Open access peer-reviewed chapter

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

By Jung Yoog Kang and Cheon Seoung Ryoo

Submitted: July 27th 2018Reviewed: December 12th 2018Published: January 23rd 2019

DOI: 10.5772/intechopen.83497

Downloaded: 149

Abstract

We introduce q-tangent polynomials and their basic properties including q-derivative and q-integral. By using Mathematica, we find approximate roots of q-tangent polynomials. We also investigate relations of zeros between q-tangent polynomials and classical tangent polynomials.

Keywords

  • q-tangent polynomials
  • q-derivative
  • q-integral
  • Newton dynamical system
  • fixed point
  • 2000 Mathematics Subject Classification: 11B68
  • 11B75
  • 12D10

1. Introduction

For a long time, studies on q-difference equations appeared in intensive works especially by F. H. Jackson [1, 2], R. D. Carmichael [3], T. E. Mason [4], and other authors [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. An intensive and somewhat surprising interest in q-numbers appeared in many areas of mathematics and applications including q-difference equations, special functions, q-combinatorics, q-integrable systems, variational q-calculus, q-series, and so on. In this paper, we introduce some basic definitions and theorems (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]).

For any nC, the q-number is defined by

nq=1qn1q,q<1.E1

Definition 1.1. [1, 2, 9, 13] The q-derivative operator of any function f is defined by

Dqfx=fxfqx1qx,x0,E2

and Dqf0=f0. We can prove that f is differentiable at 0, and it is clear that Dqxn=nqxn1.

Definition 1.2. [1, 2, 9, 13, 17] We define the q-integral as

0bfxdqx=1qbj=0qjfqjb.E3

If this function, f(x), is differentiable on the point x, the q-derivative in Definition 1.1 goes to the ordinary derivative in the classical analysis when q1.

Definition 1.3. [5, 17, 18, 21] The Gaussian binomial coefficients are defined by

mrq=mrq=0ifr>m1qm1qm11qmr+11q1q21qrifrm,E4

where m and r are non-negative integers. For r=0the value is 1 since the numerator and the denominator are both empty products. Like the classical binomial coefficients, the Gaussian binomial coefficients are center-symmetric. There are analogues of the binomial formula, and this definition has a number of properties.

Theorem 1.4. Let n, k be non-negative integers. Then we get.

  1. k=0n11+qkt=k=0nqk2nkqtk,E5

  • k=0n111qkt=k=0n+k1kqtk.

  • Definition 1.5. [5, 26] Let z be any complex number with z<1. Two forms of q-exponential functions are defined by

    eqz=n=0znnq!,eq1z=n=0znnq1!=n=0qn2znnq!.E6

    Bernoulli, Euler, and Genocchi polynomials have been studied extensively by many mathematicians(see [22, 23, 24, 25]). In 2013, C. S. Ryoo introduced tangent polynomials and he developed several properties of these polynomials (see [22, 23]). The tangent numbers are closely related to Euler numbers.

    Definition 1.6. [22, 23, 24, 25] Tangent numbers Tn and tangent polynomials Tnxare defined by means of the generating functions

    n=0Tntnn!=2e2t+1=2m=01me2mt,n=0Tntnn!=2e2t+1etx=2m=01me2m+xt.E7

    Theorem 1.7. For any positive integer n, we have

    Tnx=1nTn2x.E8

    Theorem 1.8. For any positive integer m=odd, we have

    Tnx=mni=0m11iTn2i+xm,nZ+.E9

    Theorem 1.9. For nZ+, we have

    Tnx+y=k=0nnkTkxynk.E10

    The main aim of this paper is to extend tangent numbers and polynomials, and study some of their properties. Our paper is organized as follows: In Section 2, we define q-tangent polynomials and find some properties of these polynomials. We consider q-tangent polynomials in two parameters and establish some relations between q-tangent polynomials and q-Euler or Bernoulli polynomials. In Section 3, we observe approximate roots distributions of q-tangent polynomials and demonstrate interesting phenomenon.

    2. Some properties of the q-tangent polynomials

    In this section we define the q-tangent numbers and polynomials and establish some of their basic properties. we shall also study the q-tangent polynomials involving two parameters. We shall find some important relations between these polynomials and q-other polynomials.

