Applied Mathematical Sciences, Vol. 9, 2015, no. 142, 7071 - 7079 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.510671

Symmetric Properties for the Degenerate Tangent Polynomials

Associated with p-Adic Integral on Zp

C. S. Ryoo

Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Copyright c 2015 C. S. Ryoo. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In [7], we studied the degenerate tangent numbers and polynomials. By using these numbers and polynomials, we give some interesting rela- tions between the generalized falling factorial sums and the degenerate tangent polynomials.

Mathematics Subject Classification: 11B68, 11S40, 11S80

Keywords: tangent numbers and polynomials, power sums

1 Introduction

Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, and tangent numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. L. Carlitz introduced the degenerate Bernoulli polynomials(see [1, 2]). Feng Qi et al.[3] 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 [4]), Recently, Ryoo introduced the degenerate tan- gent numbers and tangent polynomials(see [7]). By using these numbers and polynomials, we investigate some interesting relations between the alternating generalized falling factorial sums and the degenerate tangent polynomials. Let 7072 C. S. Ryoo 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 −νp(p) −1 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 |q| < 1. If q ∈ Cp, we − 1 x 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: Z pN −1 X x I−1(g) = g(x)dµ−1(x) = lim g(x)(−1) , (see [3]). (1.1) N→∞ Zp 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 [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 [8])

n n X k n X (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

∞ ∞ X tn (et − 1)m X 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 Y (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,

∞ X tn (1 + λt)x/λ = (x|λ) . (1.5) n n! n=0 Symmetric properties for degenerate tangent polynomials 7073

− 1 2x/λ For t, λ ∈ Zp such that |λt|p < p p−1 , if we take g(x) = (1 + λt) in (1.2), then we easily see that

Z 2 (1 + λt)2x/λdµ (x) = . −1 q(1 + λt)2/λ + 1 Zp

Let us define the degenerate tangent numbers Tn(λ) and polynomials Tn(x, λ) as follows: ∞ Z X tn (1 + λt)2y/λdµ (y) = T (λ) , (1.6) −1 n n! Zp n=0

∞ Z X tn (1 + λt)(2y+x)/λdµ (y) = T (x, λ) . (1.7) −1 n n! Zp n=0

The following elementary properties of the degenerate tangent numbers Tn(λ) and polynomials Tn(x, λ) are readily derived form (1.6) and (1.7)(see, for de- tails, [7]).

Theorem 1.1 For n ∈ Z+, we have Z Z (2x|λ)ndµ−1(x) = Tn(λ), (x + 2y|λ)ndµ−1(y) = Tn(x, λ). Zp Zp

Theorem 1.2 For n ≥ 0, we have

n X n T (x, λ) = T (λ)(x|λ) . n l l n−l l=0

Recently, many mathematicians have studied in the area of the analogues of the degenerate Bernoulli umbers and polynomials, Euler numbers and polyno- mials, tangent numbers and polynomials(see [1, 2, 3, 4, 7, 8]). Our aim in this paper is to obtain symmetric properties for the degenerate tangent polynomi- als. We investigate some properties which are related to tangent polynomials Tn(x, λ) and the generalized factorial sums.

2 Alternating generalized falling factorial sums

By using (1.6), we give the alternating generalized factorial sums as follows:

∞ ∞ X tn 2 X T (λ) = = 2 (−1)n(1 + λt)2n/λ. n n! (1 + λt)2/λ + 1 n=0 n=0 7074 C. S. Ryoo

From the above, we obtain

∞ ∞ X X − (−1)n(1 + λt)(2n+2k)/λ + (−1)n−k(1 + λt)2n/λ n=0 n=0 k−1 X = (−1)n−k(1 + λt)2n/λ. n=0

By using (1.6)and (1.7), we obtain

∞ ∞ ∞ k−1 ! 1 X tj 1 X tj X X tj − T (2k, λ) + (−1)−k T (λ) = (−1)−k (−1)n(2n|λ) . 2 j j! 2 j j! j j! j=0 j=0 j=0 n=0

tj By comparing coefficients of in the above equation, we obtain j!

k−1 k+1 X (−1) Tj(2k, λ) + Tj(λ) (−1)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. Then we obtain

k−1 k+1 X (−1) Tj(2k, λ) + Tj(λ) S (k − 1, λ) = (−1)n(2n|λ) = , (2.1) j j 2 n=0 where Sj(k, λ) are called the alternating generalized falling factorial sums.

