SOME PROPERTIES OF DIFFERENTIAL EQUATIONS OF HIGHER-ORDER q-FROBENIUS-TANGENT POLYNOMIALS IDREES AHMAD KHAN AND SUMIT KUMAR Abstract. The classical q-Frobenius-tangent polynomials dealt within this pa- per contains many application in various areas. We construct new types of dif- ferential equations for q-Frobenius-tangent polynomials using q-derivatives, find some properties and several difference equations of these polynomials. 1. Introduction In recent years, numerous researchers have explored the Bernoulli, Euler, Genoc- chi, Frobenius-Euler, and tangent polynomials in their classical, generalized, and uni- fied forms to investigate their properties, relationships, and applications [5, 6, 7, 9, 10]. Building on these studies, Jackson introduced the q-Bernoulli, q-Euler, and q-Genocchi polynomials [1, 2], while Kang [6, 8] and Kang and Kim [7] examined generalized q-tangent polynomials. Kang and Khan [5] studied q-Frobenius-Euler polynomials, Nisar et al. [13] introduced q-Frobenius-tangent polynomials, and Ryoo and Kang [15, 16] investigated the q-differential equation forms of Euler and Genoc- chi polynomials. These works have uncovered numerous properties, relationships, and applications in fields such as umbral calculus, p-adic analysis, and combinatorics. Let σ ∈ R , p(ψ) and g(ψ) are continuous function, the equation of Bernoulli polynomials as dϕ dψ + p(ψ)ϕ− g(ψ)ϕσ = 0, (1.1) Let σ = 0, we will get linear equation and it is not nonlinear equation. If η = ϕ1−σ in (1.1), we get differential equation of Bernoulli polynomials. dη dψ + (1− σ)p(ψ)η = (1− σ)g(ψ), Putting σ = 0, the equation (1.1) gives the differential equation of the Frobenius- tangent polynomials as follows. d dψ FTυ(ψ; η) + 1 1− η FTυ(ψ; η) + 1 1− η FT0(ψ; η)− ψυ = 0, (1.2) where FTυ(ψ; η) is the Frobenius-tangent polynomials are as follows ∞∑ υ=0 FTυ(ψ; η) φυ υ! = 1− η e(1−η)φ − η eψφ. (1.3) The corresponding Frobenius-tangent numbers have also been derived by FTυ(η) = FTυ(0; η). 2010 Mathematics Subject Classification. 