A Numerical Computation of Zeros of q -Generalized Tangent-Appell Polynomials

: The intended objective of this study is to deﬁne and investigate a new class of q -generalized tangent-based Appell polynomials by combining the families of 2-variable q -generalized tangent polynomials and q -Appell polynomials. The investigation includes derivations of generating functions, series deﬁnitions, and several important properties and identities of the hybrid q -special polynomials. Further, the analogous study for the members of this q -hybrid family are illustrated. The graphical representation of its members is shown, and the distributions of zeros are displayed.


Introduction and Preliminaries
The area of q-calculus in the last three decades act as a bridge between engineering sciences and mathematics. Recently, research in the area of q-calculus has shown worthy of attention due to its applicative diversification in various fields such as mathematics, physics, and engineering. The q-analogues of many orthogonal polynomials and functions expect a pleasant structure, and help one to remember their classical counterpart. The q-standard notations and definitions reviewed here are taken from [1].
The q-power basis is specified as (u + v) n q = n ∑ k=0 n k q q k(k−1)/2 u n−k v k , n ∈ N 0 .
The q-derivative D q of functions e q (u) and E q (u) are given by D q e q (ut) = te q (ut), D q E q (ut) = tE q (qut).
The q-derivative operator D q for any two arbitrary functions f (u) and g(u) satisfies the following product and quotient relations: D q ( f (u)g(u)) = f (qu)D q g(u) + g(u)D q f (u) = f (u)D q g(u) + g(qu)D q f (u), (9) The tangent numbers and polynomials and their q-analogue have enormous applications in analytic number theory, physics, and other related areas. Various properties of these polynomials are studied and investigated by many mathematicians, see, for example, [2][3][4]. Very recently, Yasmin et al. [5] introduced the 2-variable q-generalized tangent polynomials C n,m,q (u) and established certain interesting results for them. We recall the following definition. Definition 1. The 2-variable q-generalized tangent polynomials (qGTP) C n,m,q (u) is defined as [5]: 2 e q (mt) + 1 e q (ut)E q (vt) = ∞ ∑ n=0 C n,m,q (u, v) t n [n] q ! , |mt| < π, m ∈ R + .
A vital class of polynomial sequences known as the Appell polynomials is introduced by Appell [8]. Later, the class of q-Appell polynomials {A n,q (u)} ∞ n=0 was introduced by Al-Salam [9] and studied some of its properties. These polynomials arise in chemistry, theoretical physics, and different branches of mathematics such as numerical analysis, number theory, and in the study of polynomial expansion of analytic function. Definition 2. The q-Appell polynomials A n,q (u) are defined by the following generating function [9]: where is an analytic function at t = 0 and A n,q := A n,q (0) are q-Appell numbers.
The series representation of q-Appell polynomials is given by A significant part of the investigation of any polynomials is to discover its determinant definition. Recently, Keleshteri et al. [10] gave the determinant definition of the q-Appell polynomials, according to which the q-Appell polynomials A n,q (u) of degree n can be expressed in the following determinant form: where n = 1, 2, 3, · · · ; β 0,q = 0 and Various members of the q-Appell family can be obtained by choosing a suitable function A q (t) in the generating function expressed in Equation (13). Some of its members along with their name, generating function, and series definition, are mentioned in Table 1.
The hybrid type q-special polynomials are a subject of recent interest. In the present work, we introduce and investigate the properties of q-generalized tangent-based Appell polynomials. Their series expansion, determinant form, summation formulae, and differential recurrence relations are obtained in Section 2. In Section 3, some identities and relations involving q-generalized tangent-based Appell numbers and polynomials are derived. In the last section, the graphical representation of its members are shown for different values of indices using Matlab. Further, the distributions of zeros of these members are displayed.  [11,12] B n,q := B n,q (0) [6,12] E n,q := E n,q (0)