    Definition 2.1. For x,qC, we define q-tangent polynomials as

    n=0Tn,qxtnnq!=2qeq2t+1eqtx,t<π2.E11

    From Definition 2.1, it follows that

    n=0Tn,q0tnnq!=n=0Tn,qtnnq!=2qeq2t+1,E12

    where Tn,qis q-tangent number. If q1, then it reduces to the classical tangent polynomial(see [22, 23, 24, 25]).

    Theorem 2.2. Let x,qC. Then, the following hold.

    i.  Tn,q+k=0nnkq2nkTk,q=2qifn=00ifn0,E13
    ii.  Tn,qx+k=0nnkq2nkTk,qx=2qxn.

    Proof. From the Definition 2.1, we have

    2q=1+eq2tn=0Tn,qtnn!=n=0Tn,q+k=0nnkq2nkTk,qtnn!.E14

    Now comparing the coefficients of tn we find (i). For (ii) we use the relation

    2qeqtx=1+eq2tn=0Tn,qxtnnq!=n=0Tn,qx+k=0nnkq2nkTk,qxtnn!,E15

    and again compare the coefficients of tn.

    Theorem 2.3. Let n be a non-negative integer. Then, the following holds

    Tn,qx=k=0nnkqTnk,qxk.E16

    Proof. From the definition of the q-exponential function, we have

    n=0Tn,qxtnnq!=2qeq2t+1eqtx=n=0Tn,qtnnq!n=0xntnnq!=n=0k=0nnkqTnk,qxxktnnq!.E17

    The required relation now follows on comparing the coefficients of tn on both sides.☐

    Theorem 2.4. Let n be a non-negative integer. Then, the following holds

    Tn,q=k=0nnkq1nkqnk2Tk,qxxnk.E18

    Proof. From the property of q-exponential function, it follows that

    n=0Tn,qtnnq!=2qeq2t+1eqtxeq1tx=n=0Tn,qxtnnq!n=0qn21nxntnnq!=n=0k=0nnkq1nkqnk2Tk,qxxnktnnq!.E19

    The required relation now follows immediately.☐

    In what follows, we consider q-derivative of eqtx. Using the Mathematical Induction, we find.

    i.  k=1:Dq1eqtx=n=1xn1tnn1q!.E20
    ii.  k=i:Dqieqtx=n=ixnitnniq!.

    If (ii) is true, then it follows that.

    iii.  k=i+1:Dqi+1eqtx=Dq;x1n=ixnitnniq!=n=i+1xni+1tnni+1q!=ti+1eqtx.E21

    We are now in the position to prove the following theorem.

    Theorem 2.5. For kN, the following holds

    DqkTn,qx=nq!nkq!Tnk,qx.E22

    Proof. Considering q-derivative of eqtx, we find

    Dqi+1n=0Tn,qxtnnq!=n=0Dqi+1Tn,qxtnnq!=2qeq2t+1Dqi+1eqtx=ti+12qeq2t+1eqtx=n=0n+i+1qn+2qn+1q×Tn,qxtn+i+1n+i+1q!=n=0nqn+i+1q!Tni+1,qxtnnq!,E23

    which immediately gives the required result.☐

    Theorem 2.6. Let a, b be any real numbers. Then, we have

    abTn,qxdqx=k=0n+11n+1qTn+1,qbTn+1,qa.E24

    Proof. From Theorem 2.3, we find

    abTn,qxdqx=abk=0nnkqTk,qxnkdqx=k=0nnkqTk,q1nk+1qxnk+1ab=k=0n+1Tn+1,qbTn+1,qan+1q.E25

    Definition 2.7. For x,yC, we define q-tangent polynomial with two parameters as

    n=0Tn,qxytnnq!=2qeq2t+1eqtxeqty,t<π2.E26

    From the Definition 2.7, it is clear that

    n=0Tn,qx0tnnq!=n=0Tn,qxtnnq!=2qeq2t+1eqtx,n=0Tn,q00tnnq!=n=0Tn,qtnnq!=2qeq2t+1,E27

    where Tn,qis q-tangent number. We also note that the original tangent number, Tn,

    limq1n=0Tn,qtnnq!=n=0Tntnn!=2e2t+1,E28

    where q1.

    Theorem 2.8. Let x,ybe any complex numbers. Then, the following hold.

    i.  Tn,qxy=k=0nnkqTnk,qxyk,E29
    ii.  Tn,qxy=l=0nnkqTnl,qk=0llkqxlkyk.

