q -Generalized Tangent Based Hybrid Polynomials

: In this paper, we incorporate two known polynomials to introduce so-called 2-variable q -generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q -derivative and q -integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. Further, by using computer-aided programs (for example, Mathematica or Matlab), we draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. Lastly we provide numerous observations and questions which naturally arise amid the present investigation.

Moreover, the tangent polynomials and numbers, and their diverse extensions including their q-analogues have many applications in a number of research areas such as analytic number theory and physics (see, e.g., [3,12,13,19] and the references therein). For example, a new class of q-generalized tangent-based Appell polynomials by welding 2-variable q-generalized tangent polynomials and q-Appell polynomials is introduced and investigated in [19].
Lemma 1. If f (t) exists in a neighborhood of t = 0 and is continuous at where 0 ≤ a < b < ∞.
We recall the q-generalized tangent polynomials and numbers in [18] (Definition 2.1) whose restrictions may be slightly amended as in the following definition. Definition 1. (cf. [18]) The q-generalized tangent polynomials C n,m,q (u) in the variable u (abbreviated as qGTP) are defined by means of the generating function Here ξ is the smallest one among the absolute values of all complex zeros of e q (mt) + 1 = 0. The cases C n,m,q := C n,m,q (0) are called q-generalized tangent numbers.
Definition 3. (see [9]) The q-Apostol-Euler polynomials E (α) n,q (u, v; λ) of order α in variables u and v (abbreviated as qAEP) are defined by means of the generating function Here E n,q (0, 0; λ) are called the q-Apostol-Euler numbers.

Remark 1.
The constraints in Definitions 2-5 should and can be modified as those in Definition 6 (see also Definition 1).

q-Generalized Tangent-Apostol Type Frobenius-Euler Polynomials and Their Related Formulas
In this section, we introduce the q-generalized tangent based Apostol type Frobenius-Euler polynomials C H (α,m) n,q (u, v; ρ; λ) and investigate some of their properties. Definition 6. The 2-variable q-generalized tangent based Apostol type Frobenius-Euler polynomials C H (α,m) n,q (u, v; ρ; λ) of order α in the variables u and v (abbreviated as qGTATFEP) are defined by means of the following generating function where ξ is the same as in the restrictions of Definition 1 and η is the smallest nonzero one among the absolute values of all complex zeros of e q (t) are called the q-generalized tangent-Apostol type Frobenius-Euler numbers of order α.
By selecting suitable parameters in generating function (27), we obtain several members belonging to the family of qGTATFEP C H (α,m) n,q (u, v; ρ; λ), which are listed in Table 1.
be the generating function in (27). We present two series representations for the polynomials qGTATFEP by using series manipulation techniques in some combinations of the polynomials and numbers in Section 1 as in the following theorem. Theorem 1. Let n ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). and Proof. We find from (27) and (30) that where a series rearrangement technique (or Cauchy product for double series) of a double sequence A n,k of real or complex numbers (see, e.g., [30]): the middle double series being absolutely convergent under the given conditions, is used to give the last equality. Then, identifying the right-hand sides of (27) and (33), and equating the coefficients of t n on both sides of the resulting identity, we obtain the desired identity (31). Similarly, factoring the right member of (30) so that (8), (19) and (26) can be used, we may get (32). The details are omitted.
We establish three summation formulae for the polynomials C H (α,m) n,q (u, v; ρ; λ) as in Theorem 2.
Theorem 2. Let n ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28).
Proof. For (34), we factor the generating function (30) so that (29), (7) and (8) can be used in order and use series rearrangement technique, with the aid of (6), to get Now a similar process of the proof of Theorem 1 may be applied in (37) to obtain (34). The remaining details and proofs of the other two identities are omitted.

