EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6660 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Bell-based Frobenius-type Eulerian Polynomials and Their Applications Manoj Sharma1, Waseem Ahmad Khan 2,∗, Ugur Duran3, Mohd Farman Ali4, Ashish Sharma5, Anupma Kumari6 1 Department of Mathematics, Rustamji Institute of Technology, BSF, Academy, Tekanpur, Gwalior, India 2 Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 3 Department of Basic Sciences of Engineering, Iskenderun Technical University, Hatay 31200, Turkey 4 Department of Mathematics, Madhav University, Sirohi, Rajasthan, India 5 Amity University, Gwalior, India 6 P. K. University Shivpuri, Gwalior, India Abstract. This article considers a new class of generalized Bell-based Frobenius-type Eulerian polyno- mials of two variables. Also, diverse properties and formulae for these new polynomials are investigated and analyzed. Then, some symmetric identities and implicit summation formulae are improved. The stack of zeros and surface representations of generalized Bell-based Frobenius-type Eulerian polynomials are given for some particular values of the parameters. 2020 Mathematics Subject Classifications: Primary 11B68, 33C45, 11Y16 Key Words and Phrases: Bell polynomials, Frobenius-type Eulerian polynomials, Bell-based Frobenius- type Eulerian polynomials, Summation formulae, Symmetric identities 1. Introduction and preliminaries Some different ways, such as recurrence relations, classical and exponential generating func- tions, p-adic integrals, explicit formulae, and so on, are used to represent the definitions of special polynomials and numbers. The most significant applications of special polynomials are involved in the theory of finite differences, analytic number theory, classical analysis, and statistics. The most famous special polynomials are Bernoulli [1], Hermite [2], Euler [3], Bell [4], Frobenius-Euler [5], Eulerian [6] and so on and see the references cited therein. The Eulerian polynomials appear in combinatorial mathematics and play an important role in the theory and applications of mathematics. Thus, many number theory and combinatorics experts have extensively studied their properties and obtained diverse interesting results [7]. In recent years, Bell-based Bernoulli polynomials of order α in [8], Bell-based Appell polynomials of order α in [8], and Bell-based Frobenius-type Eulerian polynomials in complex variables in [9] have been considered, and several properties, applications, and