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A NOTE on Q-ANALOGUE of POLY-BERNOULLI NUMBERS and POLYNOMIALS†

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A NOTE on Q-ANALOGUE of POLY-BERNOULLI NUMBERS and POLYNOMIALS† J. Appl. Math. & Informatics Vol. 35(2017), No. 5 - 6, pp. 611 - 621 http://dx.doi.org/10.14317/jami.2017.611 A NOTE ON q-ANALOGUE OF POLY-BERNOULLI NUMBERS AND POLYNOMIALSy K.W. HWANG, B.R. NAM, N.S. JUNG∗ Abstract. In this paper, we define a q-analogue of the poly-Bernoulli numbers and polynomials which is generalization of the poly Bernoulli num- bers and polynomials including q-polylogarithm function. We also give the relations between generalized poly-Bernoulli polynomials. We derive some relations that are connected with the Stirling numbers of second kind. By using special functions, we investigate some symmetric identities involving q-poly-Bernoulli polynomials. AMS Mathematics Subject Classification : 11B68, 11B73, 11B75. Key words and phrases : poly-Bernoulli polynomials, q-analogue of poly- Bernoulli polynomials, Stirling numbers of the second kind, q-polylogarithm function. 1. Introduction Many mathematicians are interested in the Bernoulli numbers and polynomi- als, Euler numbers and polynomials, Genocchi numbers and polynomials, tan- gent numbers and polynomials and their applications. They possess many in- teresting properties and are treated in many areas of mathematics and physics. Due to these reasons, many applications of Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials have been studied, and recently various analogues for the above numbers and polynomials was introduced(see [1-14]). In this paper, we use the following notations. N = f1; 2; 3;::: g denotes the set of natural numbers, N0 = f0; 1; 2;::: g denotes the set of nonnegative inte- ger, Z denotes the set of integers, and C denotes the set of complex numbers, respectively. Received March 22, 2017, Revised August 20, 2017. Accepted August 28, 2017 ∗Corresponding author. yThis work was supported by the Dong-A university research fund. ⃝c 2017 Korean SIGCAM and KSCAM. 611 612 K. W. Hwang, B. R. Nam, N. S. Jung The ordinary Bernoulli polynomials Bn(x) are given by the generating func- tions: 1 t X tn ext = B (x) (see[1; 2; 6]): (1:1) et − 1 n n! n=0 (k) (k) When x = 0;Bn;q = Bn;q(0) are called poly-Bernoulli numbers. The polylogarithm function Lik is defined by 1 X xn Li (x) = ; (k 2 Z)(see[1; 2; 3; 6; 7; 8; 14]) (1:2) k nk n=1 For k ≤ 1, the polylogarithm functions are as follows x x Li1(x) = −log(1 − x); Li0(x) = ; Li−1(x) = ; ··· : 1 − x (1 − x)2 By using polylogarithm function, Kaneko defined a sequence of rational num- bers, which is refered to as poly-Bernoulli numbers, In [3] and [13], the k-th q-analogue of polylogarithm function Lik;q is intro- duced by 1 X xn Li (x) = ; (k 2 Z): (1:3) k;q [n]k n=1 q The q-analogue of polylogarithm function for k ≤ 1 is represented by a rational function, ( ) 1 Xk k qlx Li (x) = (−1)l : k;q (1 − q)k l 1 − qlx l=0 In [4] and [12], the Stirling number of the first kind is given by Xn m (x)n = S1(n; m)x ; (n ≥ 0) m=0 and 1 X tn (log(1 + t))m S (n; m) = 1 n! m! n=m where − − ··· − n − − (x)n = x(x 1)(x 2) (x n + 1) = Πk=1(x (k 1)) (1:4) is falling factorial. The Stirling numbers of the second kind is defined by 1 X tn (et − 1)m S (n; m) = : (1:5) 2 n! m! n=m In this paper, we consider a q-analogue of the poly Bernoulli polynomials containing Equation(1.3). We also find some relations between q-poly-Bernoulli polynomials and ordinary Bernoulli polynomials. And we derive several proper- ties that are connected with the Stirling numbers of the second kind. Finally, we A note on q-analogue of poly-Bernoulli numbers and polynomials 613 find some symmetric identities of the q-degenerate poly Bernoulli polynomials by using special functions. 