q-Generalized Tangent-Appell Polynomials
In this section, we introduce the q-generalized tangent-based Appell polynomials (qGTAP) by means of a generating function. Further, some properties of these polynomials are also obtained.
Using the expansion expressed in Equation (5) in the generating function expressed in Equation (13) of q-Appell polynomials and then replacing powers of u ie u 0 , u 1 , u 2 , · · · , u n by the corresponding qGTP C 0,m,q (u, v), C 1,m,q (u, v), C 2,m,q (u, v), · · · , C n,m,q (u, v) and thereafter using the generating function expressed in Equation (11) of qGTP C n,m,q (u, v) and denoting the resultant q-generalized tangent-based Appell polynomials by C A (m) n,q (u, v), the following definition is obtained: n,q (u, v) q ∈ C, 0 < |q| < 1, |mt| < π, m ∈ R + are defined by means of the following generating function: n,q are the corresponding q-generalized tangent-Appell numbers and are defined as Selecting suitable function A q (t) and appropriate values of m in the generating function expressed in Equation (18), several members belonging to the family of qGTAP C A (m) n,q (u, v) are obtained. These members are listed in Table 2.
n,q t n [n]q ! Remark 1. As for m = 2, the qGTP C n,m,q (u) reduces to the qTP T n,q (u, v). Thus, for the same choice of m, the results of qGTBP C B (m) n,q (u, v) and qGTEP C E (m) n,q (u, v) ( Table 2) reduces to the corresponding results of the q-tangent Bernoulli and q-tangent Euler polynomials.

Remark 2.
As for m = 1, the qGTP C n,m,q (u) reduces to the qEP E (m) n,q (u, v). Thus, for the same choice of m, the results of qGTBP C B (m) n,q (u, v) and qGTEP C E (m) n,q (u, v) ( Table 2) reduces to the corresponding results of the q-Euler Bernoulli and 2-iterated q-Euler polynomials.
The determinant definitions are helpful in finding solutions to general linear interpolation problems and can likewise be valuable for calculation purposes. The recent establishment of determinant definitions for various hybrid polynomials (see, for instance, [10,13]) offers inspiration to establish the determinant definition for qGTAP C A n,q (u, v) of degree n holds true: where n = 1, 2, 3, · · · ; β 0,q = 0 and k,q β n−k,q , n = 1, 2, 3, · · · . (21) Table 2 are particular members of qGTAP C A (m) n,q (u, v). Thus, by making appropriate choices for the coefficients β 0,q and β k,q (k = 1, 2, · · · , n) in the determinant definition of qGTAP C A (m) For instance, taking β 0,q = 1 and β k,q = 1 [k+1] q (k = 1, 2, · · · , n) in Equation (20), the following determinant form of qGTBP C B (m) n,q (u, v) is obtained: Next, taking β 0,q = 1 and β k,q = 1 2 (k = 1, 2, · · · , n) in Equation (20), the following determinant definition of qGTEP C E (m) n,q (u, v) is obtained: Definition 6. The following determinant form for the qGTEP C E (m) n,q (u, v) of degree n holds true: Utilizing generating function of q-Appell numbers and the relation expressed in Equation (11) in the generating function expressed in Equation (18) Another form of series representation of qGTAP is obtained by utilizing a generating function for qGTN, the expansion expressed in Equation (6), and the generating function expressed in Equation (13) in the relation expressed in Equation (18) and then employing the Cauchy product rule in the resultant expression and thereafter comparing the coefficients of similar powers of t in the resultant equation.
We then obtain the following series expansion of qGTAP C A (m) n,q (u, v): Also, utilizing the expansions expressed in Equations (5) and (6) Next, we establish the following summation formulae.
n,q (u, v) satisfies the following summation formulae: Proof. Using the expansions expressed in Equations (5) and (6) and Equation (19) in the generating function expressed in Equation (18), we obtain Now, applying the Cauchy product rule and the expansion expressed in Equation (4) in Equation (30) and then comparing the coefficients of similar powers of t in the resultant equation, we are led to the assertion expressed in Equation (27). Using the generating function expressed in Equation (18) (taking u = 0) and the expansion expressed in Equation (5) in the generating function expressed in Equation (18), we obtain which, upon employing the Cauchy product rule and the expansion expressed in Equation (3) and then comparing the coefficients of similar powers of t in the resultant equation, yields the assertion expressed in Equation (28). Using the generating function expressed in Equation (18) (taking v = 0) and the expansion expressed in Equation (6) in the generating function expressed in Equation (18), we obtain Further, employing the Cauchy product rule and the expansion expressed in Equation (3) and then comparing the coefficients of similar powers of t in the resultant equation, we are led to the assertion expressed in Equation (29).
Theorem 5. The following differential recurrence relations of qGTAP C A (m) n,q (u, v) hold true: Proof. By q-differentiating the generating function expressed in Equation (18) with respect to u, using Equation (8)  n,q (u, v) are given in Table 3. Table 3. Results for qGTBP C B (m) n,q (u, v) and qGTEP C E (m) n,q (u, v).