relations have been investigated. By mo- tivating and inspiring the above studies, in this work, we consider Bell-based Frobenius-type Eulerian polynomials of order α, and we then derive multifarious relations and identities, in- cluding some summation formulas and derivative properties. Also, we investigate some implicit ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6660 Email addresses: manoj240674@yahoo.co.in (M. Sharma). wkhan1@pmu.edu.sa (W. A. Khan), ugur.duran@iste.edu.tr (U. Duran), mohdfarmanali@gmail.com (M. F. Ali), asharma2@gwa.amity.edu.in (A. Sharma), , anupmasinghjpa@gmail.com (A. Kumari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 2 of 19 summation formulas and symmetric identities for Bell-based Frobenius-type Eulerian polyno- mials of order α. Furthermore, we provide the stack of zeros and surface representations of a generalized Bell-based Frobenius-type Eulerian polynomials for some particular values of the parameters. The Apostol-type Bernoulli, Euler and Genocchi polynomials of order α are defined by [10–12]:( ζ µeζ − 1 )α eσζ = ∞∑ ε=0 B(α) ε (σ;µ) ζε ε! for | ζ + logµ |< 2π, (1) ( 2 µeζ + 1 )α eσζ = ∞∑ ε=0 E(α) ε (σ;µ) ζε ε! for | ζ + logµ |< π, (2) and ( 2ζ µeζ + 1 )α eσζ = ∞∑ ε=0 G(α) ε (σ;µ) ζε ε! for | ζ + log µ |< π, (3) respectively. It is seen that B(α) ε (0;µ) := B(α) ε (µ),E(α) ε (0;µ) := E(α) ε (µ) and G(α) ε (0;µ) := G(α) ε (µ), are the corresponding numbers of the Apostol-type Bernoulli, Euler, and Genocchi polyno- mials of order α, respectively. Also, note that B(1) ε (σ;µ) := Bε(σ;µ), E(1) ε (σ;µ) := Eε(σ;µ) and G(1) ε (σ;µ) := Gε(σ;µ). For ε ≥ 0, the first kind of Stirling numbers are defined by the following summation formula [13–17]: (σ)ε = ε∑ θ=0 S1(ε, θ)σ θ, (4) where (σ)0 = 1, and (σ)ε = σ(σ−1) · · · (σ− ε+1) (ε ≥ 1). Also S1(ε, δ) can be represented by the following generation function: 1 δ! (log(1 + ζ))δ = ∞∑ ε=δ S1(ε, δ) ζε ε! (δ ≥ 0). (5) Similar to that of S1(ε, δ), for ε ≥ 0, the second kind of Stirling numbers are defined in the following two different ways [18, 19]: σε = ε∑ δ=0 S2(ε, δ)(σ)δ. (6) and 1 δ! (eζ − 1)δ = ∞∑ ε=δ S2(ε, δ) ζε ε! . (7) The second kind r-Stirling numbers Sr(ε, δ) are given by [9, 20]: 1 δ! erζ(eζ − 1)δ = ∞∑ ε=δ Sr(ε+ r, δ + r) ζε ε! . (8) The second kind r-Whitney numbers Wτ,r(ε, δ), for any positive integer τ , are provided by [21]: 1 τ δδ! erζ(eτσ − 1)δ = ∞∑ ε=δ Wτ,r(ε, δ) ζε ε! . (9) M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 3 of 19 The Bell polynomials