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
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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) . n,q l n n n–l,q l=
By (.), (.), (.), and Theorem ., we have the following corollary.
Corollary . For n ∈ Z+, we have n l n (–)m+lm!S (l, m) T(k)(x)= T (x)–T (x –) . n,q l [m +]k n–l n–l l= m= q
3 Some identities involving q-poly-tangent numbers and polynomials In this section, we give several combinatorics identities involving q-poly-tangent numbers and polynomials in terms of Stirling numbers, falling factorial functions, raising facto- rial functions, Beta functions, Bernoulli polynomials of higher order, and Frobenius-Euler functions of higher order. By (.) and by using the Cauchy product, we get
∞ n –t t Li ( – e ) –x T(k)(x) = k,q – –e–t n,q n! et + n= ∞ –t Li ( – e ) x + l – l = k,q –e–t et + l l= ∞ (et –)l Li ( – e–t) = x k,q e–lt l l! et + l= ∞ ∞ ∞ tn tn = x S (n, l) T(k)(–l) l n! n,q n! l= n= n= ∞ ∞ n n tn = S (i, l)T(k) (–l)x ,(.) i n–i,q l n! n= l= i=l
where x l = x(x +)···(x + l –)(l ≥ ) with x =. By comparing the coefficients on both sides of (.), we have the following theorem.
Theorem . For n ∈ Z+, we have
∞ n n T(k)(x)= x S (i, l)T(k) (–l). n,q i l n–i,q l= i=l
By using the Jackson q-integral (see []) and Theorem .,weget n n T(k)(x) d x = T(k)xn–l d x n,q q l l,q q l= n n (k) = Tl,q .(.) l [n – l +]q l=
By (.)andTheorem., we have the following theorem. Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 7 of 14
Theorem . For any positive integer n, we have n ∞ n n (k) n (k) l ˆ Tl,q = S(i, l)Tn–i,q(–l)(–) cl,q, l [n – l +]q i l= l= i=l
where cˆl,q are q-Cauchy numbers of the second kind (see []).
By (.) and by using the Cauchy product, we get
∞ n –t t Li ( – e ) x T(k)(x) = k,q et – + n,q n! et + n= ∞ –t Li ( – e ) x l = k,q et – et + l l= ∞ (et –)l Li ( – e–t) = (x) k,q l l! et + l= ∞ ∞ ∞ tn tn = (x) S (n, l) T(k) l n! n,q n! l= n= n= ∞ ∞ n n tn = (x) S (i, l)T(k) . (.) i l n–i,q n! n= l= i=l
By comparing the coefficients on both sides of (.), we have the following theorem.
Theorem . For n ∈ Z+ and ≤ q <,we have
∞ n n T(k)(x)= (x) S (i, l)T(k) . n,q i l n–i,q l= i=l
By (.)andTheorem., we have the following theorem.
Theorem . For any positive integer n, we have n (k) ∞ n n Tn–l,q n (k) = S(i, l)Tn–i,qcl,q, l [l +]q i l= l= i=l
where cl,q are q-Cauchy numbers of the first kind (see []).
By Theorem .,wenotethat n n ynT(k)(x + y) dy = yn T(k) (x)yl dy n,q l n–l,q l= n n = T(k) (x) yn+l dy l n–l,q l= n n = T(k) (x) .(.) l n–l,q n + l + l= Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 8 of 14
From (.)andTheorem.,wenotethat
n (k) (k) n (k) y Tn+,q(x + y) n– Tn+,q(x + y) y Tn,q(x + y) dy = – ny dy n + n +
(k) Tn+,q(x +) n n– (k) = – y Tn+,q(x + y) dy n + n + (k) T (x +) n n+ n + = n+,q – yn– T(k)(x)yn+–l dy n + n + l l,q l= (k) T (x +) n n+ n + = n+,q – T(k)(x) .(.) n + n + l l,q n – l + l=
Therefore, by (.)and(.), we obtain the following theorem.
Theorem . For n ∈ Z+, we have n+ n + n n n n + T(k) (x +)= T(k)(x) + T(k) (x) . n+,q l l,q n – l + l n–l,q n + l + l= l=
By (.), (.), (.),andbyusingtheCauchyproduct,weget
∞ tn Li ( – e–t) T(k)(x) = k,q ext n,q n! et + n= ∞ (et –)r r! t r tn = ext T(k) r! tr et – n,q n! n= ∞ ∞ (et –)r tn tn r! = B(r)(x) T(k) r! n n! n,q n! tr n= n= ∞ n n n–l n – l tn = l S (l + r, r) B(r)(x)T(k) . l+r i i n–l–i,q n! n= l= r i=
By comparing the coefficients on both sides, we have the following theorem.