    Proof. From the Definition 2.7, we have

    n=0Tn,qxytnnq!=2qeq2t+1eqtxeqty=n=0Tn,qxtnnq!n=0yntnnq!.E30

    Using Cauchy’s product and the method of coefficient comparison in the above relation, we find (i). Next, we transform q-tangent polynomials in two parameters as

    n=0Tn,qxytnnq!=2qeq2t+1eqtxeqty=n=0Tn,qtnnq!n=0xntnnq!n=0yntnnq!.E31

    Now following same procedure as in (i), we obtain (ii).☐

    Theorem 2.9. Setting y=2in q-tangent polynomials with two parameters, the following relation holds

    2qxn=Tn,qx2+Tn,qx.E32

    Proof. Using q-tangent polynomials and its polynomials with two parameters, we have

    n=0Tn,qx2tnnq!+n=0Tn,qxtnnq!=2qeq2teq2t+1eqtx+2qeq2t+1eqtx=2qeqtxE33

    Now from the definition of q-exponential function, the required relation follows.☐

    Theorem 2.9 is interesting as it leads to the relation

    xn=Tn,qx2+Tn,qx2q.E34

    Theorem 2.10. Let q<1. Then, the following holds

    Tn,qx=k=0nnkq1kTk,1q2xnk.E35

    Proof. To prove the relation, we note that

    e1q2t=Eq2t,E36

    where Eqt=eq1t. Using the above equation we can represent the q-tangent polynomials as

    n=0Tn,qxtnnq!=2qeq2t+1eqtx=2q1+Eq2tEq2teqtx=2qe1q2t+1e1q2teqtx=n=0Tn,1q2tnnq!n=0xntnnq!=n=0k=0nnkq1kTk,1q2xnktnnq!,E37

    which leads to the required relation immediately.☐

    Now we shall find relations between q-tangent polynomials and others polynomials. For this, first we introduce well known polynomials by using q-numbers.

    Definition 2.11. We define q-Euler polynomials, En,qx, and q-Bernoulli polynomials, Bn,qx, as

    n=0En,qxtnnq!=2qeqt+1eqtx,t<π,n=0Bn,qxtnnq!=teqt1eqtx,t<2π.E38

    Theorem 2.12. For x,yC, the following relation holds

    Tn,qxy=12ql=0nnkqTnl,qxml+k=0nlnlkqTk,qxmnkEl,qmy.E39

    Proof. Transforming q-tangent polynomials containing two parameters, we find

    2qeq2t+1eqtxeqty=2qeqtm+1eqtyeqtm+12q2qeq2t+1eqtx.E40

    Thus, for the relation between q-tangent polynomials of two parameters and q-Euler polynomials, we have

    n=0Tn,qxytnnq!=n=0En,qmytnmnnq!n=0Tn,qxtnnq!n=012qtnmnnq!+12q=12qn=0l=0nnlqEl,qmyk=0nlnlkqTk,qxmnktnnq!+12qn=0l=0nnlqEl,qmyTnl,qxmltnnq!,E41

    which on comparing the coefficients immediately gives the required relation. ☐

    Corollary 2.13. From Theorem 2.12, the following hold.

    i.  Tn,qxy=12ql=0nnlqTnl,qxml+k=0nlnlkqTk,qxmnkEl,qmy.E42
    ii.  Tnxy=12l=0nnlTnlxml+k=0nlnlkTkxmnkElmy.

    Theorem 2.14. For x,yC, the following relation holds

    Tn1,qxy=1nql=0nnkqk=0nlnlkqTk,qxmnkTnl,qxmlBl,qmy.E43

    Proof. We note that

    2qeq2t+1eqtxeqty=teqtm1eqtyeqtm1t2qeq2t+1eqtx.E44

    Thus as in Theorem 2.12, we have

    n=0Tn,qxytnnq!=n=0tn1mnnq!1tn=0Bn,qmytnmnnq!n=0Tn,qxtnnq!=n=0l=0nnlqk=0nlnlkqTk,qxmnkBl,qmytn1nq!n=0l=0nnlqTnl,qxmlBl,qmytn1nq!.E45

    The required relation now follows on comparing the coefficients. ☐

    Corollary 2.15. From the Theorem 2.14, the following relations hold.

    i.  Tn1,qxy=1nql=0nnlqk=0nlnlkqTk,qxmnkTnl,qxmlBl,qmy.E46
    ii.  Tn1xy=1nl=0nnlk=0nlnlkTkxmnkTnlxmlBlmy.

