. If q ∈ Cp , we normally assume that |q −1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. For 1 g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic invariant integral on Zp is defined by Kim as follows: I−1 (g) = Zp N −1 p g(x)dµ−1 (x) = lim N →∞ g(x)(−1)x , (see [2, 3]). (1.1) x=0 If we take g1 (x) = g(x + 1) in (1.1), then we see that I−1 (g1 ) + I−1 (g) = 2g(0), (see [2, 3]). (1.2) We recall that the classical Stirling numbers of the first kind S1 (n, k) and S2 (n, k) are defined by the relations (see [7]) (x)n = n S1 (n, k)x and x = k n k=0 n S2 (n, k)(x)k , k=0 respectively. Here (x)n = x(x −1) · · · (x −n+1) denotes the falling factorial polynomial of order n. We also have ∞ ∞ tn tn (et − 1)m (log(1 + t))m S2 (n, m) = S1 (n, m) = and . n! m! n! m! n=m n=m (1.3) The generalized falling factorial (x|λ)n with increment λ is defined by (x|λ)n = n−1 (x − λk) (1.4) k=0 for positive integer n, with the convention (x|λ)0 = 1. We also need the binomial theorem: for a variable x, (1 + λt)x/λ = ∞ tn (x|λ)n . n! (1.5) n=0 For t, λ ∈ Zp such that |λt|p < p − p−1 , if we take g(x) = q x (1 + λt)2x/λ in (1.2), then we easily see that 2 q x (1 + λt)2x/λ dµ−1 (x) = . q(1 + λt)2/λ + 1 Zp 1 Symmetric properties for the degenerate q-tangent polynomials 2821 Let us define the degenerate q-tangent numbers Tn,q (λ) and polynomials Tn,q (x, λ) as follows: ∞ tn y 2y/λ (1.6) q (1 + λt) dµ−1 (y) = Tn,q (λ) , n! Zp n=0 ∞ tn y (2y+x)/λ q (1 + λt) dµ−1 (y) = Tn,q (x, λ) . (1.7) n! Zp n=0 By (1.6) and (1.7), we obtain the following Witt’s formula. Theorem 1.1. For n ≥ 0, we have Tn,q (x, λ) = n n l=0 l Tl,q (λ)(x|λ)n−l . Theorem 1.2. For n ∈ Z+ , we have q x (2x|λ)n dµ−1 (x) = Tn,q (λ), Z p q y (x + 2y|λ)n dµ−1 (y) = Tn,q (x, λ). Zp Recently, many mathematicians have studied in the area of the q-analogues of the degenerate Bernoulli umbers and polynomials, Euler numbers and polynomials, tangent numbers and polynomials (see [2, 3, 5, 7]). Our aim in this paper is to obtain symmetric properties for the degenerate q-tangent numbers and polynomials. We investigate some properties which are related to degenerate q-tangent polynomials Tn,q (x, λ) and the generalized factorial sums. 2. The alternating generalized factorial sums and q-tangent polynomials In this section, we assume that q ∈ C with |q| < 1. By using (1.6), we give the alternating generalized factorial sums as follows: ∞ n=0 ∞ 2 tn Tn,q (λ) = (−1)n q n (1 + λt)2n/λ . = 2 2/λ n! q(1 + λt) + 1 n=0 From the above, we obtain − = ∞ ∞ (−1)n q n (1 + λt)(2n+2k)/λ + (−1)n−k q (n−k) (1 + λt)2n/λ n=0 k−1 (−1)n−k q (n−k) (1 + λt)2n/λ . n=0 n=0 2822 C. S. Ryoo By using (1.6) and (1.7), we obtain ∞ ∞ 1 1 tj tj −k −k − Tj,q (2k) + (−1) q Tj,q 2 j! 2 j! j =0 j =0 k−1 ∞ tj . (−1)−k q −k (−1)n q n (2n|λ)j = j! j =0 By comparing coefficients of n=0 tj in the above equation, we obtain j! k−1 (−1)k+1 q k Tj,q (2k) + Tj,q . (−1)n q n (2n|λ)j = 2 n=0 By using the above equation we arrive at the following theorem: Theorem 2.1. Let k be a positive integer and q ∈ C with |q| < 1. Then we obtain Sj,q (k − 1, λ) = k−1 (−1)k+1 q k Tj,q (2k) + Tj,q . (−1)n q n (2n|λ)j = 2 (2.1) n=0 Remark 2.2. For the alternating generalized factorial sums, we have lim Sj,q (k − 1) = q→1 k−1 (−1)k+1 Tj (2k) + Tj (−1)n (2n|λ)j = , 2 n=0 where Tj (x) and Tj denote the tangent polynomials and the tangent numbers, respectively (see [6]). 