81P15, 11B83; 33B10; 34A34. Key words and phrases. q-numbers, q-derivative, q-Frobenius-tangent polynomials, differential equation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3218 Article History: Received: 21-02-2025 Revised: 25-03-2025 Accepted: 29-04-2025 2 IDREES AHMAD KHAN AND SUMIT KUMAR By the above method, the first order differential equation of q-Bernoulli differential can written as Dqϕ+ p(ψ)ϕ− g(ψ)ϕσ = 0 in q-calculus. Again, if σ = 0, the equation (1.1) will give the first order q-differential equation of q-Frobenius-tangent polynomials as D (1) q,ψFTυ,q(ψ; η) + (1− η)−1 (FT0,q(ψ; η) + FTυ,q(ψ; η))− ψυ = 0, (1.4) and Dq is called the q-derivative and FTυ,q(ψ; η) is the q-Frobenius-tangent poly- nomials. Let υ ∈ Z0 and η ∈ Z, the q-Frobenius-tangent polynomials are defined by (see [7]) 1− η eq((1− η)φ)− η eq(ψφ) = ∞∑ υ=0 FTυ,q(ψ; η) φυ [υ]q! . (1.5) The corresponding q-Frobenius-tangent numbers have also been derived by FTυ,q(η) = FTυ,q(0; η). It is worthy note that if q → 1 then (1.5) becomes (1.2). The main purpose of this paper is to establish higher-order differential equations for the q-Frobenius-tangent polynomials as defined by (1.5). Building on this concept, we will apply the theory of q-calculus throughout the paper. Let us begin by introducing some definitions from q-calculus theory. The shifted factorial (ϕ)υ in term of q-analogue is given by [3, 4] (ϕ; q)0 = 1, (ϕ; q)υ = υ−1∏ σ=0 (1− qσϕ), υ ∈ N. The factorial function in q-calculus theory given by [ϕ]q = 1− qϕ 1− q , q ∈ C− {1};ϕ ∈ C, [υ]q! = υ∏ σ=1 [σ]q = [1]q[2]q · · · [υ]q = (q; q)υ (1− q)υ , q ̸= 1; υ ∈ N, [0]q! = 1, q ∈ C; 0 < q < 1. The definition q-binomial coefficient of Gauss ( υ ψ ) q is given by( υ ψ ) q = [υ]q! [ν]q![υ − ν]q! = (q; q)υ (q; q)ν(q; q)υ−ν , ν = 0, 1, · · · , υ. The function (ψ + ϕ)υq is given by (ψ + ϕ)υq = υ∑ ν=0 ( υ ν ) q qν(ν−1)/2ψυ−νϕν , υ ∈ N0. (1.6) The definition of exponential function in q-calculus theory is given by eq(ψ) = ∞∑ υ=0 ψυ [υ]q! = 1 ((1− q)ψ; q)∞ , 0 <| q |< 1; | ψ |<| 1− q |−1, (1.7) For ψ ̸= 0, the definition of q-derivative Dq,ψf(ψ) as Dq,ψf(ψ) = Dqf(ψ) = f(ψ)− f(qψ) (1− q)ψ , (1.8) and Dqf(0) = f ′ (0). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3219 SOME PROPERTIES OF DIFFERENTIAL EQUATIONS OF HIGHER-ORDER 3 Here the function f is differentiable at zero, and it is obvious thatDqψ υ = [υ]qψ υ−1. Let us point out that D (ν) q,ψf(ψ) converges to f (ν)(ψ) as q goes to 1. By(1.8), the some formulae of q-derivative. (i) Dq (f(ψ)g(ψ)) = q(ψ)Dqf(ψ) + f(qψ)Dqg(ψ) = f(ψ)Dqg(ψ) + g(qψ)Dqf(ψ), (ii) Dq ( f(ψ) g(ψ) ) = g(qψ)Dqf(ψ)− f(qψ)Dqg(ψ) g(ψ)g(qψ) = g(ψ)Dqf(ψ)− f(ψ)Dqg(ψ) g(ψ)g(qψ) , (iii) for any constant a and b, Dq (af(ψ) + bg(ψ)) = aDqf(ψ) + bDqg(ψ). The q-Bernoulli Bυ,q(ψ), the q-Euler Eυ,q(ψ) and q-Genocchi polynomials Gυ,q(ψ) are defined by (see [11, 12, 14]): φ eq(φ)− 1 eq(ψφ) = ∞∑ υ=0 Bυ,q(ψ) φυ [υ]q! (| φ |< 2π), (1.9) 2 eq(φ) + 1 eq(ψφ) = ∞∑ υ=0 Eυ,q(ψ) φυ [υ]q! (| φ |< π), (1.10) 2φ eq(φ) + 1 eq(ψφ) = ∞∑ υ=0 Gυ,q(ψ) φυ [υ]q! (| ψ |< π), (1.11) respectively. Clearly, we have Bυ,q = Bυ,q(0),Eυ,q = Eυ,q(0),Gυ,q = Gυ,q(0). The main purpose of this paper, we find some differential equation for q-analogue of Frobenius-tangent numbers and polynomials. Based on these polynomials, we con- struct some differential equation of these polynomials. Also, we derive differential equations associated with symmetric properties. 2. Differential equations of q-analogue of Frobenius-tangent polynomials This section is dedicated to deriving some fundamental higher-order q-differential equations for q-Frobenius-tangent polynomials through the use of q-calculus theory. Through the application of q-derivatives, we will obtain several related differential equations that connect to the q-analogue of Frobenius-tangent polynomials, based on definition (1.5). Furthermore, we will establish a q-differential equation that captures the symmetric property of these polynomials via q-derivatives. Theorem 2.1. A solutions of the following differential equation (i)FTυ−ν,q(ψ; η) = [υ − ν]q! [υ]q! D (ν) q,ψFTυ,q(ψ; η), (2.1) (ii)FTυ−ν,q(q−1ψ; η) = qν [υ − ν]q! [υ]q! D (ν) q,ψFTυ,q(q −1ψ; η). (2.2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3220 4 IDREES AHMAD KHAN AND SUMIT KUMAR Proof. Using (1.5) and (1.8), we note that D (1) q,ψ ∞∑ υ=0 Tυ,q(ψ; η) φυ [υ]q! = 1− η eq((1− η)φ)− η D (1) q,ψeq(ψφ) = ∞∑ υ=0 [υ]qTυ−1,q(ψ; η) φυ [υ]q! . (2.3) From the equation (2.3), we get D (1) q,ψTυ,q(ψ; η) = [υ]qTυ−1,q(ψ; η). In similar method, we find D (2) q,ψTυ,q(ψ; η) = [υ]q[υ − 1]qTυ−2,q(ψ;u). Therefore, we have D (ν) q,ψTυ,q(x; η) = [υ]q[υ − 1]q · · · [υ − (ν − 1)]qTυ−1,q(ψ; η). Hence, we find the desired result at once. (ii) Similarly, we can proof of Theorem 2.1 (ii), so we omit the proof. □ Theorem 2.2. The differential