S. No.
Results

Identities Involving q-Generalized Tangent-Appell Polynomials
In this section, we derive some identities involving qGTAP C A (m) n,q (u, v).

Theorem 6. The following identities of qGTAP C A (m)
n,q (u, v) hold true: Proof. By q-differentiating the generating function expressed in Equation (18) with respect to u, we obtain which, upon using Equation (18) on the left hand side and then comparing the coefficients of similar powers of t in the resultant equation, we obtain the assertion expressed in Equation (37). Now, taking the definite q-integral of Equation (37), we obtain which gives the assertion expressed in Equation (38).
Similarly q-differentiating the generating function expressed in Equation (18) with respect to v, we obtain the following result.

Theorem 7. The following identities of qGTAP C A (m)
n,q (u, v) hold true: In order to derive our next result, we first recall the 2D q-Appell polynomials A n,q (u, v).

Definition 7.
The 2D q-Appell polynomials A n,q (u, v) are defined by the following generating function [10]: where A q (t) is an analytic function at t = 0 given by Equation (14) and A n,q := A n,q (0, 0) are q-Appell numbers.
Multiplying both side of the above identity by A q (t)e q (ut)E q (vt), we obtain 2 eq(mt)+1 A q (t)e q (ut)E q (vt)e q (mt) + 2 eq(mt)+1 A q (t)e q (ut)E q (vt) = 2A q (t)e q (ut)E q (vt).
Now, using Equations (5), (18), and (43), we obtain which on employing the Cauchy product rule and comparing the coefficients of similar powers of t gives the assertion expressed in Equation (44).
Putting v = 0 in Theorem 8, we have the following corollary.

Corollary 1. The following identity of one variable qGTAP C A (m)
n,q (u) holds true: Putting u = v = 0 in Theorem 8, we have the following corollary. n,q (u, v) and q-Bernoulli polynomials B n,q (u) holds true: Proof. Consider the generating function expressed in Equation (18) in the form Making use of the generating function of q-Bernoulli polynomials B n,q (u) in Table 1 (I) and the generating function of qGTAP C A (m) n,q (u, v) expressed in Equation (18) in suitable forms gives Simplifying and employing the Cauchy product rule and thereafter comparing the coefficients of similar powers of t gives the assertion expressed in Equation (50).

Theorem 10. The following identity of qGTAP C A (m)
n,q (u, v) and q-Bernoulli polynomials B n,q (u) holds true:

Proof. Consider the generating function expressed in Equation (18) in the form
Making use of the expansion expressed in Equation (5), the generating function of q-Bernoulli polynomials B n,q (u) in Table 1 (I), and the generating function of qGTAP C A (m) n,q (u, v) expressed in Equation (18), we obtain Simplifying and employing the Cauchy product rule and thereafter comparing the coefficients of similar powers of t gives the assertion expressed in Equation (53).

Theorem 11. The following identity of qGTAP C A (m)
n,q (u, y) and q-Euler polynomials E n,q (u) holds true: (56)

Proof. Consider the generating function expressed in Equation (18) in the form
Making use of the generating function of q-Euler polynomials E n,q (u) in Table 1 (II) and the generating function of qGTAP C A (m) n,q (u, v) expressed in Equation (18) in suitable forms gives Further, employing the Cauchy product rule and comparing the coefficients of similar powers of t gives the assertion expressed in Equation (56).

Theorem 12.
The following identity of qGTAP C A (m) n,q (u, v) and q-Euler polynomials E n,q (u) holds true: Proof. Consider the generating function expressed in Equation (18) in the form Making use of the expansion expressed in Equation (5), the generating function of q-Euler polynomials E n,q (u) in Table 1 (II), and the generating function of qGTAP C A (m) n,q (u, v) expressed in Equation (18), we obtain Simplifying and employing the Cauchy product rule and thereafter comparing the coefficients of similar powers of t gives the assertion expressed in Equation (59). n,q (u, v) are given in Table 4. n,q (u, v) and qGTEP C E (m) n,q (u, v).

S. No. Identities Involving q-Generalized Tangent Identities Involving q-Generalized Tangent -Bernoulli Polynomials qGTBP C B
(m) In the next section, the graph of qGTAP C A (m) n,q (u, v) are displayed by using Matlab. The analysis of the zeros of these polynomials are also carried out using numerical computations.