3 Symmetry property of the deformed fermionic integral on Zp In this section, we obtain recurrence identities the degenerate tangent poly- nomials and the alternating generalized falling factorial sums. By using (1.1), we have n−1 n−1 X n−1−k I−1(gn) + (−1) I−1(g) = 2 (−1) g(k), k=0 where n ∈ N, gn(x) = g(x + n). If n is odd from the above, we obtain

n−1 X k I−1(gn) + I−1(g) = 2 (−1) g(k) (see [1], [2], [3], [5]). (3.1) k=0 Symmetric properties for degenerate tangent polynomials 7075

It will be more convenient to write (3.1) as the equivalent integral form Z Z n−1 X k g(x + n)dµ−1(x) + g(x)dµ−1(x) = 2 (−1) g(k). (3.2) Zp Zp k=0 Substituting g(x) = (1 + λt)2x/λ into the above, we obtain Z Z (2x+2n)/λ 2x/λ (1 + λt) dµ−1(x) + (1 + λt) dµ−1(x) Zp Zp n−1 (3.3) X = 2 (−1)j(1 + λt)2j/λ. j=0 After some calculations, we have Z 2 (1 + λt)2x/λdµ (x) = , −1 (1 + λt)2/λ + 1 Zp (3.4) Z 2 (1 + λt)(2x+2n)/λdµ (x) = (1 + λt)2n/λ . −1 (1 + λt)2/λ + 1 Zp By using (3.3) and (3.4), we have Z Z 2(1 + (1 + λt)2n/λ) (1 + λt)(2x+2n)/λdµ (x) + (1 + λt)2x/λdµ (x) = . −1 −1 (1 + λt)2/λ + 1 Zp Zp From the above, we get Z Z (2x+2n)/λ 2x/λ (1 + λt) dµ−1(x) + (1 + λt) dµ−1(x) Zp Zp R 2x/λ (3.5) 2 (1 + λt) dµ−1(x) = Zp . R 2nx/λ (1 + λt) dµ−1(x) Zp By (3.3), we obtain ∞ ! X Z Z tm (2x + 2n|λ) dµ (x) + (2x|λ) dµ (x) m −1 m −1 m! m=0 Zp Zp ∞ n−1 ! X X tm = 2 (−1)j(2j|λ) m m! m=0 j=0 tm By comparing coefficients in the above equation, we obtain m! m X m Z Z (2n|λ) (2x|λ) dµ (x) + (2x|λ) dµ (x) k m−k k −1 m −1 k=0 Zp Zp n−1 X j = 2 (−1) (2j|λ)m j=0 7076 C. S. Ryoo

By using (2.1), we have

m X m Z Z (2n)m−k (2x)kdµ (x) + (2x)mdµ (x) k −1 −1 k=0 Zp Zp (3.6)

= 2Sm(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

R 2x/λ ∞ 2 (1 + λt) dµ−1(x) m p X t Z = (2S (n − 1, λ)) . (3.7) R 2nx/λ m (1 + λt) dµ−1(x) m! Zp m=0

Let w1 and w2 be odd positive integers. By using (3.7), we have

R R (w12x1+w22x2+w1w2x)/λ (1 + λt) dµ−1(x1)dµ−1(x2) Zp Zp R 2w w x/λ (1 + λt) 1 2 dµ−1(x) Zp (3.8) 2(1 + λt)w1w2x/λ (1 + λt)2w1w2/λ + 1
= ((1 + λt)2w1/λ + 1)((1 + λt)2w2/λ + 1)

By using (3.7) and (3.8), after elementary calculations, we obtain

! R 2x2w2/λ ! 1 Z 2 (1 + λt) dµ−1(x2) a = (1 + λt)(w12x1+w1w2x)/λdµ (x ) Zp −1 1 R 2w w x/λ 2 (1 + λt) 1 2 dµ−1(x) Zp Zp ∞ ! ∞ ! 1 X  λ  tm X  λ  tm = T w x, wm 2 S w − 1, wm . 2 m 2 w 1 m! m 1 w 2 m! m=0 1 m=0 2 (3.9) By using Cauchy product in the above, we have