equation of the q-Frobenius tangent polynomials as follows (1− η)υ−1 [υ]q! D (υ) q,ψFTυ,q(ψ; η)+ (1− η)υ−2 [υ − 1]q! D (υ−1) q,ψ FTυ,q(ψ; η)+ (1− η)υ−3 [υ − 2]q! D (υ−2) q,ψ FTυ,q(ψ; η) + · · ·+ (1− η)3 [4]q! D (4) q,ψFTυ,q(ψ; η) + (1− η)2 [3]q! D (3) q,ψFTυ,q(ψ; η) + (1− η) [2]q! D (2) q,ψFTυ,q(ψ; η)+D (1) q,ψFTυ,q(ψ; η)+(1−η)−1(FT0,q(ψ; η)−ηFTυ,q(ψ; η)−ψυ = 0. Proof. By using (2.1), we see that (1− η)eq(ψφ) = ∞∑ υ=0 FTυ,q(ψ; η) φυ [υ]q! (eq((1− η)φ)− η) = ∞∑ υ=0 FTυ,q(ψ; η) φυ [υ]q! ( ∞∑ ν=0 (1− η)ν φν [ν]q! − η ) = ∞∑ υ=0 ( υ∑ ν=0 ( υ ν ) q (1− u)νFTυ−ν,q(ψ; η)− ηFTυ,q(ψ; η) ) φυ [υ]q! . (2.4) and (1− η)eq(ψφ) = (1− η) ∞∑ υ=0 ψυ φυ [υ]q! . (2.5) Therefore, by (2.4) and (2.5), we get υ∑ ν=0 ( υ ν ) q (1− η)νFTυ−ν,q(ψ; η) = (1− η)ψυ. (2.6) Taking the ν − th derivative of above equation, we obtain υ∑ ν=0 (1− η)ν−1 [ν]q! D (ν) q,ψFTυ,q(ψ; η) = η(1− η)−1 FTυ,q(ψ; η) + (1− η)ψυ = 0. Hence, we find the desired result at once. □ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3221 SOME PROPERTIES OF DIFFERENTIAL EQUATIONS OF HIGHER-ORDER 5 Corollary 2.1. As q approaches 1 in Theorem 2.2, we derive. (1− η)υ−1 υ! dυ dψυ FTυ(ψ; η)+ (1− η)υ−2 [υ − 1]! dυ−1 dψυ−1 F Tυ(ψ; η)+ (1− η)υ−3 [υ − 2]! dυ−2 dψυ−2 F Tυ(ψ; η) + · · ·+ (1− η)3 4! d4 dψ4 F Tυ(ψ; η) + (1− η)2 3! d3 dψ3 F Tυ(ψ; η) + (1− η) 2! d2 dψ2 F Tυ(ψ; η)+ d dψ FTυ(ψ; η)+ (1− η)−1(FT0(ψ; η)− ηFTυ(ψ; η)−ψυ = 0. Theorem 2.3. Let υ ≥ 0. Then FTυ,q(1; η)− ηTυ,q(η) [υ]q! D (υ) q,ψFTυ,q(ψ; η)+ FTυ−1,q(1; η)− ηFTυ−1,q(η) [υ − 1]q! D (υ−1) q,ψ FTυ,q(ψ; η)+ · · ·+ FT2,q(1; η)− ηFT2,q(η) [2]q! D (2) q,ψFTυ,q(1; η)+(FT1,q(1; η)−ηFT1,q(η))D (1) q,ψFTυ,q(ψ; η) +(FT0,q(1; η)− ηFT0,q(η)− (1− η))FTυ,q(ψ; η) = 0. Proof. From (2.1), we have ∞∑ υ=0 FTυ,q(ψ;u) φυ [υ]q! = 1− η eq((1− η)φ)− η eq(ψφ) = 1 1− η ( 1− η eq((1− η)φ)− η eq((1− η)φ)− η 1− η eq((1− η)φ)− η ) 1− η eq((1− η)φ)− η eq(ψφ). (1−η) ∞∑ υ=0 FTυ,q(ψ; η) φυ [υ]q! = ∞∑ υ=0 ( υ∑ ν=0 ( υ ν ) q (FTν,q(1; η)− ηFTν,q(η)) FTυ−ν,q(ψ; η) ) φυ [υ]q! υ∑ ν=0 ( υ ν ) q (FTν,q(1; η)− ηFTν,q(η)) FTυ−ν,q(ψ; η)− (1− η)FTυ,q(ψ; η) = 0. (2.7) Replacing FTυ−ν,q(ψ; η) with D(ν) q,ψFTυ,q(ψ; η) in equation (2.7), we have υ∑ ν=0 (FTν,q(1; η)− ηFTν,q(η)) [ν]q! D (ν) q,ψFTυ,q(ψ; η)− (1− η)FTυ,q(ψ; η) = 0. Hence, we find the desired result at once. □ Corollary 2.2. As q approaches 