Theorem . For n ∈ Z+ and r ∈ N, we have n n n–l n – l T(k)(x)= l S (l + r, r) T(k) B(r)(x). n,q l+r i n–l–i,q i l= r i=
From (.)andTheorem.,wenotethat
n (k) y Tn,q(x + y) dy
n (k) n– (k) y Tn+,q(x + y) ny Tn+,q(x + y) = – dy n + n + (k) ∞ T (x +) n l+ l + = n+,q – (–)iT (x + y – i)yn– dy n + n + [l +]k i n+ l= q i= Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 9 of 14
T(k) (x +) = n+,q n + ∞ n l+ n+ l+ n+ – i l (–)n++iT ( – x + i) yn–( – y)j dy n + [l +]k n+–j l= i= j= q
T(k) (x +) = n+,q n + ∞ n l+ n+ l+ n+ – i l (–)n++iT ( – x + i)B(n, j + ), (.) n + [l +]k n+–j l= i= j= q
where B(n, j)isthebetaintegral(see[]). Therefore, by (.)and(.), we obtain the following theorem.
Theorem . For n ∈ Z+, we have (k) ∞ n+ n + T (x) l+ n+ l+ n+ l,q = i l (–)n++iT ( – x + i)B(n, j +). l n – l + [l +]k n+–j l= l= i= j= q
By (.), (.), (.),andbyusingtheCauchyproduct,weget
∞ tn Li ( – e–t) T(k)(x) = k,q ext n,q n! et + n= (et – u)r –u r Li ( – e–t) = ext k,q ( – u)r et – u et + ∞ tn r r Li ( – e–t) = H(r)(u, x) eit(–u)r–i k,q n n! i ( – u)r et + n= i= ∞ ∞ r r tn tn = (–u)r–i H(r)(u, x) T(k)(i) ( – u)r i n n! n,q n! i= n= n= ∞ r r n n tn = (–u)r–i H(r)(u, x)T(k) (i) . ( – u)r i l l n–l n! n= i= l=
By comparing the coefficients on both sides, we have the following theorem.
Theorem . For n ∈ Z+ and r ∈ N, we have r n r n T(k)(x)= (–u)r–iH(r)(u, x)T(k) (i). n,q ( – u)r i l l n–l,q i= l=
For n ∈ N with n ≥ , we obtain
(k) (k) n (k) n+ Tn,q(x + y) n+ Tn–,q(x + y) y Tn,q(x + y) dy = y – ny dy n + n +
(k) Tn,q(x +) n, q n+ (k) = – y Tn–,q(x + y) dy n + n + Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 10 of 14
(k) T(k)(x +) nT (x +) = n,q – n–,q n + (n +)(n +)
n n – n+ (k) +(–) y Tn–,q(x + y) dy n +n + (k) T(k)(x +) nT (x +) = n,q +(–) n–,q n + (n +)(n +) (k) n n –T (x +) +(–) n–,q n +n + n + (k) n n –n –T (x +) +(–) n–,q n +n +n + n +
n n –n –n – n+ (k) +(–) y Tn–,q(x + y) dy. n +n +n +n +
Continuing this process, we obtain
n (k) Tn,q(x +) y Tn,q(x + y) dy = n + n n(n –)···(n – l + )(–)l– + T(k) (x +) (n +)(n +)···(n + l) n–l+,q l=
n n! n (k) +(–) ··· y T,q(x + y) dy.(.) (n +)(n +) (n)
Hence, by (.)and(.), we have the following theorem.