∞ m ! X X m  λ   λ  tm a = T w x, wjS w − 1, wm−j . (3.10) j j 2 w 1 m−j 1 w 2 m! m=0 j=0 1 2

By using the symmetry in (3.9), we have

! R 2x1w1/λ ! 1 Z 2 (1 + λt) dµ−1(x1) a = (1 + λt)(w22x2+w1w2x)/λdµ (x ) Zp −1 2 R 2w w x/λ 2 (1 + λt) 1 2 dµ−1(x) Zp Zp ∞ ! ∞ ! 1 X  λ  tm X  λ  tm = T w x, wm 2 S w − 1, wm . 2 m 1 w 2 m! m 2 w 1 m! m=0 2 m=0 1 Symmetric properties for degenerate tangent polynomials 7077

Thus we have

∞ m ! X X m  λ   λ  tm a = T w x, wjS w − 1, wm−j . (3.11) j j 1 w 2 m−j 2 w 1 m! m=0 j=0 2 1

tm By comparing coefficients in the both sides of (3.10) and (3.11), we arrive m! at the following theorem:

Theorem 3.2 Let w1 and w2 be odd positive integers. Then we obtain

m X m  λ   λ  T w x, S w − 1, wjwm−j j j 1 w m−j 2 w 2 1 j=0 2 1 m X m  λ   λ  = T w x, S w − 1, wjwm−j, j j 2 w m−j 1 w 1 2 j=0 1 2 where Tk(x, λ) and Sm(s, λ) denote the degenerate tangent polynomials and the alternating generalized falling factorial sums, respectively.

By using Theorem 1.2, we have the following corollary:

Corollary 3.3 Let w1 and w2 be odd positive integers. Then we obtain

m j        X X m j m−j j λ λ w1 w2 w1x Tk Sm−j(w2 − 1) j k w2 w2 j=0 k=0 j−k m j        X X m j j m−j λ λ = w1w2 w2x Tk Sm−j(w1 − 1). j k w1 w1 j=0 k=0 j−k

By using (3.8), we have

! 1 Z a = (1 + λt)w1w2x/λ (1 + λt)2x1w1/λdµ (x ) 2 −1 1 Zp

R 2x2w2/λ ! 2 (1 + λt) dµ−1(x2) × Zp R 2w w x/λ (3.12) (1 + λt) 1 2 dµ−1(x) Zp ∞ w1−1   ! n X X j 2jw2 λ n t = (−1) T w1 w x + , w . n 2 w w 1 n! n=0 j=0 1 1 7078 C. S. Ryoo

By using the symmetry property in (3.12), we also have ! 1 Z a = (1 + λt)w1w2x/λ (1 + λt)2x2w2/λdµ (x ) 2 −1 2 Zp

R 2x1w1/λ ! 2 (1 + λt) dµ−1(x1) × Zp R 2w w x/λ (3.13) (1 + λt) 1 2 dµ−1(x) Zp ∞ w2−1   ! n X X 2jw1 λ t = (−1)jT w x + , wn . n 1 w w 2 n! n=0 j=0 2 2

tn By comparing coefficients in the both sides of (3.12) and (3.13), we have n! the following theorem.

Theorem 3.4 Let w1 and w2 be odd positive integers. Then we obtain

w1−1   X 2jw2 λ (−1)jT w x + , wn n 2 w w 1 j=0 1 1 (3.14) w2−1   X 2jw1 λ = (−1)jT w x + , wn. n 1 w w 2 j=0 2 2

Observe that if λ → 0, then (3.14) reduces to Theorem 3.4 in [6]. Substituting w1 = 1 into (3.14), we have the following corollary.

Corollary 3.5 Let w2 be odd positive integer. Then we obtain

w −1 X2 x + 2j λ  T (x, λ) = wn (−1)jT , . n 2 n w w j=0 2 2

References

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[3] 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. Symmetric properties for degenerate tangent polynomials 7079

[4] Tae Kyun Kim, Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput., 258 (2015), 556-564. http://dx.doi.org/10.1016/j.amc.2015.02.040

[5] C. S. Ryoo, A note on the tangent Numbers and Polynomials, Adv. Studies Theor. Phys., 7 (2013), no. 9, 447 - 454.

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Received: November 9, 2015; Published: December 12, 2015