Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1567-1574 © Research India Publications http://www.ripublication.com/gjpam.htm

Modified degenerate tangent numbers and polynomials

Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea.

Abstract In this paper, we introduce the modified degenerate tangent numbers and poly- nomials. We also obtain some explicit formulas for modified degenerate tangent numbers and polynomials.

AMS subject classification: 11B68, 11S40, 11S80. Keywords: Tangent numbers and polynomials, degenerate tangent numbers and polynomials, modified degenerate tangent numbers and polynomials.

1. Introduction Recently, many mathematicians have studied in the area of degenerate Bernoulli poly- nomials and Euler polynomials (see [1, 2, 3, 4, 6]). For example, in [2], L. Carlitz introduced the degenerate Bernoulli polynomials. In [3], Feng Qi et al. studied the par- tially degenerate Bernoull polynomials of the first kind in p-adic field. In this paper, we introduce the modified degenerate tangent numbers and polynomials. We also establish some interesting properties for modified degenerate tangent numbers and polynomials. Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers and Z+ = N ∪{0}, C denotes the set of complex numbers. The Tangent numbers are defined by means of the following generating function: ∞  
tn 2 π T = |t| < . (1.1) n n! e2t + 1 2 n=0 In [5], we defined Tangent polynomials by multiplying ext on the right side of the Eq. (1.1) as follows: ∞    
tn 2 π T (x) = ext |t| < . (1.2) n n! e2t + 1 2 n=0 1568 Cheon Seoung Ryoo

For more theoretical properties of the tangent numbers and polynomials, the readers may refer to [4, 6, 7]. 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. The numbers S2(n, m) also admit a representation in terms of a generating function ∞
tn (et − 1)m S (n, m) = .(1.3) 2 n! m! n=m We also have ∞
tn (log(1 + t))m S (n, m) = .(1.4) 1 n! m! n=m

The generalized falling factorial (x|λ)n with increment λ is defined by

n−1 (x|λ)n = (x − λk) (1.5) k=0 for positive integer n, with the convention (x|λ)0 = 1; we may also write

n n−k k (x|λ)n = S1(n, k)λ x .(1.6) k=0

Note that (x|λ) is a homogeneous polynomials in λ and x of degree n,soifλ = 0 then n −1 n (x|λ)n = λ (λ x|1)n. Clearly (x|0)n = x . We also need the binomial theorem: for a variable x, ∞
tn (1 + λt)x/λ = (x|λ) .(1.7) n n! n=0 For a variable t, we consider the degenerate tangent polynomials which are given by the generating function to be

∞ 2
tn (1 + λt)x/λ = T (x, λ) .(1.8) (1 + λt)2/λ + 1 n n! n=0

When x = 0, Tn(0,λ)= Tn(λ) are called the degenerate tangent numbers(see [6]). Note that (1 + λt)1/λ tends to et as λ → 0. In [6], we investigated some properties which are related to degenerate tangent numbers Tn(λ) and polynomials Tn(x, λ). Modified degenerate tangent numbers and polynomials 1569

2. On modified degenerate tangent numbers and polynomials

In this section, we define the modified degenerate tangent numbers and polynomials, and we obtain explicit formulas for them. For a variable t, we consider the modified degenerate tangent polynomials which are given by the generating function to be

∞ 2
tn (1 + λ)xt/λ = T (x) .(2.1) (1 + λ)2t/λ + 1 n,λ n! n=0

When x = 0, Tn,λ(0) = Tn,λ are called the modified degenerate tangent numbers. Note that (1 + λ)t/λ tends to et as λ → 0. From (2.1) and (1.2), we note that

∞ n t 2 xt/λ lim Tn,λ(x) = lim (1 + λ) λ→0 n! λ→0 (1 + λ)2t/λ + 1 n=0 ∞
tn = T (x) . n n! n=0

Thus, we have

lim Tn,λ(x) = Tn(x), (n ≥ 0). λ→0

From (2.1) and (1.5), we have

∞
tn 2 T (x) = (1 + λ)xt/λ n,λ n! (1 + λ)2t/λ + 1 n=0    ∞ ∞  
tm
log(1 + λ) n tn = T xn (2.2) m,λ m! λ n! m=0 n=0      
∞
n + l n n log(1 λ) l t = T − x . l n l,λ λ n! n=0 l=0

Therefore, by (2.2), we obtain the following theorem.