2. A q-analogue of the poly-Bernoulli polynomials (k) In this section, we define a q-analogue of poly-Bernoulli numbers Bn;q and (k) polynomials Bn;q(x) by the generating functions. From the definition, we get some identities that is similar to the ordinary Bernoulli polynomials. Definition 2.1. For n ≥ 0; n; k 2 Z; 0 ≤ p < 1, we introduce a q-analogue of poly-Bernoulli polynomials by: 1 Li (1 − e−t) X tn k;q ext = B(k)(x) (2:1) et − 1 n;q n! n=0 where 1 X tn Li (t) = k;q [n]k n=0 q is the k-th q-polylogarithm function. (k) (k) When x = 0, Bn;q = Bn;q(0) are called a q-analogue of poly-Bernoulli numbers. (k) (k) Note that limq!1[n]q = n; and limq!1Bn;q(x) = Bn (x): From Equation(2.1), we have the following relation between q-poly-Bernoulli numbers and q-poly-polynomials. Theorem 2.2. Let n ≥ 0; n; k 2 Z; 0 < p < 1: We have ( ) Xn n B(k)(x) = B(k)xn−l: (2:2) n;q l l;q l=0 Proof. For n ≥ 0; n; k 2 Z; 0 < p < 1; we easily get: 1 X tn Li (1 − e−t) B(k)(x) = k;q ext n;q n! et − 1 n=0 ! 1 ( ) X Xn n tn = B(k)xn−l : l l;q n! n=0 l=0 Therefore, we have ( ) Xn n B(k)(x) = B(k)xn−l: n;q l l;q l=0 By Equation(2.2), we obtain an addition theorem. Theorem 2.3. For n ≥ 0; n; k 2 Z; 0 < p < 1; we have ( ) Xn n B(k)(x + y) = B(k)(x)yn−l: n;q l l;q l=0 614 K. W. Hwang, B. R. Nam, N. S. Jung Proof. Let n ≥ 0; n; k 2 Z. Then we obtain 1 X tn Li (1 − e−t) B(k)(x + y) = k;q e(x+y)t n;q n! et − 1 n=0 ! 1 ( ) X Xn n tn = B(k)(x)yn−l : l l;q n! n=0 l=0 Thus, we get ( ) Xn n B(k)(x + y) = B(k)(x)yn−l: n;q l l;q l=0 By using the definition of the q-analogue of polylogarithm function Lik;q in Equation(1.3), we have next relation which is connected with the ordinary Bernoulli polynomials. Theorem 2.4. Let n ≥ 0; n; k 2 Z; 0 ≤ p < 1: We obtain 1 ( ) X 1 Xl+1 l + 1 B(k)(x) = (−1)aB (x − a): n;q [l + 1]k a n l=0 q a=0 Proof. For n ≥ 0; n; k 2 Z; 0 ≤ p < 1; 1 Li (1 − e−t) X (1 − e−t)l ext k;q ext = et − 1 [l]k et − 1 l=1 q 1 ( ) X 1 Xl+1 l + 1 e(x−a)t = (−1)a [l + 1]k a et − 1 n=0 q a=0 ! 1 1 ( ) X X 1 Xl+1 l + 1 B (x − a) tn = (−1)a n+1 [l + 1]k a n + 1 n! n=0 l=0 q a=0 Comparing the coefficient of the result, we easily get next equation: 1 ( ) X 1 Xl+1 l + 1 B (x − a) B(k)(x) = (−1)a n+1 : n;q [l + 1]k a n + 1 l=0 q a=0 From the binomials series and the Equation(1.3), we derive the following Theorem. Theorem 2.5. If n; k 2 Z; n ≥ 0and 0 ≤ p < 1; then we have 1 ( ) X Xl mX+1 m + 1 (−1)a(l − m − a + x)n B(k)(x) = : n;q a [m + 1]n l=0 m=0 a=0 q A note on q-analogue of poly-Bernoulli numbers and polynomials 615 Proof. Let n; k 2 Z; n ≥ 0 and 0 ≤ p < 1: Using the Equation(1.3), we have ! ! 1 1 Li (1 − e−t) X X (1 − e−t)l+1 k;q ext = emt ext t − k e 1 [l + 1]q m=0 l=0 ! ! 1 ( ) X Xl e(l−m)t mX+1 m + 1 = (−1)ae(x−a)t [m + 1]k a l=0 m=0 q a=0 1 ( ) X Xl mX+1 m + 1 (−1)ae(l−m−a+x)t = a [m + 1]k l=0 m=0 a=0 q 1 1 ( ) X X Xl mX+1 m + 1 (−1)a(l − m − a + x)n tn = a [m + 1]k n! n=0 l=0 m=0 a=0 q Hence, we have 1 ( ) X Xl mX+1 m + 1 (−1)a(l − m − a + x)n B(k)(x) = : n;q a [m + 1]n l=0 m=0 a=0 q 3. Some relation involving the Stirling numbers of the second kind In this section, using well-known Stirling numbers, we find several identities of the q-poly-Bernoulli polynomials. Note that the Stirling numbers of the second kind is defined 1 (et − 1)m X tn = S (n; m) (see[4; 12; 13; 14]): (3:1) m! 2 n! n=m The q-analogue of polylogarithm function Lik;q; in the Equation(1.3), is repre- sented as below 1 X (1 − e−t)l Li (1 − e−t) = k;q [l]k l=1 q 1 X (e−t − 1)l = (−1)l [l]k n=1 q 1 X Xn (−1)l+n tn = l!S (n; l) [l]k 2 n! n=1 l=1 q Therefore, we obtain 1 1 X nX+1 (−1)l+n+1 S (n + 1; l) tn Li (1 − e−t) = l! 2 : (3:2) t k;q [l]k n + 1 n! n=0 l=1 q Using the Equation(3.2), we get next theorem.
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