Graphical Representation and Computation of Zeros
This section intends to exhibit the benefit of employment of numerical investigation and to discover a new, interesting pattern of the zeros of the qGTAP and to support theoretical prediction.
The qGTBP C B (m) n,q (u, v) can be determined explicitly. A few of them are as follows: We display the shapes of the qGTBP C B (m) n,q (u, v) and investigate its zeros. We plot the graph of qGTBP C B (m) n,q (u, v) for n = 1, 2, 3, · · · , 10 combined together. The shape of qGTBP C B (m) n,q (u, v) for −3 ≤ u ≤ 3, v = 2, m = 3, and q = −1/3 are displayed in Figure 1.
Our numerical results for the number of real and complex zeros of the qGTBP C B (m) n,q (u, v) for v = 2, m = 3 and q = −1/3 are listed in Table 5.
Using computers, several values of n were verified. However, it remains unknown whether the following conjecture is true or false for all values of n (see Tables 5 and 6 and Figure 2).
n,q (u, b), u ∈ C, has Im(u) = 0 reflection symmetry. However, n,q (u, b) has no Re(u) = a reflection symmetry for a ∈ R.

Stacks of zeros of qGTBP C B
(m) n,q (u, v) = 0 for v = 2, m = 10, q = −1/3 and 1 ≤ n ≤ 20 form a 3-D structure and are presented in Figure 3. Next, we plot the real zeros of the qGTBP C B (m) n,q (u, v) = 0 for u ∈ R, v = 2, m = 10, q = −1/3 and 1 ≤ n ≤ 20 in Figure 4.  The qGTEP C E (m) n,q (u, v) can be determined explicitly. A few of them are as follows: We display the shapes of the qGTEP C E (m) n,q (u, v) and investigate its zeros. We plot the graph of qGTEP C E (m) n,q (u, v) for n = 1, 2, 3, · · · , 10 combined together. The shape of qGTEP C E (m) n,q (u, v) for −5 ≤ u ≤ 5, v = 2, m = 3, and q = 1/3 are displayed in Figure 5. We observed a remarkable regular structure of zeros of the qGTBP C B (m) n,q (u, v) = 0 and hope to verify the same kind of remarkable regular structure of zeros of the qGTEP C E (m) n,q (u, v) = 0. Our numerical results for the number of real and complex zeros of the qGTEP C E (m) n,q (u, v) are listed in Tables 7 for v = 2, m = 3, and q = 1/3. Table 7. Numbers of real and complex zeros of C E (m) n,q (u, v). Next, we calculated an approximate solution satisfying the qGTEP C E (m) n,q (u, v) = 0 for u ∈ R, v = 2, m = 3, and q = 1/3. The results are given in Table 8. We investigate the beautiful zeros of the qGTEP C E (m) n,q (u, v) = 0 by using a computer. The zeros of the qGTEP C E (m) n,q (u, v) = 0 for u ∈ C are displayed in Figure 6. In Figure 6 (top-left), we choose n = 20, m = 10, q = 1/3, and v = 2. In Figure 6 (top-right), we choose n = 20, m = 20, q = 1/3, and v = 2. In Figure 6 (bottom-left), we choose n = 20, m = 30, q = 1/3, and v = 2. In Figure 6 (bottom-right), we choose n = 20, m = 40, q = 1/3, and v = 2.

Degree n Number of Real Zeros Number of Complex Zeros
Stacks of zeros of C E (m) n,q (u, v) = 0 for v = 2, m = 10, q = 1/3, and 1 ≤ n ≤ 26 form a 3-D structure and are presented in Figure 7. Next, we plot the real zeros of the qGTEP C E (m) n,q (u, v) = 0 for u ∈ R, v = 2, m = 10, q = 1/3, and 1 ≤ n ≤ 26 in Figure 8. From all the numerical computations done in this research work, we give the following conjectures: n,q (u, b), u ∈ C, has Im(u) = 0 reflection symmetry. However, C E (m) n,q (u, b) has not Re(u) = a reflection symmetry for a ∈ R. Using computers, several values of n have been verified. However, it is still remains unknown if these conjectures hold true or not for any value of n (see Tables 7 and 8 and Figure 6). We expect that the research in this direction will be a new approach using numerical methods for the study of the qGTAP C A (m) n,q (u, v).
Author Contributions: All authors contributed equally to this work. All authors have read and approved the final manuscript.