    3. The observation of scattering zeros of the q-tangent polynomials

    In this section, we will find the approximate structure and shape of the roots according to the changes in n and q. We will extend this to identify the fixed points and try to understand the structure of the composite function using the Newton method.

    The first five q-tangent polynomials are:

    T0,qx=1+q2,T1,qx=121+q1+x,T2,qx=121+q1+q1+x+xx2,T3,qx=121+q1+q22+qqx+q3x1+q+q2x2+x3,T4,qx=121+q(1+q1+q1+4+qq1+q+q21+q21+3+qq1+q2x+1+q1+q21+q+q2x21+q1+q2x3+x4).E47

    Using Mathematica, we will examine the approximate movement of the roots. In Figure 1, the x-axis means the numbers of real zeros and the y-axis means the numbers of complex zeros in the q-tangent polynomials. When it moves from left to right, it changes to n = 30, 40, 50, and when it is fixed at q = 0.1, the approximate shape of the root appears to be almost circular. The center is identified as the origin, and it has 2.0 as an approximate root, which is unusual.

    Figure 1.

    Zeros of T n , 0.1 x for n = 30, 40, 50.

    Figure 2 shows the shape of the approximate roots when n is changed to the above conditions and fixed at q = 0.5.

    Figure 2.

    Zeros of T n ,0.5 x for n = 30, 40, 50.

    In Figure 2, the shape of the root changes to an ellipse, unlike the q = 0.1 condition, and the widening phenomenon appears when the real number is 0.5. In addition, like the previous Figure 1, we can see that it has a common approximate root at 2.0. In the following Figure 3, n of the far-left figure is 30, and it increases by 10 while moving to the right, and the far-right figure shows the shape of the root when n = 50 and is fixed at q = 0.9.

    Figure 3.

    Zeros of T n ,0.9 x for n = 30, 40, 50.

    In Figure 3, the roots have a general tangent polynomial shape with similar properties (see [22, 23, 24, 25]). If each approximate root obtained in the previous step is piled up according to the value of n, it will appear as shown in Figure 4. The left Figure 4 is q = 0.1 with n from 1 to 50. The middle Figure 4 is q = 0.5 with n from 1 to 50. The right Figure 4 is q = 0.9 with n from 1 to 50.

    Figure 4.

    Zeros of T n , q x for q = 0.1, 0.5, 0.9, 1 ≤ n ≤ 50.

    Let f:DDbe a complex function, with D as a subset of C. We define the iterated maps of the complex function as the following:

    fr:z0f(f((frz0)))E48

    The iterates of f are the functions f,ff,fff,,which are denoted f1,f2,f3,If zC, and then the orbit of z0 under f is the sequence <z0,fz0,ffz0,>.

    We consider the Newton’s dynamical system as follows [12, 15, 20]:

    C:Rx=xTxTx.E49

    R is called the Newton iteration function of T. It can be considered that the fixed points of R are the zeros of Tand all the fixed points of R are attracting. R may also have one or more attracting cycles.

    For xC, we consider T4,qx, and then this polynomial has four distinct complex numbers, aii=1,2,3,4such that T4,qai=0. Using a computer, we obtain the approximate zeros (Table 1) as follows:

    iq = 0.1q = 0.5q = 0.9
    1−0.672809−0.581881 − 0.412941i−1.10249
    2−0.0821877 − 0.710388i−0.581881 + 0.412941i−0.158841
    3−0.0821877 + 0.710388i0.9070241.84004
    41.948182.131742.86029

    Table 1.

    Approximate zeros of T4,qx.

    In Newton’s method, the generalized expectation is that a typical orbit {R(x)} will converge to one of the roots of T4,qxfor x0C. If we choose x0, which is sufficiently close to ai, then this proves that

    limrRx0=ai,fori=1,2,3,4.E50

    When it is given a point x0 in the complex plane, we want to determine whether the orbit of x0 under the action of R(x) converges to one of the roots of the equation. The orbit of x0 under the action of R also appears by calculating until 30 iterations or the absolute difference value of the last two iterations is within 10−6.

    The output in Figure 5 is the last calculated orbit value. We construct a function, which assigns one of four colors for each point according to the outcome of R in the plane. If an orbit of x0 for q = 0.1 converges to −0.672809, −0.0821877 − 0.710388i, −0.0821877 + 0.710388i and 1.94818, then we denote the red, blue, yellow, and sky-blue, respectively(the left figure). For example, the yellow region for the left figure represents the part of the basin of attraction of a3 = −0.0821877 + 0.710388i.