1 in Theorem 2.3, we derive FTυ(1; η)− ηTυ(η) υ! dυ dxυ FTυ(ψ; η) + FTυ−1(1; η)− ηFTυ−1(η) (υ − 1)! dυ−1 dψυ−1 FTυ(ψ; η)+ · · ·+ FT2(1; η)− ηFT2(η) 2! d2 dψ2 FTυ(ψ; η) + (FT1(1; η)− ηT1(η)) d dψ FTυ(ψ; η) +(FT0(1; η)− ηFT0(η)− (1− η))FTυ(ψ; η) = 0. Theorem 2.4. Let υ ≥ 0. Then υ−1∑ ν=0 (1− η)υ−ν−1 FTν,q(η) [υ − ν − 1]q![ν]q! D (υ−1) q,ψ FTυ−1,q(ψ; η)+ υ−2∑ ν=0 (1− η)υ−ν−2qFTν,q(η) [υ − ν − 2]q![ν]q! D (υ−2) q,ψ FTυ−1,q(ψ; η)+· · · + 2∑ ν=0 (1− η)2−νqυ−3 FTν,q(η) [2− ν]q![ν]q! D (2) q,ψFTυ−1,q(ψ; η)+ 1∑ ν=0 (1− η)1−νqυ−2 FTν,q(η) [1− ν]q![ν]q! D (1) q,ψFTυ−1,q(ψ; η) + ( qυ−1 FT0,q(η)− qυψ ) FTυ−1,q(ψ; η) + FTυ,q(qψ; η) = 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3222 6 IDREES AHMAD KHAN AND SUMIT KUMAR Proof. Let qψ → ψ in (1.5), we have Dq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = eq(qφψ)Dq,φ ( 1− η eq((1− η)φ)− η ) + 1− η eq((1− η)φ)− η Dq,φeq(qφψ) = ∞∑ υ=0 qυFTυ,q(ψ; η) φυ [υ]q! ( qψ − ∞∑ υ=0 ( υ∑ ν=0 ( υ ν ) q (1− η)υ−νFTν,q(η) ) φυ [υ]q! ) = ∞∑ υ=0 ( qυ+1ψFTυ,q(η)− υ∑ θ=0 θ∑ ν=0 ( υ θ ) q ( θ ν ) q (1− η)θ−νqυ−θFTν,q(η)FTυ−θ,q(ψ; η) ) φυ [υ]q! (2.8) Dq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = ∞∑ υ=0 [υ]qq υψFTυ−1,q(ψ; η) φυ [υ]q! − ∞∑ υ=0 [υ]q ( υ−1∑ θ=0 θ∑ ν=0 ( υ − 1 θ ) q ( θ ν ) q (1− η)θ−νqυ−θ−1 FTν,q(η)FTυ−θ−1,q(ψ; η) ) φυ [υ]q! . (2.9) On the other hand, we have φDq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = ∞∑ υ=0 [υ]qFTυ,q(qψ; η) φυ [υ]q! . (2.10) By (2.9) and (2.10), we have υ−1∑ θ=0 θ∑ ν=0 ( υ − 1 θ ) q ( θ ν ) q (1− η)θ−νqυ−θ−1 FTν,q(η)FTυ−θ−1,q(ψ; η) = qυψFTυ−1,q(ψ; η)− FTυ,q(qψ; η). (2.11) In Theorem 2.1 (i), we get FTυ−ν−1,q(ψ; η) = [υ − ν − 1]q! [υ − 1]q! D (ν) q,ψFTυ−1,q(ψ; η). (2.12) υ−1∑ θ=0 θ∑ ν=0 ( υ − 1 θ ) q ( θ ν ) q (1− η)θ−νqυ−θ−1 FTν,q(η)FTυ−l−1,q(ψ; η) = υ−1∑ θ=0 θ∑ ν=0 (1− η)θ−νqυ−θ−1 FTν,q(η) [θ − ν]q![ν]q! D (θ) q,ψFTυ−1,q(ψ; η). (2.13) Therefore, we acquire at the desired result. □ Corollary 2.3. As q approaches 1 in Theorem 2.4, we derive υ−1∑ ν=0 (1− η)υ−ν−1 FTν(η) [υ − ν − 1]![ν]! d(υ−1) dψυ−1 FTυ−1(ψ; η)+ υ−2∑ ν=0 (1− η)υ−ν−2 FTν(η) [υ − ν − 2]![ν]! d(υ−2) dψυ−2 FTυ−1(ψ; η)+· · · + 2∑ ν=0 (1− η)2−νFTν(η) [2− ν]![ν]! d2 dψ2 FTυ−1(ψ; η) + θ∑ ν=0 (1− η)θ−νFTν(η) [θ − ν]![ν]! d dψ FTυ−1(ψ; η) + (FT0(η)− ψ) FTυ−1(ψ; η) + FTυ(ψ; η) = 0. Theorem 2.5. Let υ ≥ 0. Then