Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 DOI 10.1186/s13662-017-1275-2 R E S E A R C H Open Access Some identities involving q-poly-tangent numbers and polynomials and distribution of their zeros CS Ryoo1* and RP Agarwal2 *Correspondence: [email protected] 1Department of Mathematics, Abstract Hannam University, Daejeon, 306-791, Korea In this paper we introduce the q-poly-tangent polynomials and numbers. We also Full list of author information is give some properties, explicit formulas, several identities, a connection with available at the end of the article poly-tangent numbers and polynomials, and some integral formulas. Finally, we investigate the zeros of the q-poly-tangent polynomials by using a computer. MSC: 11B68; 11S40; 11S80 Keywords: tangent numbers and polynomials; q-poly-tangent numbers and polynomials; Cauchy numbers; Stirling numbers; Frobenius-Euler polynomials; q-poly-Euler polynomials 1 Introduction Many mathematicians have studied in the area of the Bernoulli numbers and polynomi- als, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent num- bers and polynomials, poly-Bernoulli numbers and polynomials, poly-Euler numbers and polynomials, and poly-tangent numbers and polynomials (see [–]). In this paper, we define q-poly-tangent polynomials and numbers and study some properties of the q-poly- tangent polynomials and numbers. Throughout this paper, we always make use of the fol- lowing notations: N denotes the set of natural numbers and Z+ = N∪{}. We recall that the classical Stirling numbers of the first kind S(n, k)andS(n, k) are defined by the relations (see []) n n k n (x)n = S(n, k)x and x = S(n, k)(x)k, (.) k= k= respectively. Here (x)n = x(x –)···(x – n + ) denotes the falling factorial polynomial of order n.ThenumbersS(n, m) also admit a representation in terms of a generating func- tion, ∞ n m t et – = m! S (n, m) .(.) n! n=m © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 Page 2 of 14 We also have ∞ n t m m! S (n, m) = log( + t) .(.) n! n=m We also need the binomial theorem: for a variable x, ∞ c + n – = tn.(.) ( – t)c n n= (k) For ≤ q <,theq-poly-Bernoulli numbers Bn were introduced by Mansour []byusing the following generating function: ∞ Li ( – e–t) tn k,q = B(k) (k ∈ Z), (.) –e–t n,q n! n= where ∞ tn Li (t)= (.) k,q [n]k n= q –qn is the kth q-poly-logarithm function, and [n]q = –q is the q-integer (cf. []). (k) The q-poly-Euler polynomials En,q(x) are defined by the generating function ∞ Li ( – e–t) tn k,q ext = E(k) (x) (k ∈ Z). (.) et + n,q n! n= The familiar tangent polynomials Tn(x) are defined by the generating function [–] ∞ tn ext = T (x) |t| < π .(.) et + n n! n= (r) When x =,Tn() = Tn are called the tangent numbers. The tangent polynomials Tn (x) of order r are defined by ∞ r tn ext = T(r)(x) |t| < π .(.) et + n n! n= It is clear that for r = we recover the tangent polynomials Tn(x). (r) The Bernoulli polynomials Bn (x)oforderr are defined by the following generating function: ∞ t r tn ext = B(r)(x) |t| <π . (.) et – n n! n= (r) The Frobenius-Euler polynomials of order r,denotedbyHn (u, x), are defined as ∞ –u r tn ext = H(r)(u, x) . (.) et – u n n! n= Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 Page 3 of 14 The values at x = are called Frobenius-Euler numbers of order r;whenr =,thepoly- nomials or numbers are called ordinary Frobenius-Euler polynomials or numbers. (k) The poly-tangent polynomials Tn,q(x) are defined by the generating function ∞ Li ( – e–t) tn k ext = T(k)(x) (k ∈ Z), (.) et + n n! n= ∞ n where Li (t)= t is the kth poly-logarithm function (see []). k n= nk Many kinds of generalizations of these polynomials and numbers have been presented in the literature (see [–]). In the following section, we introduce the q-poly-tangent polynomials and numbers. After that we will investigate some their properties. We also give some relationships both between these polynomials and tangent polynomials and between these polynomials and q-cauchy numbers. Finally, we investigate the zeros of the q-poly-tangent polynomials by using a computer. 