    Figure 5.

    Orbit of x0 under the action of R for T 4 , q x for q = 0.1, 0.5, 0.9.

    If we use T3,0.1xto draw a figure using the Newton method, we can obtain Figure 6. The picture on the left shows three roots, and the colors are blue, red, and ivory in the counterclockwise direction. When we examine the area closely, we can see that it converges to an approximate value in each color area. The convergence value in the blue area is 0.379202+0.523651i, that in the red area is 0.3792020.523651i, and that in the ivory area is 1.8684. We can also see that it shows self-similarity at the boundary point as divided into three areas. The figure on the right is obtained by 2-times iterated q-tangent polynomials, T3,0.12x, and the area is divided into nine colors “gray (x=2.31831), scarlet (x=1.76736+0.216319i), light brown (x=0.137247+0.59473i), sky blue (x=0.604153+1.19884i), blue (x=0.794606+0.378411i), red (x=0.7946060.378411i), ivory (x=0.6041531.19884i), green (x=0.1372470.59473i), and navy blue (x=1.767360.216319i) in the counterclockwise direction. This also shows self-similarity at the boundary.

    Figure 6.

    Orbit of x0 under the action of R for T 3 , 0.1 x , T 3,0.1 2 x .

    In Figure 7, we express the coloring for T3,0.12x.

    Figure 7.

    Palette for escaping points.

    Conjecture 3.1. The q-tangent polynomials always have self-similarity at the boundary.

    We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of C, and f has a fixed point at z0D. Then z0 is said to be (see [6, 16, 20]):

    1. an attracting fixed point if fz0<1;

    2. a repelling fixed point if fz0>1;

    3. a neutral fixed point if fz0=1.

    For example, T3,0.1xhas three points satisfying T3,0.1x=x.

    That is, x0=0.967484,0.33466,2.41214. Since

    ddtT3,0.10.967484=0<1,ddtT3,0.10.33466=0<1E51

    Theorem 3.2. T3,0.1xfor q = 0.1 has two attracting fixed points.

    Using Mathematica, we can separate the numerical results for fixed points of Tn,0.1x. From Table 2, we know that Tn,0.1xhave no neutral fixed point for 1n4. We can also reach Conjecture 3.3.

    Degree nAttractorRepellorNeutral
    1010
    2110
    3210
    4130
    5140

    Table 2.

    Numbers of fixed points of Tn,0.1x.

    Conjecture 3.3. The q-tangent polynomials for n2have at least one attracting fixed point except for infinity.

    In Table 3, we denote RTn,qrxas the numbers of real zeros for rth iteration and RFTn,qrxas the numbers of attracting fixed point on real number. From this table, we can know that number of real fixed points of T3,qrxare less than two. Here, we can suggest Conjecture 3.4.

    rRT3,0.1rxRFT3,0.1rx
    132
    232
    332
    4232
    522
    611

    Table 3.

    The numbers of RT3,0.1rxand RFT3,0.1rxfor 1r6.

    Conjecture 3.4. The q-tangent polynomials that are iterated, T3,0.1rx, have real fixed point, α=0.33466.

    In the top-left of Figure 8, we can see the forms of 3D structure related to stacks of fixed points of T3,0.1rxfor 1r6. When we look at the top-left of Figure 8 in the below position, we can draw the top-right figure. The bottom-left of Figure 8 shows that image and n-axes exist but not real axis in three dimensions. In three dimensions, the bottom-right of Figure 8 is the right orthographic viewpoint for the top-left figure,-that is, there exist real and n-axes but there is no image axis (Figure 8).

    Figure 8.

    Stacks of fixed point of T 3 , 0.1 r x for 1 ≤ r ≤ 6.

    4. Conclusion

    We can see that when q comes closer to 0, the approximate shape of the roots become increasingly more circular. Also in this situation, we can observe scattering of zeros in q-tangent polynomials around 2 in three-dimension. When q comes closer to 1, it has properties that are more symmetrical. We can also assume that the property that appears when iterating Tn,qxhas self-similarity. By iterating, we can conjecture some properties about fixed points. This property warrants further study so that we can create a new property.

    Acknowledgments

    This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2017R1E1A1A03070483).

    Conflict of interests

    The authors declare that there is no conflict of interests regarding the publication of this paper.

    © 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Jung Yoog Kang and Cheon Seoung Ryoo (January 23rd 2019). A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials, Polynomials - Theory and Application, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.83497. Available from:

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