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 polyno- mials 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. −νp(p) −1 Let νp be the normalized exponential valuation of Cp with |p|p = p = p . 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 1 − − x |q| < 1. If q ∈ Cp, we normally assume that |q −1|p <p p 1 so that q = exp(x log q) for |x|p ≤ 1. For g ∈ UD(Zp) ={g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic invariant integral on Zp is defined by Kim as follows: pN −1 x I−1(g) = g(x)dµ−1(x) = lim g(x)(−1) ,(see [2, 3]). (1.1) Z N→∞ p 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]) n n k n (x)n = S1(n, k)x and x = S2(n, k)(x)k, k=0 k=0 respectively. Here (x)n = x(x−1) ···(x−n+1) denotes the falling factorial polynomial of order n. We also have ∞ ∞ tn (et − 1)m tn (log(1 + t))m S (n, m) = and S (n, m) = .(1.3) 2 n! m! 1 n! m! n=m n=m The generalized falling factorial (x|λ)n with increment λ is defined by n−1 (x|λ)n = (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, ∞ tn (1 + λt)x/λ = (x|λ) .(1.5) n n! n=0 1 − − x 2x/λ For t,λ ∈ Zp such that |λt|p <p p 1 , if we take g(x) = q (1 + λt) in (1.2), then we easily see that x 2x/λ 2 q (1 + λt) dµ− (x) = . 1 + 2/λ + Zp q(1 λt) 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: ∞ n y 2y/λ t q (1 + λt) dµ−1(y) = Tn,q(λ) ,(1.6) Z n! p n=0 ∞ n y (2y+x)/λ t q (1 + λt) dµ−1(y) = Tn,q(x, λ) .(1.7) Z n! p n=0 By (1.6) and (1.7), we obtain the following Witt’s formula. Theorem 1.1. For n ≥ 0, we have n n T (x, λ) = T (λ)(x|λ) − . n,q l l,q n l l=0 Theorem 1.2. For n ∈ Z+, we have x q (2x|λ)ndµ−1(x) = Tn,q(λ), Z p y q (x + 2y|λ)ndµ−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: ∞ ∞ tn 2 T (λ) = = 2 (−1)nqn(1 + λt)2n/λ. n,q n! q(1 + λt)2/λ + 1 n=0 n=0 From the above, we obtain ∞ ∞ + − − − (−1)nqn(1 + λt)(2n 2k)/λ + (−1)n kq(n k)(1 + λt)2n/λ n=0 n=0 k−1 − − = (−1)n kq(n k)(1 + λt)2n/λ. n=0 2822 C. S. Ryoo By using (1.6) and (1.7), we obtain ∞ ∞ 1 tj 1 tj − T ( k) + (− )−kq−k T j,q 2 ! 1 j,q ! 2 = j 2 = j j 0 j 0 ∞ k−1 j − − t = (−1) kq k (−1)nqn(2n|λ) . j j! j=0 n=0 tj By comparing coefficients of in the above equation, we obtain j! − k 1 (−1)k+1qkT (2k) + T (−1)nqn(2n|λ) = j,q j,q . 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 − k 1 (−1)k+1qkT (2k) + T S (k − 1,λ)= (−1)nqn(2n|λ) = j,q j,q .(2.1) j,q j 2 n=0 Remark 2.2. For the alternating generalized factorial sums, we have k−1 − k+1T + T n ( 1) j (2k) j lim Sj,q(k − 1) = (−1) (2n|λ)j = , q→1 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 n−1 n−1 n−1−k I−1(gn) + (−1) I−1(g) = 2 (−1) g(k), k=0 where n ∈ N,gn(x) = g(x + n).Ifn is odd from the above, we obtain n−1 n−1−k I−1(gn) + I−1(g) = 2 (−1) g(k) (see [2], [3], [4], [5]).(3.1) k=0 Symmetric properties for the degenerate q-tangent polynomials 2823 It will be more convenient to write (3.1) as the equivalent integral form n−1 n−1−k g(x + n)dµ−1(x) + g(x)dµ−1(x) = 2 (−1) g(k). (3.2) Z Z p p k=0 Substituting g(x) = qx(1 + λt)2x/λ into the above, we obtain (x+n) (2x+2n)/λ x 2x/λ q (1 + λt) dµ−1(x) + q (1 + λt) dµ−1(x) Zp Zp n−1 (3.3) = 2 (−1)j qj (1 + λt)2j/λ. j=0 After some calculations, we have x 2x/λ 2 q (1 + λt) dµ−1(x) = , Z q(1 + λt)2/λ + 1 p (3.4) (x+n) (2x+2n)/λ n 2n/λ 2 q (1 + λt) dµ− (x) = q (1 + λt) . 1 + 2/λ + Zp q(1 λt) 1 By using (3.3) and (3.4), we have (x+n) (2x+2n)/λ x 2x/λ q (1 + λt) dµ−1(x) + q (1 + λt) dµ−1(x) Zp Zp 2(1 + qn(1 + λt)2n/λ) = . q(1 + λt)2/λ + 1 From the above, we get (x+n) (2x+2n)/λ x 2x/λ q (1 + λt) dµ−1(x) + q (1 + λt) dµ−1(x) Z Z p p x 2x/λ (3.5) 2 Z q (1 + λt) dµ−1(x) = p . qnx(1 + λt)2nx/λdµ− (x) Zp 1 By (3.3), we obtain ∞ m (x+n) x t q (2x + 2n|λ)mdµ−1(x) + q (2x|λ)mdµ−1(x) Z Z m! m=0 p p ∞ − n 1 tm = 2 (−1)j qj (2j|λ) m m! m=0 j=0 2824 C. S. Ryoo tm By comparing coefficients in the above equation, we obtain m! m n m x x q (2n|λ)m−k q (2x|λ)kdµ−1(x) + q (2x|λ)mdµ−1(x) k Z Z k=0 p p n−1 j j = 2 (−1) q (2j|λ)m j=0 By using (2.1), we have m n m m−k x k x m q (2n) q (2x) dµ−1(x) + q (2x) dµ−1(x) k Z Z k=0 p p (3.6) = 2Sm,q(n − 1, λ).