Theorem 2.1. For n ≥ 0, we have    
n + l n log(1 λ) l T (x) = T − x . n,λ l n l,λ λ l=0 1570 Cheon Seoung Ryoo

By (1.4), (2.1), and using Cauchy product, we see that

∞
tn 2 T (x) = (1 + λ)xt/λ n,λ n! (1 + λ)2t/λ + 1 n=0    ∞ ∞  
tk
tx λn = T xm k,λ k! λ n! k=0 m=0 m  ∞ ∞  
tk

m tx l λm = T S (m, l) k,λ k! 1 λ m! k=0 m=0 l=0  ∞ ∞ ∞   (2.3)
tk

tx l λm = T S (m, l) k,λ k! 1 λ m! k=0 l=0 m=l   ∞ ∞ ∞  
tk

x l λm tl = T S (m, l) l! k,λ k! 1 λ m! l! k=0 l=0 m=l  ∞ n   ∞  

n
x l λm tn = S (m, l) l! T − . l 1 λ m! n l,λ n! n=0 l=0 m=l

tl Comparing the coefficients of in (2.3), we obtain the following theorem. l!

Theorem 2.2. For n ≥ 0, we have

n ∞    

n x l λm T (x) = T − S (m, l) l! . n,λ l n l,λ 1 λ m! l=0 m=l

By theorem 2.1 and Theorem 2.2, we have the following corollary.

Corollary 2.3. For n ≥ 0, we have

n ∞     n    

l m
+ l n x λ n log(1 λ) l T − S (m, l) l! = T − x . l n l,λ 1 λ m! l n l,λ λ l=0 m=l l=0 Modified degenerate tangent numbers and polynomials 1571

From (2.1), we can derive the following recurrence relation:

∞
tn 2 = ((1 + λ)2t/λ + 1) T n,λ n! n=0 ∞ ∞
tn
tn = (1 + λ)2t/λ T + T n,λ n! n,λ n!  n=0 n=0  ∞   ∞ ∞ (2.4)
log(1 + λ) l tl
tm
tn = 2l T + T λ l! m,λ m! n,λ n! l=0 m=0 n=0      
∞
n + l n n l log(1 λ) t = 2 T − + T . l λ n l,λ n,λ n! n=0 l=0

tn By comparing of the coefficients on the both sides of (2.4), we have the following n! theorem.

Theorem 2.4. For n ∈ Z+,wehave    
n + l n l log(1 λ) 2, if n = 0, 2 T − + T = l λ n l,λ n,λ 0, if n = 0. l=0 Also, we can obtain similar recurrence relation:  
n
∞ − n 2ll!λm l 2, if n = 0, T − S (m, l) + T = l n l,λ 1 m! n,λ 0, if n = 0. l=0 m=l By (2.1), we have

∞ ∞
tn
tn T (x + 2) + T n,λ n! n,λ n! n=0 n=0 2 + 2 = (1 + λ)(x 2)t/λ + (1 + λ)xt/λ (1 + λ)2t/λ + 1 (1 + λ)2t/λ + 1 (2.5) = 2(1 + λ)xt/λ ∞  
log(1 + λ) n tn = 2 xn . λ n! n=0

tn By comparing of the coefficients on the both sides of (2.4), we obtain the following n! theorem. 1572 Cheon Seoung Ryoo

Theorem 2.5. For n ∈ Z+,wehave   log(1 + λ) n T (x + 2) + T (x) = 2 xn. n,λ n,λ λ

By (2.1), we get

∞ n t 2 − T (x) = (1 + λ)(xt 2t)/λ n,λ n! (1 + λ)−2t/λ + 1 n=0 2 − = (1 + λ)(2 x)t/λ (2.6) (1 + λ)−2t/λ + 1 ∞
tn = (−1)nT (2 − x) . n,λ n! n=0

tn By comparing of the coefficients on the both sides of (2.6), we have the following n! theorem.