Theorem . For n ∈ N with n ≥ , we have n n T(k) (x) l n–l,q n + l + l= T(k)(x +) n n(n –)···(n – l + )(–)l– = n,q + T(k) (x +). n + (n +)(n +)···(n + l) n–l+,q l=
4 Zeros of the q-poly-tangent polynomials This section aims to demonstrate the benefit of using a numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the poly- (k) (k) tangent polynomials Tn,q(x). The q-poly-tangent polynomials Tn,q(x) can be determined explicitly. A few of them are
(k) T,q(x)=, (k) T,q (x)=, T(k) x x ,q( )=–+ k + , []q T(k) x x x ,q( )=– k + k – – k + , []q []q []q Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 11 of 14
T(k) x x ,q( )=+ k – k + k + – k + k []q []q []q []q []q x x – – k + , []q T(k) x x ,q ( )=–– k + k – k + k + + k – k + k []q []q []q []q []q []q []q x x x + – k + k – + k + . []q []q []q
(k) We investigate the beautiful zeros of the q-poly-tangent polynomials Tn,q(x)byusing (k) a computer. We plot the zeros of the q-poly-tangent polynomials Tn,q(x)forn =,q = /, –/, k = –, and x ∈ C (Figure ). In Figure (top-left), we choose n =,q = /, and k =.InFigure (top-right), we choose n =,q = –/, and k =.InFigure (bottom- left), we choose n =,q = /, and k =–.InFigure (bottom-right), we choose n =, q = –/, and k =–. (k) Stacks of zeros of Tn,q(x)for≤ n ≤ from a -D structure are presented (Figure ). In Figure ,wechoosek =,q = /. Our numerical results for approximate solutions of (k) real zeros of Tn,q(x) are displayed (Tables , ).
(k) Figure 1 Zeros of Tn,q(x). Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 12 of 14
(k) Figure 2 Stacks of zeros of Tn,q(x)for2≤ n ≤ 40.
(k) Table 1 Numbers of real and complex zeros of Tn,q(x)
Degree nk=3, q =1/2 k =–3, q =–1/2 Real zeros Complex zeros Real zeros Complex zeros 21 0 1 0 32 0 2 0 43 0 1 2 54 0 0 4 61 4 1 4 72 4 0 6 83 4 1 6 94 4 2 6 10 5 4 1 8 1146 010 1238 110
(k) Table 2 Approximate solutions of Tn,q(x)=0,k =3,q =1/2
Degree nx 2 1.2037 3 0.24060, 2.1668 4 –0.47700, 1.2291, 2.8590 5 –0.96664, 0.23158, 2.2563, 3.2936 6 1.2308 7 0.23043, 2.2296
(k) The plot of real zeros of Tn,q(x)forthe≤ n ≤ structure is presented (Figure ). In Figure ,wechoosek =. We observe a remarkable regular structure of the real roots of the q-poly-tangent poly- (k) nomials Tn,q(x). We also hope to verify a remarkable regular structure of the real roots of (k) the q-poly-tangent polynomials Tn,q(x)(Table). Next, we calculated an approximate solution satisfying q-poly-tangent polynomials (k) Tn,q(x)=forx ∈ R. The results are given in Table and Table . By numerical computations, we will present a series of conjectures. Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 13 of 14
(k) Figure 3 Real zeros of Tn,q(x)for2≤ n ≤ 40.
(k) Table 3 Approximate solutions of Tn,q(x)=0,k =–3,q = –1/2
Degree nx 2 1.3750 3 0.91289, 1.8371 4 2.6383 5- 6 1.8993 7-
(k) Conjecture . Prove that Tn,q(x), x ∈ C, has Im(x)=reflection symmetry analytic com- (k) plex functions. However, Tn,q(x) has no Re(x)=a reflection symmetry for a ∈ R.
Using computers, many more values of n have been checked. It still remains unknown if the conjecture fails or holds for any value n (see Figures , , ). (k) We are able to decide if Tn,q(x)=hasn – distinct solutions (see Tables , , ).
(k) Conjecture . Prove that Tn,q(x)=has n –distinct solutions.
(k) Since n –isthedegreeofthepolynomialTn,q(x), the number of real zeros R (k) lying Tn,q(x) on the real plane Im(x)=isR (k) = n – C (k) ,whereC (k) denotes complex zeros. Tn,q(x) Tn,q(x) Tn,q(x) See Table for tabulated values of R (k) and C (k) . The authors have no doubt that Tn,q(x) Tn,q(x) investigations along these lines will lead to a new approach employing numerical method (k) in the research field of the poly-tangent polynomials Tn,q(x), which appear in mathematics and physics.
Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).
Competing interests The authors declare that they have no competing interests.
Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript.
Author details 1Department of Mathematics, Hannam University, Daejeon, 306-791, Korea. 2Department of Mathematics, Texas A&M University, Kingsville, USA.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Ryoo and Agarwal Advances in Difference Equations (2017) 2017:213 Page 14 of 14
Received: 8 June 2017 Accepted: 10 July 2017
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