υ−1∑ ν=0 (1− η)υ−1Hν,q(η) [υ − ν − 1]q![ν]q! D (υ−1) q,ψ FTυ−1,q(ψ; η)+ υ−2∑ ν=0 (1− η)υ−2qHν,q(η) [υ − ν − 2]q![ν]q! D (υ−2) q,ψ FTυ−1,q(ψ; η)+· · · Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3223 SOME PROPERTIES OF DIFFERENTIAL EQUATIONS OF HIGHER-ORDER 7 + 2∑ ν=0 (1− η)2qυ−3Hν,q(η) [2− ν]q![ν]q! D (2) q,ψFTυ−1,q(ψ; η)+ 1∑ ν=0 (1− η)qυ−2Hν,q(η) [1− ν]q![ν]q! D (1) q,ψFTυ−1,q(ψ; η) + ( q−1H0,q − ψ ) qυFTυ−1,q(ψ; η) + FTυ,q(qψ; η) = 0. Proof. Using (1.5), we have Dq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = ∞∑ υ=0 qυFTυ,q(ψ; η) φυ [υ]q! ( qψ − ∞∑ υ=0 (1− η)υHυ,q(η) φυ [υ]q! ∞∑ υ=0 (1− η)υ φυ [υ]q! ) = ∞∑ υ=0 ( qυ+1ψFTυ,q(ψ; η)− υ∑ θ=0 θ∑ ν=0 ( υ θ ) q ( θ ν ) q (1− η)θqυ−θHν,q(η)FTυ−θ,q(ψ; η) ) φυ [υ]q! . Therefore, we have υ−1∑ θ=0 θ∑ ν=0 (1− η)θqυ−θ−1Hν,q(η) [θ − ν]q![ν]q! D (θ) q,ψFTυ−θ,q(ψ; η)−qυψFTυ−θ,q(ψ; η)+FTυ,q(qψ; η) = 0, (2.14) which is the desired result. □ Corollary 2.4. As q approaches 1 in Theorem 2.5, we derive υ−1∑ ν=0 (1− η)υ−1Hν(η) [υ − ν − 1]![ν]! D (υ−1) ψ FTυ−1(ψ; η)+ υ−2∑ ν=0 (1− η)υ−2Hν(η) [υ − ν − 2]![ν]! D (υ−2) ψ FTυ−1(ψ; η)+· · · + 2∑ ν=0 (1− η)2Hν(η) [2− ν]![ν]! D (2) ψ FTυ−1(ψ; η) + 1∑ ν=0 (1− η)Hν(η) [1− ν]!ν! D (1) ψ FTυ−1(ψ; η) + (H0 − ψ) FTυ−θ(ψ; η) + FTυ(ψ; η) = 0. Theorem 2.6. Let υ ≥ 0. Then FTυ−1,q(1− η) [υ − 1]q! D (υ−1) q,ψ FTυ−1,q(ψ; η) + qFTυ−2,q(1− η) [υ − 2]q! D (υ−2) q,ψ FTυ−1,q(ψ; η) + · · · + qυ−4 FT3,q(1− η) [3]q! D (3) q,ψFTυ−1,q(ψ; η) + qυ−3 FT2,q(1− η) [2]q! D (2) q,ψFTυ−1,q(ψ; η) +qυ−2 FT1,q(1−η)D(1) q,ψFTυ−1,q(ψ; η)+ ( q−1 FT0,q(1− η)− ψ ) qυFTυ−1,q(ψ; η)+FTυ,q(qψ; η) = 0. Proof. By using (1.5), (1.8) and (2.8), we have Dq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = ∞∑ υ=0 ( qυ+1ψFTυ,q(ψ; η)− υ∑ ν=0 ( υ ν ) q qυ−νFTυ,q(1− η)FTυ−ν,q(ψ; η) ) φυ [υ]q! . (2.15) On multiplying φ in the above equation, we get φDq,φ ∞∑ υ=0 FTυ,q(qψ; η) φυ [υ]q! = ∞∑ υ=0 [υ]qq υψFTυ−1,q(q −1ψ; η) φυ [υ]q! Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3224 8 IDREES AHMAD KHAN AND SUMIT KUMAR − ∞∑ υ=0 [υ]q υ−1∑ ν=0 ( υ − 1 ν ) q qυ−ν−1 FTν,q(1− η)FTυ−ν−1,q(ψ; η) φυ [υ]q! . (2.16) By (2.15) and (2.16), we attain υ−1∑ ν=0 ( υ − 1 ν ) q qυ−ν−1 FTν,q(1− η)FTυ−ν−1,q(ψ; η) = qυψFTυ−1,q(q −1ψ; η)− qυψFTυ−1,q(ψ; η)− FTυ,q(qψ; η). (2.17) Applying a relation between Dυ q,ψFTυ,q(ψ; η) and