3. Symmetry properties of the q-deformed fermionic integral on Zp In this section, we assume that q ∈ Cp . In this section, we obtain recurrence identities the degenerate q-tangent polynomials and the alternating generalized factorial sums. By using (1.1), we have I−1 (gn ) + (−1) n−1 n−1 I−1 (g) = 2 (−1)n−1−k g(k), k=0 where n ∈ N, gn (x) = g(x + n). If n is odd from the above, we obtain n−1 (−1)n−1−k g(k) (see [2], [3], [4], [5]). I−1 (gn ) + I−1 (g) = 2 k=0 (3.1) Symmetric properties for the degenerate q-tangent polynomials 2823 It will be more convenient to write (3.1) as the equivalent integral form Zp g(x + n)dµ−1 (x) + Zp n−1 g(x)dµ−1 (x) = 2 (−1)n−1−k g(k). (3.2) k=0 Substituting g(x) = q x (1 + λt)2x/λ into the above, we obtain (x+n) (2x+2n)/λ q (1 + λt) dµ−1 (x) + q x (1 + λt)2x/λ dµ−1 (x) Zp Zp (3.3) n−1 (−1)j q j (1 + λt)2j/λ . =2 j =0 After some calculations, we have 2 , q x (1 + λt)2x/λ dµ−1 (x) = q(1 + λt)2/λ + 1 Zp q (x+n) (1 + λt)(2x+2n)/λ dµ−1 (x) = q n (1 + λt)2n/λ Zp By using (3.3) and (3.4), we have (x+n) (2x+2n)/λ q (1 + λt) dµ−1 (x) + Zp = 2(1 + q n (1 + λt)2n/λ ) q(1 + λt)2/λ + 1 = Zp q x (1 + λt)2x/λ dµ−1 (x) By (3.3), we obtain ∞ (x+n) q (2x + 2n|λ)m dµ−1 (x) + = q x (1 + λt)2x/λ dµ−1 (x) (3.5) 2 Z q x (1 + λt)2x/λ dµ−1 (x) p . nx 2nx/λ dµ (x) −1 Zp q (1 + λt) m=0 ∞ m=0 Zp 2 n−1 tm j j (−1) q (2j |λ)m m! j =0 (3.4) . From the above, we get (x+n) (2x+2n)/λ q (1 + λt) dµ−1 (x) + Zp Zp 2 . q(1 + λt)2/λ + 1 Zp q x (2x|λ)m dµ−1 (x) tm m! 2824 C. S. Ryoo tm in the above equation, we obtain m! m m x n q x (2x|λ)m dµ−1 (x) q (2x|λ)k dµ−1 (x) + (2n|λ)m−k q k Zp Zp By comparing coefficients k=0 n−1 (−1)j q j (2j |λ)m =2 j =0 By using (2.1), we have m m n m−k x k q (2x) dµ−1 (x) + q x (2x)m dµ−1 (x) (2n) q k Zp Zp k=0 (3.6) = 2Sm,q (n − 1, λ). By using (3.5) and (3.6), we arrive at the following theorem: Theorem 3.1. Let n be odd positive integer. Then we obtain ∞ 2 Zp q x (1 + λt)2x/λ dµ−1 (x) tm . 2S (n − 1, λ) = m,q nx 2nx/λ dµ (x) m! −1 Zp q (1 + λt) (3.7) m=0 Let w1 and w2 be odd positive integers. By using (3.7), we have (w1 x1 +w2 x2 ) (1 + λt)(w1 2x1 +w2 2x2 +w1 w2 x)/λ dµ (x )dµ (x ) −1 1 −1 2 Zp Zp q w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x) −1 Zp q (3.8) 2(1 + λt)w1 w2 x/λ q w1 w2 (1 + λt)2w1 w2 /λ + 1 = w (q 1 (1 + λt)2w1 /λ + 1)(q w2 (1 + λt)2w2 /λ + 1) By using (3.7) and (3.8), after elementary calculations, we obtain 1 a= q w1 x1 (1 + λt)(w1 2x1 +w1 w2 x)/λ dµ−1 (x1 ) 2 Zp 2 Zp q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 ) × w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x) −1 Zp q ∞ ∞ m m 1 t t λ λ = Tm,q w1 w2 x, Sm,q w2 w1 − 1, w1m w2m 2 . 2 w1 m! w2 m! m=0 m=0 (3.9) By using Cauchy product in the above, we have m ∞ λ λ m tm j m−j Tj,q w1 w2 x, w1 Sm−j,q w2 w1 − 1, w2 . a= w1 w2 m! j m=0 j =0 (3.10) Symmetric properties for the degenerate q-tangent polynomials 2825 By using the symmetry in (3.9), we have 1 a= q w2 x2 (1 + λt)(w2 2x2 +w1 w2 x)/λ dµ−1 (x2 ) 2 Zp 2 Zp q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 ) × w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x) −1 Zp q ∞ ∞ m m 1 t t λ λ = 2 . Tm,q w2 w1 x, Sm,q w1 w2 − 1, w2m w1m 2 w2 m! w1 m! m=0 m=0 Thus we have a= ∞ m=0 m λ λ m tm j m−j Tj,q w2 w1 x, w2 Sm−j,q w1 w2 − 1, w1 w2 w1 m! j j =0 (3.11) tm in the both sides of (3.10) and (3.11), we