Theorem 2.6. For n ∈ Z+,wehave

n Tn,λ(x) = (−1) Tn,λ(2 − x),

In particular, n Tn,λ = (−1) Tn,λ(2). For d ∈ N with d ≡ 1(mod 2), we have

∞
tn 2 T (x) = (1 + λ)xt/λ n,λ n! (1 + λ)2t/λ + 1 n=0 − 2
d 1 = (1 + λ)xt/λ (−1)l(1 + λ)2lt/λ + 2dt/λ + (1 λ) 1 =  l 0  ∞ −  

d 1 2l + x tn = dn (−1)lT . n,λ d n! n=0 l=0

tm By comparing coefficients of in the above equation, we have the following theorem: m!

Theorem 2.7. For d ∈ N with d ≡ 1(mod 2) and n ∈ Z+,wehave

−  
d 1 2l + x T (x) = dn (−1)lT . n,λ n,λ d l=0 Modified degenerate tangent numbers and polynomials 1573

In particular, −  
d 1 2l T = dn (−1)lT . n,λ n,λ d l=0 From (2.1), we have

∞ n t 2 + T (x + y) = (1 + λ)(x y)t/λ n,λ n! (1 + λ)2t/λ + 1 n=0 2 = (1 + λ)xt/λ(1 + λ)yt/λ + 2t/λ + (1 λ) 1  ∞ ∞   (2.7)
tn
log(1 + λ) n tn = T (x) yn n,λ n! λ n! n=0 n=0      
∞
n + l n n log(1 λ) l t = T − (x) x . l n l,λ λ n! n=0 l=0 Therefore, by (2.7), we have the following theorem.

Theorem 2.8. For n ∈ Z+,wehave   
n + l n log(1 λ) l T (x + y) = x T − (x). n,λ l λ n l,λ l=0 By replacing t by (e − 1)t and λ by e − 1 in (2.1), we obtain

∞ − n n 2 xt (e 1) t e = T − (x) 2t + n,e 1 ! e 1 = n n 0 (2.8)
∞ n n t = T − (x)(e − 1) . n,e 1 n! n=0 Thus, by (2.8) and (1.2), we obtain the following theorem.

Theorem 2.9. For n ∈ Z+,wehave

m Tm(x) = Tm,e−1(x)(e − 1) . By replacing t by log(1 + λ)t/λ in (1.2), we have
∞ n 1 2 T (x) log(1 + λ)t/λ = (1 + λ)xt/λ n n! (1 + λ)2t/λ + 1 n=0 ∞ (2.9)
tm = T (x) . m,λ m! m=0 1574 Cheon Seoung Ryoo and  
∞
∞ n n n 1 log(1 + λ) t T (x) log(1 + λ)t/λ = T (x) . (2.10) n n! n λ n! n=0 n=0 Thus, by (2.9) and (2.10), we have the following theorem.

Theorem 2.10. For n ∈ Z+,wehave   log(1 + λ) n T (x) = T (x). m,λ λ n By (1.8) and (2.1), we have the following theorem.

Theorem 2.11. For n ∈ Z+,wehave

m m m−n Tm,e−1(x)(e − 1) = Tn,λ(x)λ S2(m, n). n=0

References [1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15(1979), 51–88. [2] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7(1956), 28– 33. [3] F. Qi, D. V.Dolgy, T. Kim, C. S. Ryoo, On the partially degenerate Bernoulli polyno- mials of the first kind, Global Journal of Pure and Applied Mathematics, 11(2015), 2407–2412. [4] T. Kim, Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258(2015), 556–564 [5] C. S. Ryoo, A Note on the Tangent Numbers and Polynomials, Adv. Studies Theor. Phys., 7(2013), no. 9, 447–454 [6] C. S. Ryoo, Notes on degenerate tangent polynomials, Global Journal of Pure and Applied Mathematics, 11(2015), no. 5, 3631–3637 [7] P.T.Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theorey., 128(2008), 738–758