FTυ,q(ψ; η) in the left-hand side of (2.17), we obtain υ−1∑ ν=0 ( υ − 1 ν ) q qυ−ν−1 FTν,q(1− η)FTυ−ν−1,q(ψ; η) = υ−1∑ ν=0 qυ−ν−1 FTν,q(1− η) [ν]q! D (ν) q,ψFTυ−1,q(ψ; η). (2.18) Hence, complete the proof. □ Corollary 2.5. As q approaches 1 in Theorem 2.6, we derive FTυ−1(1− η) [υ − 1]! D (υ−1) ψ FTυ−1(ψ; η) + FTυ−2(1− η) [υ − 2]! D (υ−2) ψ FTυ−1(ψ; η) + · · · + FT3(1− η) [3]! D (3) ψ FTυ−1(ψ; η) + FT2(1− η) 2! D (2) ψ FTυ−1(ψ; η) +FT1(1− η)D (1) ψ FTυ−1(ψ; η) + (FT0(1− η)− ψ) FTυ−1(ψ; η) + FTυ(ψ; η) = 0. Theorem 2.7. Let υ ≥ 0. Then FTν,q(b−1ζ; η) [υ]q! D (υ) q,ψFTυ,q(a−1ψ; η) + b−1 FTυ−1,q(b −1ζ; η) [υ − 1]q! D (υ−1) q,ψ FTυ,q(a−1ψ; η) + · · · +b1−υFT1,q(b −1ζ; η)D (1) q,ψFTυ,q(a−1ψ; η) + b−υFT0,q(b −1ζ; η)FTυ,q(a−1ψ; η) = FTν,q(a−1ζ; η) [υ]q! D (υ) q,ψFTυ,q(b−1ψ; η)+ a−1 FTυ−1,q(a −1ζ; η) [υ − 1]q! D (υ−1) q,ψ FTυ,q(b−1ψ; η)+· · · +a1−υFT1,q(a −1ζ; η)D (1) q,ψFTυ,q(b−1ψ; η) + a−υFT0,q(a −1ζ; η)FTυ,q(b−1ψ; η). Proof. Let A(φ) = (1− η)2eq(ab(ψ + η)φ) (eq((1− η)aφ)− η)(eq((1− η)bφ)− η) . Using the definition (1.5) and Cauchy products, then A(φ) = ∞∑ υ=0 ( υ∑ ν=0 ( υ ν ) q aυ−νbνFTν,q(b−1ζ; η)FTυ−ν,q(a−1ψ; η) ) φυ [υ]q! . (2.19) Similarly, we have A(φ) = ∞∑ υ=0 ( υ∑ ν=0 ( υ ν ) q bυ−νaνFTν,q(a−1ζ; η)FTυ−ν,q(b−1ψ; η) ) φυ [υ]q! . (2.20) By (2.19) and (2.20), we have υ∑ ν=0 ( υ ν ) q aυ−νbνFTν,q(b−1ζ; η)FTυ−ν,q(a−1ψ; η) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3225 SOME PROPERTIES OF DIFFERENTIAL EQUATIONS OF HIGHER-ORDER 9 = υ∑ ν=0 ( υ ν ) q bυ−νaνFTν,q(a−1ζ; η)FTυ−ν,q(b−1ψ; η). (2.21) Applying a relation between D (υ) q,ψFTυ,q(ψ; η) and FTυ,q(ψ; η) in (2.21), we have bν−υFTν,q(b−1ζ; η) [ν]q! D (ν) q,ψFTυ−ν,q(a−1ψ; η) = aν−υFTν,q(a−1ζ; η) [ν]q! D (ν) q,ψFTυ−ν,q(b−1ψ; η). Hence, complete the proof. □ Corollary 2.6. Letting a = 1 in Theorem 2.7, we have FTν,q(b−1ζ; η) [υ]q! D (υ) q,ψFTυ,q(ψ; η) + b−1 FTυ−1,q(b −1ζ; η) [υ − 1]q! D (υ−1) q,ψ FTυ,q(ψ; η) + · · · +b1−υFT1,q(b −1ζ; η)D (1) q,ψFTυ,q(ψ; η) + b−υFT0,q(b −1ζ; η)FTυ,q(ψ; η) = FTν,q(ζ; η) [υ]q! D (υ) q,ψFTυ,q(b−1ψ; η) + a−1 FTυ−1,q(ζ; η) [υ − 1]q! D (υ−1) q,ψ FTυ,q(b−1ψ; η) + · · · +FT1,q(ζ; η)D (1) q,ψFTυ,q(b−1ψ; η) + FT0,q(ζ; η)FTυ,q(b−1ψ; η). Corollary 2.7. As q approaches 1 in Theorem 2.7, we derive FTν(b−1ζ; η) [υ]! D (υ) ψ FTυ(a−1ψ; η) + b−1 FTυ−1(b −1ζ; η) [υ − 1]! D (υ−1) ψ FTυ(a−1ψ; η) + · · · +b1−υFT1(b −1ζ; η)D (1) ψ FTυ(a−1ψ; η) + b−υFT0(b −1ζ; η)FTυ(a−1ψ; η) = FTν(a−1ζ; η) [υ]! D (υ) ψ FTυ(b−1ψ; η) + a−1 FTυ−1(a −1ζ; η) [υ − 1]! D (υ−1) ψ FTυ(b−1ψ; η) + · · · +a1−υFT1(a −1ζ; η)D (1) ψ FTυ(b−1ψ; η) + a−υFT0(a −1ζ; η)FTυ(b−1ψ; η). 