Theorem 4. (Addition formula)
Let n ∈ N 0 and β ∈ R + . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28).
Proof. Factor the generating function Then, with the aid of (25)- (27) and (29), using the similar process of the proofs of the previous theorems, we can obtain (39). The details are omitted.
Theorem 5. Let n ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). Then Proof. Consider the identity 2 e q (mt) in the generating function (27)

Explicit Representations
In this section, we present explicit expressions for some numbers and polynomials which are chosen from the previous sections as in the following remarks. Remark 3. Let n ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). Then the q-generalized tangent numbers C n,m,q in Definition 1 are explicitly given by where the sum is over all nonnegative integers 1 , 1 , . . ., n that satisfy 1 + 2 2 + · · · + n n = n, and k = 1 + 2 + · · · + n . The first few of them are

Remark 5.
Let n ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). Then, from Definition 6, the q-generalized tangent-Apostol type Frobenius-Euler numbers C H (α,m) n,q (ρ; λ) of order α are given by The first few of them are We find from (7) and (8) that where The first few of E n,q (u, v) are From Definition 6, we find that the 2-variable q-generalized tangent based Apostol type Frobenius-Euler polynomials C H (α,m) n,q (u, v; ρ; λ) of order α in the variables u and v are given by The first few of them are

q-Derivative and q-Integral Formulas
In this section, we establish q-derivative and q-integral formulas for the polynomials C H (α,m) n,q (u, v; ρ; λ), which are in the following theorems.
Theorem 6. Let n, , r ∈ N 0 . Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). Then n−r− ,q (u, q r v; ρ; λ). (52) Proof. Use (7) to expand the left member of (27), and q-differentiate both sides of the resulting series term-by-term with respect to u with the aid of the first formula in (13), and match the coefficients of t n on both sides of the last resultant identity to give (48). A successive use of the process of the proof of (48), r times is found to easily provide (49). So the details of the proof of (49) including (50)-(52) are omitted.
A q-derivative formula of the polynomials C H (α,m) n,q (u, v; ρ; λ) with respect to m is established as in the following theorem. Theorem 7. Let n ∈ N. Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28). Then Proof. Using (13) and (15), we have D q,m 2 e q (mt) + 1 = − 2t e q (qmt) + 1 + 2t e q (mt) + 1 e q (qmt) + 1 .
Further, q-differentiating both sides of (27) termwise with respect to m, with the aid of (54), Definition 1, and we obtain Using C n,m,q := C n,m,q (0) in (19) and (27) in the last expression, we get Setting n + 1 = n in the last summations and dropping the prime on n, we find Employing the following series manipulation for a double sequence B n,k of real or complex numbers (both sides are absolutely convergent) in the last double series, we get Finally, upon matching the coefficients of t n on both sides of (56) yields (53).
Two q-integral formulas are presented in the following theorems.
Theorem 8. Let 0 ≤ a < b < ∞, 0 < q < 1, and n ∈ N. Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28).
Proof. Employing the formula for a fundamental theorem of q-calculus (18) in the first identity in (13), we can obtain b a e q (ut) d q u = 1 t e q (bt) − e q (at) .
On q-integrating both sides of (27) with respect to the variable u and using (58), we get Finally, equating the coefficients of t n on both sides of (59) leads to the formula (57).

Theorem 9.
Let 0 ≤ a < b < ∞, 0 < q < 1, and n ∈ N. Moreover, let the other parameters and variables in the identities below be assumed to satisfy the restrictions (28).
Proof. Employing the formula for a fundamental theorem of q-calculus (18) in the second identity in (13) On q-integrating both sides of (27) with respect to the variable v and using (61), we get Hence, identifying the coefficients of t n on both sides of (62) produces the formula (60).

Graphical Representations and Locations of Zeros
In this section, by using Mathematica, we draw graphs of C H (α,m) n,q (u, v; ρ; λ) for some chosen n and particular parameters to examine several of their properties such as shapes, surface plot, zeros. In particular, we observe their zeros in several ways.
The numbers of real and complex zeros of C H (1,2) n,1/2 (u, 0; 5; 1) along with its approximate values are listed in Table 3. If each approximate real zeros of C H (1,2) n,1/2 (u, 0; 5; 1), (u ∈ R) is piled up according to the value of n for 1 ≤ n ≤ 20, it will appear as shown in Figure 6. The values of real zeros for 1 ≤ n ≤ 9 are listed in Table 3.