arrive at the By comparing coefficients m! following theorem: Theorem 3.2. Let w1 and w2 be odd positive integers. Then we obtain m m λ λ j m−j Tj,q w2 w1 x, Sm−j,q w1 w2 − 1, w2 w1 w2 w1 j j =0 m λ λ m j m−j = Tj,q w1 w2 x, Sm−j,q w2 w1 − 1, w1 w2 , w1 w2 j j =0 where Tk,q (x) and Tm,q (k) denote the degenerate q-tangent polynomials and the alternating generalized factorial sums, respectively (see [5]). By using Theorem 2, we have the following corollary: Corollary 3.3. Let w1 and w2 be odd positive integers. Then we obtain j m m j j =0 k=0 = j k m−j j w1 w2 j m m j j =0 k=0 j k λ w1 x w2 j m−j w1 w2 j −k λ w2 x w1 Tk,q w2 λ Sm−j,q w1 (w2 − 1) w2 j −k Tk,q w1 λ Sm−j,q w2 (w1 − 1). w1 2826 C. S. Ryoo By using (3.8), we have 1 a= (1 + λt)w1 w2 x/λ q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 ) 2 Zp 2 Zp q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 ) × w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x) −1 Zp q 1 = q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 ) (1 + λt)w1 w2 x/λ 2 Zp w 1 −1 × 2 (−1)j q w2 j (1 + λt)2j w2 /λ j =0 = w 1 −1 (−1)j q w2 j Zp j =0 = ∞ w 1 −1 2j w2 2x1 +w2 x+ w1 q w1 x1 (1 + λt) j =0 n=0 (w1 )/λ dµ−1 (x1 ) (−1)j q w2 j Tn,q w1 w2 x + (3.12) 2j w2 λ tn , w1n . w1 w1 n! By using the symmetry property in (3.12), we also have 1 q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 ) (1 + λt)w1 w2 x/λ a= 2 Zp w x 2 Zp q 1 1 (1 + λt)2x1 w1 /λ dµ−1 (x1 ) × w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x) −1 Zp q 1 = (1 + λt)w1 w2 x/λ q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 ) 2 Zp w 2 −1 × 2 (−1)j q w1 j (1 + λt)2j w1 /λ j =0 = w 2 −1 (−1)j q w1 j j =0 = ∞ n=0 w 2 −1 j =0 Zp 2j w1 2x2 +w1 x+ w2 q w2 x2 (1 + λt) (−1)j q w1 j Tn,q w2 w1 x + (3.13) (w2 )/λ dµ−1 (x1 ) 2j w1 λ tn , w2n . w2 w2 n! tn By comparing coefficients in the both sides of (3.12) and (3.13), we have the following n! theorem. Symmetric properties for the degenerate q-tangent polynomials 2827 Theorem 3.4. Let w1 and w2 be odd positive integers. Then we obtain w 1 −1 j w2 j (−1) q T n,q w1 j =0 = w 2 −1 2j w2 λ , w2 x + w1n w1 w1 2j w1 λ , w2n . w1 x + w2 w2 j w1 j (−1) q Tn,q w2 j =0 Observe that if λ → 0, then Theorem 9 reduces to Theorem 3.4 in [4]. Substituting w1 = 1 into Theorem 9, we have the following corollary. Corollary 3.5. Let w2 be odd positive integer. Then we obtain Tn,q (x) = w2n w 2 −1 j =0 (−1) q Tn,q w2 j j x + 2j λ , . w2 w2 Acknowledgment This work was supported by 2016 Hannam University Research Fund References [1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15(1979), 51–88. [2] F. Qi, D. V. Dolgy, T. Kim, C. S. Ryoo, On the partially degenerate Bernoulli polynomials of the first kind, Global Journal of Pure and Applied Mathematics, 11(2015), 2407–2412. [3] T. Kim, Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258(2015), 556–564 [4] C. S. Ryoo, On the symmetric properties for the q-tangent polynomials, Int. Journal of Math. Analysis, 7(2013), 3027–3033. [5] C. S. Ryoo, On the degenerate q-tangent numbers and polynomials associated with the p-adic integral on Zp , International Journal of Mathematical Analysis, 9(2015), 2571–2579. [6] C. S. Ryoo, A Note on the Tangent Numbers and Polynomials, Adv. Studies Theor. Phys., 7(9)(2013), 447–454. [7] P. T.Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theory, 128(2008), 738–758.