3. Conclusion We have constructed the q-analogue of Frobenius-tangent polynomials and num- bers, and derived several differential equations with these polynomials as solutions. Additionally, we identified differential equations that combine q-Frobenius and q- tangent polynomials. Several properties of the q-analogue of Frobenius-tangent poly- nomials and numbers were also established. The results obtained in this paper are broadly general and may lead to potential applications in the theory of special func- tions. Moreover, the main findings are significant, as they allow us to deduce im- portant integral formulas for specific parameter values, which could be particularly useful in laser technology. Acknowledgement Authors are thankful to integral University, Lucknow for providing Manuscript Communication Number (MCN): IU/R D/2024-MCN0003247 References [1] Alshejari, A, Khan, W. A, Duran, U, Ryoo, C. S. A study on differential equations associated with (q, h)-Frobenius-Genocchi polynomials. Journal of Mathematics and Computer Science, 2025. In Press. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3226 10 IDREES AHMAD KHAN AND SUMIT KUMAR [2] Aledamat, A, Khan, W. A, Duran, U, Kirmani, S. A. K, Ryoo, C. S. A study on differential equations associated with (q, h)-Frobenius-Euler polynomials. Journal of Mathematics and Computer Science, 2025, 36(3), 386-398. [3] Jackson, H.F. q-Difference equations. Am. J. Math. 1910, 32, 305-314. 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Properties of q-differential equations of higher order and visualization of fractal using q-Bernoulli polynomials. Mathematics 2022, 6, 296. Department of Mathematics and Statistics, Faculty of Science, Integral University, Lucknow- 226026, India Email address: khanidrees077@gmail.com Department of Mathematics and Statistics, Faculty of Science, Integral University, Lucknow- 226026, India Email address: lect.sumit@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) https://internationalpubls.com 3227