Concluding Remarks, Further Observations, and Posing Questions
Recently, due mainly to their importance and diverse applications, a growing number of polynomials and numbers, and their variants and generalizations have been introduced and investigated. In the wake of this trend, by combining the polynomials in Definitions 1 and 5, we introduced the 2-variable q-generalized tangent based Apostol type Frobenius-Euler polynomials C H (α,m) n,q (u, v; ρ; λ) of order α in the variables u and v. Then we presented a number of properties and formulas for these polynomials such as explicit representations, series representations, summation formulas, addition formula, q-derivative and q-integral formulas. Moreover, using computer-aid programs (e.g., Mathematica, or Matlab), we tried to draw graphs of certain specialized polynomials introduced here. Through those graphs, a number of questions about certain unexpected properties of the polynomials (for example, their zeros) are found to be naturally occurred.
We tried to apply these newly-introduced polynomials to a real world problem (for example, computational fluid dynamics [32,33]). However, we find that it will take a longer period to be familiar with such topics. It remains to be a future investigation.

Observations and Questions
(i) It may be important to find complex zeros of the following equations λe q (t) − ρ = 0 and e q (mt) + 1 = 0 (63) from Definitions 1 and 6 (see also generating functions in Definitions 2, 3, and 5), in particular, in order to determine the ξ and η there exactly. When q = 1, the zeros of two equations in (63) are easily given, respectively, by where ϑ is an argument of ρ λ . Question 1: Find or approximate the zeros of two equations in (63). For Question 1, we tried to draw graph of λe q (t) − ρ (for λ = 1, q = 1 2 and ρ = 5) as follow Figures 8 and 9: Graph of e q (mt) + 1 (for m = 2 and q = 1 2 ) as follows:  (ii) To approximate zeros of some functions or polynomials, we can use Newton-Raphson's theorem (see, e.g., [34] (pp. 262-263); for a use of this theorem, one may consult with [11] (Section 6)). (iii) It may follow from (47) that C H (α,m) n,q (u, v; ρ; λ) are polynomials in both u and v of the same degree n.
(iv) As shown in Figure 5, all zeros of the polynomials C H (α,m) n,q (u, b; ρ; λ) (b ∈ R) with the other parameters being real are found to be symmetrically located with respect to the real axis of u (that is, (u) = 0). Indeed, if u 0 is among its zeros, then, in view of (47), we have which implies that the complex conjugate u 0 of u 0 is also zero. One may also recall the reflection principle (see, e.g., [35] (p. 57)).
(v) In Figure 5, as m becomes larger, the corresponding absolute values (distances from the origin) of zeros of C H (1,m) 20,1/2 (u, 0; 5; 1) are getting greater (become more distant from the origin). Question 2: Prove or disprove that this observation is true as m ↑ ∞. Question 3: Prove or disprove truth of this observation for C H (α,m) n,q (u, 0; ρ; λ) where m ∈ R + becomes larger and (n ∈ N, 0 < q < 1, α, λ, ρ ∈ R). This can be observed graphically. For several different values of m (−10,000, −1000, 1000, 10,000), graphs of zeros of C H (1,2) 20,1/2 (u, 0; 5; 1) are demonstrated in Figure 10. (vi) From Figure 6, the number of real zeros of C H (1,2) n,1/2 (u, 0; 5; 1) (1 ≤ n ≤ 20) is observed to range from 1 to 4. Question 4: Prove or disprove that this observation is true for general n ∈ N. Question 5: Prove or disprove truth of this observation for C H (α,m) n,q (u, 0; ρ; λ) where n ∈ N varies and (0 < q < 1, α, λ, ρ ∈ R, m ∈ R + ). For C H (1,2) n,1/2 (u, 0; 5; 1), it is observed experimentally (Mathematica) for n up to 200 that for even values of n ≥ 10, number of real zeros are 2 and for odd values of n ≥ 10, number of real zero is 1. For n < 10, number of zeros are mentioned in Table 3. (vii) In each of Definitions 1-5 and Definition 6, the ordinary Taylor (Maclaurin) series expansion is employed, even though each generating function is involved in q-analogues. Question 6: In the above definitions, it may be really interesting and speculative to see the possible resulting series if the q-Taylor series expansion (see, e.g., [26,28] (Theorem 6.3)) is used, instead of the ordinary Taylor series expansion. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.