2 q-Poly-tangent numbers and polynomials In this section, we define q-poly-tangent numbers and polynomials and provide some of their relevant properties. (k) For ≤ q <,theq-poly-tangent polynomials Tn,q(x) are defined by the generating func- tion: ∞ Li ( – e–t) tn k,q ext = T(k)(x) (k ∈ Z). (.) et + n,q n! n= (k) (k) When x =,Tn,q() = Tn,q(x) are called the q-poly-tangent numbers. Observe that (k) (k) limq→ Tn,q(x)=Tn (x). By (.), we get ∞ tn Li ( – e–t) T(k)(x) = k,q ext n,q n! et + n= ∞ ∞ tn tn = T(k) xn n,q n! n! n= n= ∞ n n tn = T(k)xn–l . (.) l l,q n! n= l= By comparing the coefficients on both sides of (.), we have the following theorem. Theorem . For n ∈ Z+, we have n n T(k)(x)= T(k)xn–l. n,q l l,q l= (k) The following elementary properties of the q-poly-tangent numbers Tn,q and polynomi- (k) als Tn,q(x) are readily derived form (.). We, therefore, choose to omit the details involved. Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 Page 4 of 14 Theorem . For k ∈ Z, we have n n () T(k)(x + y)= T(k)(x)yn–l, n,q l l,q l= n n () T(k)( – x)= (–)l T(k) ()xl. n,q l n–l,q l= Theorem . For any positive integer n, we have n n () T(k)(mx)= T(k)(x)(m –)n–lxn–l (m ≥ ), n,q l l,q l= n– n () T(k)(x +)–T(k)(x)= T(k)(x), n,q n,q l l,q l= (.) d () T(k)(x)=nT (k) (x), dx n,q n–,q x (k) (k) (k) () Tn,q(x)=Tn,q + n Tn–,q(t) dt. From (.), (.), and (.), we get ∞ ∞ tn Li ( – e–t) ( – e–t)l+ ext T(k)(x) = k,q ext = n,q n! et + [l +]k et + n= l= q ∞ l+ l + e(x–i)t = (–)i [l +]k i et + l= q i= ∞ ∞ l+ l + tn = (–)i T (x – i) [l +]k i n n! l= q i= n= ∞ ∞ l+ l + tn = (–)iT (x – i) .(.) [l +]k i n n! n= l= q i= By comparing the coefficients on both sides of (.), we have the following theorem. Theorem . For n ∈ Z+, we have ∞ l+ l + T(k)(x)= (–)jT (x – j). n,q [l +]k j n l= q j= By using the definition of tangent polynomials and Theorem ., we have the following corollary. Corollary . For any positive integer n, we have ∞ l+ l + T(k)( – x)=(–)n (–)jT (x + j). n,q [l +]k j n l= q j= Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 Page 5 of 14 By (.), we note that ∞ ∞ ∞ tn ( – e–t)l+ T(k)(x) = (–)lelt ext n,q n! [l +]k n= l= l= q ∞ l i+ l+j–i i+ (–) j = e(l–i–j+x)t [i +]k l= i= j= q ∞ ∞ l i+ l+j–i i+ n n (–) j (l –i – j + x) t = . [i +]k n! n= l= i= j= q Comparing the coefficients on both sides, we have the following theorem. Theorem . For n ∈ Z+, we have ∞ l i+ l+j–i i+ (–) j T(k)(x)= (l –i – j + x)n. n,q [i +]k l= i= j= q By (.), (.), and (.) and by using the Cauchy product, we get ∞ tn Li ( – e–t) (et +) T(k)(x) = k,q ext n,q n! et + et + n= ∞ ∞ tn tn = E(k) (x) T () + T n,q n! n n n! n= n= ∞ n n = T () + T E(k) (x) .(.) l n n n–l,q n= l= By comparing the coefficients on both sides of (.), we have the following theorem related the q-poly-Euler polynomials and tangent polynomials. Theorem . For n ∈ Z+, we have n n T(k)(x)= T () + T E(k) (x). n,q l n n n–l,q l= By (.), (.), and (.) and by using the Cauchy product, we have ∞ tn Li ( – e–t) ( – e–t) T(k)(x) = k,q ext n,q n! –e–t et + n= ∞ ∞ tn tn = B(k) T (x)–T (x –) n,q n! n n n! n= n= ∞ n n = T (x)–T (x –) B(k) .(.) l n n n–l,q n= n= By comparing the coefficients on both sides of (.), we have the following theorem related the q-poly-Bernoulli polynomials and tangent polynomials. Ryoo and Agarwal Advances in Difference Equations (2017)2017:213 Page 6 of 14 Theorem . For n ∈ Z+, we have n n T(k)(x)= T (x)–T (x –) B(k) .