Belε(σ) are defined by [4]: eσ(e ζ−1) = ∞∑ ε=0 Belε(σ) ζε ε! . (10) When σ = 1, Belε = Belε(1), (ε ≥ 0) are called the Bell numbers. By (7) and (10), we observe that Belε(σ) = ε∑ δ=0 S2(ε, δ)σ δ (ε ≥ 0). (11) Recently, Duran et al. [8] introduced the partially degenerate Bell-based Bernoulli polyno- mials of of the BelB (α) ε (σ; ρ):( ζ eζ − 1 )α eσζ+η(eζ−1) = ∞∑ ε=0 BelB(α) ε (σ; ρ) ζε ε! . (12) For α = 0 in (12), we get the bivariate Bell polynomials, [8]. eσζ+ρ(eζ−1) = ∞∑ ε=0 Belε(σ; ρ) ζε ε! . (13) Let µ ∈ C with µ ̸= 1, α ∈ C and ς ∈ R. The Frobenius-type Eulerian polynomials A(α) ε (σ;µ) of order α are given by [6]:( 1− µ eζ(µ−1) − µ )α eσζ = ∞∑ ε=0 A(α) ε (σ;µ) ζε ε! , ∣∣∣∣ζ + ln ( µ− 1 µ )∣∣∣∣ < 2π. (14) At the point σ = 0, A(α) ε (0;µ) := A(α) ε (µ) are called the Frobenius-type Eulerian numbers of order α. By (14), we observe that A(α) ε (σ;µ) = ε∑ ν=0 ( ε ν ) A(α) ν (µ)σε−ν , (15) and A(α) ε (σ;µ) = (µ− 1)εH(α) ε ( σ µ− 1 |µ ) , (16) where H(α) ε (σ|µ) are the εth Frobenius-Euler polynomials of order α [10]. Recently, Khan et al.[16] considered Bell-based Frobenius-type sine-Eulerian polynomials and Bell-based Frobenius-type cosine-Eulerian polynomials, respectively, as follows:( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1)sinχζ = ∞∑ ε=0 BelA(α,s) ε (σ, ρ, χ;µ) ζε ε! and ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1)cosχζ = ∞∑ ε=0 BelA(α,c) ε (σ, ρ, χ;µ) ζε ε! . (17) Then, they derived several properties and relations, and also they gave some applications of these polynomials, [4, 7]. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 4 of 19 2. Properties of Bell-based Frobenius-type Eulerian polynomials Here, we consider Bell-based Frobenius-type Eulerian polynomials of order α and then we analyze some properties. Definition 1. Let µ ∈ C with µ ̸= 1, and σ, ρ ∈ R. The Bell-based Frobenius-type Eulerian polynomials BelA (α) ε (σ, ρ;µ) of order α are defined by ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! , ∣∣∣∣ζ + ln ( µ− 1 µ )∣∣∣∣ < 2π. (18) Remark 1. On picking σ = 0 in (18), we get a new type of Bell-based Frobenius-type Eulerian polynomials BelA (α) ε (ρ;µ) of order α as: ( 1− µ eζ(µ−1) − µ )α eρ(e ζ−1) = ∞∑ ε=0 BelA(α) ε (ρ;µ) ζε ε! . (19) Remark 2. Upon laying ρ = 0 in (18), the Bell-based Frobenius-type Eulerian polynomials BelA (α) ε (σ, ρ;µ) of order α reduces to familiar Frobenius-type Eulerian polynomials A(α) ε (σ;µ) of order α in (14). Theorem 1. For ε ≥ 0, we have BelA(α) ε (σ, ρ;µ) = ε∑ u=0 ( ε u ) A(α) u (µ)Belε−u(σ; ρ), (20) BelA(α) ε (σ, ρ;µ) = ε∑ u=0 ( ε u ) A(α) u (σ;λ)Belε−u(ρ), (21) BelA(α) ε (σ, ρ;µ) = ε∑ u=0 ( ε u ) BelA(α) u (ρ;λ)σε−u. (22) Proof. By (10), (12), (14), and (18) and using the Cauchy product rule, we readily get the representations (20)-(22). Theorem 2. For ε ≥ 0, we have BelA(α+β) ε (σ + w, ρ+ u;µ) = ε∑ s=0 ( ε s ) BelA(β) s (u,w;µ)BelA (α) ε−s(σ, ρ;µ). (23) Proof. Using (14) and (18), we have ∞∑ ε=0 BelA(α+β) ε (σ + w, ρ+ u;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α+β e(σ+u)ζ+(ρ+w)(eζ−1) = ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! ∞∑ s=0 BelA(β) s (u,w;λ) ζs s! = ∞∑ ε=0 ε∑ s=0 ( ε s ) BelA(β) s (u,w;µ)BelA (α) ε−s(σ, ρ;µ) ζε ε! , (24) which means the asserted result (23). M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 5 of 19 Remark 3. For u = β = 0 in Theorem 2, we get BelA(α) ε (σ + w, ρ;µ) = ε∑ s=0 ( ε s ) A(α) ε−s(σ;µ)Bels(ρ;w). (25) Theorem 3. The following differentiation formulas hold: ∂BelA (α) ε (σ, ρ;µ) ∂σ = εBelA (α) ε−1(σ, ρ;µ), (26) ∂BelA (α) ε (σ, ρ;µ) ∂ρ = BelA(α) ε (σ + 1, ρ;µ)− BelA(α) ε (σ, ρ;µ). (27) Proof. By (18), we have ∞∑ ε=1 ∂ ∂σ BelA(α) σ (σ, ρ;µ) ζε ε! = ∂ ∂σ ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = ( 1− µ eζ(µ−1) − µ )α ∂ ∂σ eσζ+ρ(eζ−1) = ( 1− µ eζ(µ−1) − µ )α ζeσζ+η(eζ−1) = ∞∑ ε=1 BelA (α) ε−1(σ, ρ;µ) ζε ε! , (28) which gives the claimed result (26). Again, using (18), we note that ∞∑ ε=0 ∂ ∂ρ BelA(α) ε (σ, ρ;µ) ζε ε! = ∂ ∂ρ ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = ( 1− µ eζ(µ−1) − µ )α ∂ ∂ρ eσζ+ρ(eζ−1) = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1)(eζ − 1) = ∞∑ ε=0 BelA(α) ε (σ + 1, ρ;µ) ζε ε! − ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! , (29) which provides the asserted result (27). Theorem 4. Let ε ≥ 0. Then (2µ− 1) ε∑ δ=0 ( ε δ ) Aδ(σ;µ)BelAε−δ(σ, ρ; 1− µ) = λBelAε(σ, ρ;µ)− (1− µ)BelAε(σ, ρ; 1− µ). (30) Proof. We set (2µ− 1) (eζ(µ−1) − µ)(eζ(µ−1) − (1− µ)) = 1 eζ(µ−1) − µ − 1 eζ(µ−1) − (1− µ) . We observe that (2µ−1) (1− µ)eσζ(1− (1− µ))eρ(e ζ−1) (eζ(µ−1) − µ)(eζ(µ−1) − (1− µ)) = (1− µ)eρ(e ζ−1)µeσζ eζ(µ−1) − µ −(1− µ)eρ(e ζ−1)µeσζ(1− (1− µ)) eζ(µ−1) − (1− µ) , and then (2µ− 1) ( ∞∑ δ=0 Aδ(σ;µ) ζδ δ! )( ∞∑ ε=0 BelAε(ρ; 1− µ) ζε ε! ) = µ ∞∑ ε=0 BelAε(σ, ρ;µ) ζε ε! − (1− µ) ∞∑ ε=0 BelAε(σ, ρ; 1− µ) ζε ε! , which implies the desired result. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 6 of 19 Theorem 5. For ε ≥ 0, we have µBelAε(σ, ρ;µ) = ε∑ δ=0 ( ε δ ) BelAε−δ(σ, ρ;µ)(1− µ)δ − (1− µ)Belε(σ; ρ). (31) Proof. Consider µ (eζ(µ−1) − µ)eζ(µ−1) = 1 (eζ(µ−1) − µ) − 1 eζ(µ−1) . We find µ(1− µ)eσζ+ρ(eζ−1) (eζ(µ−1) − µ)eζ(µ−1) = (1− µ)eσζ+ρ(eζ−1) (eζ(µ−1) − µ) − (1− µ)eσζ+ρ(eζ−1) eζ(µ−1) µ ∞∑ ε=0 BelAε(σ, ρ;µ) ζε ε! = ∞∑ ε=0 BelAε(σ, ρ;µ) ζε ε! ∞∑ δ=0 (1− µ)δ ζδ δ! − (1− µ) ∞∑ ε=0 Belε(σ; ρ) ζε ε! . (32) Therefore, by (32), we get (31). Theorem 6. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = 1 1− µ ε∑ δ=0 ( ε δ )[ Aε−δ(µ)BelA (α) δ ((1− µ)σ, ρ;µ)− µAε−δ(µ)BelA (α) δ (σ, ρ;µ) ] . (33) Proof. In (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )( eζ(µ−1) − µ 1− µ )( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = 1 1− µ [( 1− µ eζ(µ−1) − µ ) e(µ−1)ζ ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) −µ ( 1− µ eζ(µ−1) − µ )( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) ] = 1 1− µ [ ∞∑ ε=0 Aε(µ) ζε ε! ∞∑ δ=0 BelA (α) δ ((µ− 1)σ, ρ;µ) ζδ δ! − µ ∞∑ ε=0 Aε(µ) ζε ε! ∞∑ δ=0 BelA (α) δ (σ, ρ;µ) ζδ δ! ] . (34) By (18) and (34), we obtain (33). Theorem 7. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = ε∑ s=0 s∑ δ=0 ( ε s ) (σ)δS2(s, δ)BelA(α) ε (σ;µ). (35) Proof. By (18), we note that ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eρ(e ζ−1)[eζ−1+1]σ = ( 1− µ eζ(µ−1) − µ )α eρ(e ζ−1) ∞∑ δ=0 (σ)δ (eζ − 1)δ δ! = ∞∑ ε=0 BelA(α) ε (ρ;µ) ζε ε! ∞∑ s=0 s∑ δ=0 (σ)δS2(s, δ) ζs s! = ∞∑ ε=0 ( ε∑ s=0 s∑ δ=0 ( ε s ) (σ)δS2(s, δ)BelA(α) ε (ρ;µ) ) ζε ε! . (36) In view of (18) and (36), we get (35). M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 7 of 19 3. Summation formulae Here, we investigate several implicit formulas and symmetric relations for Bell-based Frobenius- type Eulerian polynomials of order α. Theorem 8. The following formula is correct: BelA (α) h+f (σ, ρ;µ) = h,f∑ ε,s=0 ( f s )( h ε ) (σ − ζ)ε+s BelA (α) h+f−ε−s(ζ, η;µ). (37) Proof. By replacing z by z + w in (18), we see that( 1− µ e(µ−1)(z+w) − µ )α eη(e z+w−1) = e−ξ(z+w) ∞∑ h,f=0 BelA (α) h+f (σ, ρ;µ) zh h! wf f ! . (38) Also by replacing ξ by ζ in (38), we acquire e−ζ(z+w) ∞∑ h,f=0 BelA (α) h+f (ζ, ρ;µ) zh h! wf f ! = ( 1− µ e(µ−1)(z+w) − µ )α eρ(e z+w−1) (39) e(σ−ζ)(z+w) ∞∑ h,f=0 BelA (α) h+f (ζ, ρ;µ) zh h! wf f ! = ∞∑ h,f=0 BelA (α) h+f (σ, ρ;µ) zh h! wf f ! (40) ∞∑ N=0 [(σ − ζ)(z + w)]N N ! ∞∑ h,f=0 BelA (α) h+f (ζ, η;µ) zh h! wf f ! = ∞∑ h,f=0 BelA (α) h+f (σ, ρ;µ) zh h! wf f ! . (41) Using the formula [16] ∞∑ N=0 f(N) (ζ + η)N N! = ∞∑ ε,δ=0 f(ε+ δ) ζε ε! ηδ δ! , (42) we then obtain ∞∑ j,s=0 (ξ − ζ)j+szjws j!s! ∞∑ h,f=0 BelA (α) h+f (ζ, η;µ) zh h! wf f ! = ∞∑ h,f=0 BelA (α) h+f (σ, ρ;µ) zh h! wf f ! . (43) Thus we have ∞∑ h,f=0 h,f∑ ε,s=0 (σ − ζ)ε+s ε!s! BelA (α) h+f−ε−s(ζ, ρ;µ) zh (h− j)! wf (f − s)! = ∞∑ h,f=0 BelA (α) h+f (σ, ρ;µ) zh h! wf f ! , (44) which implies the claimed result. Remark 4. Permitting f = 0 in (37), we get BelA (α) θ (σ, ρ;µ) = h∑ ε=0 ( h ε ) (σ − ζ)εBelA (α) h−ε(ζ, ρ;µ) (ε ≥ 0). (45) Remark 5. Replace σ → σ + ζ and setting ρ = 0 in (37), we obtain M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 8 of 19 BelA (α) h+f (σ + ζ;µ) = h,f∑ ε,δ=0 ( f δ )( h ε ) σε+s BelA (α) h+f−ε−δ(ζ;µ). (46) Again, by permitting σ = 0 in (37), we obtain BelA (α) h+f (ρ;µ) = h,f∑ ε,s=0 ( f δ )( h ε ) (−ζ)ε+δ BelA (α) h+f−ε−δ(ζ, ρ;µ). Theorem 9. Let ε ≥ 0. Then BelA(α+1) ε (σ, ρ;µ) = ε∑ d=0 ( ε d ) Aε−d(µ)BelA (α) d (σ, ρ;µ). (47) Proof. By (18), we have 1− µ e(µ−1)ζ − µ ( 1− µ e(µ−1)ζ − µ )α eσζ+ρ(eζ−1) = 1− µ e(µ−1)ζ − µ ∞∑ d=0 BelA (α) d (σ, ρ;µ) ζd d! ( 1− µ e(µ−1)ζ − µ )α+1 eσζ+ρ(eζ−1) = 1− µ e(µ−1)ζ − µ ∞∑ d=0 BelA (α) d (σ, ρ;µ) ζd d! = ∞∑ ε=0 Aε(µ) ζε ε! ∞∑ d=0 BelA (α) d (σ, ρ;µ) ζd d! = ∞∑ ε=0 ( ε∑ d=0 ( ε d ) Aε−d(µ)BelA (α) d (σ, ρ;µ) ) ζε ε! , which implies the result (47). Theorem 10. Let ε ≥ 0. Then BelA(α) ε (σ + 1, ρ;µ) = ε∑ δ=0 ( ε δ ) BelA (α) δ (σ, ρ;µ). (48) Proof. Using definition (18), we have ∞∑ ε=0 BelA(α) ε (σ + 1, ρ;µ) ζε ε! − ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ e(µ−1)ζ − µ )α eσζ+ρ(eζ−1)(eζ − 1) = ( ∞∑ δ=0 BelA (α) δ (σ, ρ;µ) ζδ δ! )( ∞∑ ε=0 ζε ε! ) − ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ∞∑ ε=0 ε∑ δ=0 ( ε δ ) BelA (α) δ (σ, ρ;µ) ζε ε! − ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! , which gives the claimed result (48). M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 9 of 19 Theorem 11. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = ε∑ δ=0 ε∑ θ=ϑ ( α+ θ − 1 θ ) θ! ( ε ϑ ) S2(ϑ, θ)(µ− 1)ϑ−θBelε−ϑ(σ; ρ). (49) Proof. In (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = eσζ+ρ(eζ−1) ( 1 + eζ(µ−1) − 1 1− µ )−α = ∞∑ θ=0 ( −α θ )( eζ(µ−1) − 1 1− µ )θ ∞∑ ε=0 Belε(σ; ρ) ζε ε! = ∞∑ ε=0 ( ε∑ θ=0 ε∑ ϑ=θ ( α+ θ − 1 θ ) θ! ( ε ϑ ) S2(ϑ, θ)(µ− 1)ϑ−θBelε−ϑ(σ; ρ) ) ζε ε! , which yields the result (49). Theorem 12. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = ∞∑ δ=0 δ∑ θ=0 ( δ θ ) Belε(σ + (µ− 1)θ; ρ) (α)δ δ! (µ− 1)−α(−1)δ−θ. (50) Proof. By (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = eσζ+ρ(eζ−1) ( 1− eζ(µ−1) − 1 µ− 1 )−α = eσζ+ρ(eζ−1) ( µ− eζ(µ−1) µ− 1 )−α = eσζ+ρ(eζ−1) ∞∑ δ=0 (α)δ 1 δ! ( eζ(µ−1) − 1 µ− 1 )δ = eσζ+ρ(eζ−1) ∞∑ δ=0 (α)δ 1 δ! (µ− 1)−α δ∑ θ=0 ( δ θ ) eζ(µ−1)θ(−1)δ−θ ∞∑ θ=0 θ∑ δ=0 (α)δ(µ− 1)θ−δSδ l ( σ µ− 1 ) ζθ θ! = ∞∑ ε=0 ∞∑ δ=0 δ∑ θ=0 ( δ θ ) Belε(σ + (µ− 1)θ; ρ) (α)δ δ! (µ− 1)−α(−1)δ−θ ζ ε ε! = ∞∑ ε=0 ( ε∑ θ=0 θ∑ δ=0 ( ε θ ) (α)δ(µ− 1)θ−δSδ l ( σ µ− 1 ) Belε−θ(ς) ) ζε ε! , which means the claimed formula (50). Theorem 13. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = 1 ε+ 1 ε+1∑ δ=0 ( ε+ 1 δ )( µ δ∑ θ=0 ( δ θ ) Bδ−θ(σ;µ)− Bθ(σ;µ) ) BelA (α) ε−δ+1(0, ρ;µ). (51) Proof. By (1) and (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) ( ζ µeζ − 1 )( µeζ − 1 ζ ) M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 10 of 19 = 1 ζ ( µ ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Bδ(σ;µ) ζδ δ! ∞∑ θ=0 ζθ θ! − ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Bδ(σ;µ) ζδ δ! ) , (52) which yields the claimed result (51). Theorem 14. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = 1 2 ε∑ δ=0 ( ε δ )( µ δ∑ θ=0 ( δ θ ) Eδ−θ(σ;µ) + Eθ(σ;µ) ) BelA (α) ε−δ(0, ρ;µ). (53) Proof. From (2) and (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) ( 2 µeζ + 1 )( µeζ + 1 2 ) = 1 2 ( µ ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Eδ(σ;µ) ζδ δ! ∞∑ θ=0 ζθ θ! + ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Eδ(σ;µ) ζδ δ! ) , (54) which yields the result (53). Theorem 15. Let ε ≥ 0. Then BelA(α) ε (σ, ρ;µ) = 1 2(ε+ 1) ε+1∑ δ=0 ( ε+ 1 δ )( µ δ∑ θ=0 ( δ θ ) Gδ−θ(σ;µ) +Gθ(σ;µ) ) BelA (α) ε−δ+1(0, ρ;µ). (55) Proof. By (3) and (18), we have ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! = ( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) ( 2ζ µeζ + 1 )( µeζ + 1 2ζ ) = 1 2ζ ( µ ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Gδ(σ;µ) ζδ δ! ∞∑ θ=0 ζθ θ! + ∞∑ ε=0 BelA(α) ε (0, ρ;µ) ζε ε! ∞∑ δ=0 Gδ(σ;µ) ζδ δ! ) , which implies the claimed result (55). 4. Symmetric Identities Here, we give some symmetric identities of the Bell-based Frobenius-type Eulerian polyno- mials of order α. Theorem 16. Let ε ≥ 0 and a, b,> 0 with a ̸= b . Then ε∑ δ=0 ( ε δ ) bδaε−δ BelA (α) ε−δ(bσ, ρ;µ)BelA (α) δ (aσ, ρ;µ) = ε∑ δ=0 ( ε δ ) aδbε−δ BelA (α) ε−δ(aσ, ρ;µ)BelA (α) δ (bσ, ρ;µ). (56) M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 11 of 19 Proof. Let A(ζ) = ( (1− µ)2 (e(µ−1)aζ − µ)(e(µ−1)bζ − µ) )α e2abσζ+ρ(eaζ−1)+ρ(ebζ−1), (57) A(ζ) = ∞∑ ε=0 BelA(α) ε (bσ, ρ;µ) (aζ)ε ε! ∞∑ δ=0 BelA (α) δ (aσ, ρ;µ) (bζ)δ δ! = ∞∑ ε=0 ( ε∑ δ=0 ( ε δ ) bδaε−δ BelA (α) ε−δ(bσ, ρ;µ)BelA (α) δ (aσ, ρ;µ) ) ζε ε! . Similarly, we can show that A(ζ) = ∞∑ ε=0 BelA(α) ε (aσ, ρ;µ) (bζ)ε ε! ∞∑ δ=0 BelA (α) δ (bσ, ρ;µ) (aζ)δ δ! = ∞∑ ε=0 ( ε∑ δ=0 ( ε δ ) aδbε−δ BelA (α) ε−δ(aσ, ρ;µ)BelA (α) δ (bσ, ρ;µ) ) ζε ε! , which implies the claimed result (56). Remark 6. For α = 1 in (56), we have ε∑ δ=0 ( ε δ ) bδaε−δ BelAε−δ(bσ, ρ;µ)BelAδ(aσ, ρ;µ) = ε∑ δ=0 ( ε δ ) aδbε−δ BelAε−δ(aσ, ρ;µ)BelAδ(bσ, ρ;µ). (58) Theorem 17. Let ε ≥ 0. Then ε∑ δ=0 ( ε δ ) a−1∑ θ=0 b−1∑ ϑ=0 (−µ)θ+ϑaε−δbδδBelA (α) ε−δ ( bσ + b a θ + ϑ, ρ;µ ) BelA (α) k (aσ, ρ;µ) = ε∑ δ=0 ( ε δ ) b−1∑ θ=0 a−1∑ ϑ=0 (−λ)θ+ϑbε−δaδBelA (α) ε−δ ( aσ + a b θ + ϑ, ρ;µ ) BelA (α) δ (bσ, ρ;µ). (59) Proof. Let B(ζ) = ( (1− µ)2 (e(µ−1)aζ − µ)(e(µ−1)bζ − µ) )α ( 1− ( −µebζ )a) ( 1− ( −µeaζ )b) (µeaζ + 1)(µebζ + 1) e2abσζ+ρ(eaζ−1)+ρ(ebζ−1). Then, we have B(ζ) = ( 1− µ e(µ−1)aζ − µ )α eabσζ+ρ(eaζ−1) ( 1− ( −µebζ )a µebζ + 1 )( 1− µ e(µ−1)bζ − µ )α × ( 1− ( −µeaζ )b µeaζ + 1 ) eabσζ+ρ(ebζ−1) = ( 1− µ e(µ−1)aζ − µ )α eabσζ+ρ(eaζ−1) a−1∑ θ=0 (−µ)θebζθ ( 1− µ e(µ−1)bζ − µ )α eabσζ+ρ(ebζ−1) b−1∑ ε=0 (−µ)ϑeaζϑ M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 12 of 19 = ( 1− µ e(µ−1)aζ − µ )α eρ(e aζ−1) a−1∑ θ=0 b−1∑ ϑ=0 (−µ)θ+ϑe(bσ+ b a θ+ϑ)aζ ∞∑ δ=0 BelA (α) δ (aσ, ρ;µ) (bζ)δ δ! = ∞∑ ε=0 a−1∑ θ=0 b−1∑ ϑ=0 (−µ)θ+ϑ BelA(α) ε ( bσ + b a θ + ϑ, ρ;µ ) (aζ)ε ε! ∞∑ δ=0 BelA (α) δ (aσ, ρ;µ) (bζ)δ (δ)! = ∞∑ ε=0 ε∑ δ=0 ( ε δ ) a−1∑ θ=0 b−1∑ ϑ=0 (−µ)θ+ϑaε−δbδBelA (α) ε−δ ( bσ + b a θ + ϑ, ρ;µ ) × BelA (α) δ (aσ, ρ;µ) ζε ε! . (60) On the other hand, we have B(ζ) = ∞∑ ϵ=0 ε∑ δ=0 ( ε δ ) b−1∑ θ=0 a−1∑ ϑ=0 (−µ)θ+ϑbε−δaδBelA (α) ε−δ ( aσ + a b θ + ϑ, ρ;µ ) × BelA (α) δ (bσ, ρ;µ) ζε ε! . (61) By (60) and (61), we arrive at the desired result (59). 5. Stack of zeros and surface representations of Bell-based Frobenius-type Eulerian polynomials Certain zero values of the Bell-based Frobenius-type Eulerian polynomials are discussed, and some graphical representations are presented in this section. Let’s remember the definition of Bell-based Frobenius-type Eulerian polynomials from (18):( 1− µ eζ(µ−1) − µ )α eσζ+ρ(eζ−1) = ∞∑ ε=0 BelA(α) ε (σ, ρ;µ) ζε ε! ∣∣∣∣ζ + ln ( µ− 1 µ )∣∣∣∣ < 2π.) The first few polynomials of BelA (α) ε (σ, ρ;µ) are as follows: M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 13 of 19 BelA (α) 0 (σ, ρ;µ) = 1, BelA (α) 1 (σ, ρ;µ) = α+ ρ+ σ, BelA (α) 2 (σ, ρ;µ) = α2 + αµ+ ρ+ 2αρ+ ρ2 + 2ασ + 2ρσ + σ2, BelA (α) 3 (σ, ρ;µ) = α3 + αµ+ 3α2µ+ αµ2 + ρ+ 3αρ+ 3α2ρ+ 3αµρ + 3ρ2 + 3αρ2 + ρ3 + 3α2σ + 3αµσ + 3ρσ + 6αρσ + 3ρ2σ + 3ασ2 + 3ρσ2 + σ3, BelA (α) 4 (σ, ρ;µ) = −α+ α4 + 24α (1− µ)4 − 36α (1− µ)3 + 14α (1− µ)2 − α 1− µ − 10αµ+ 4α2µ + 6α3µ− 96αµ (1− µ)4 + 144αµ (1− µ)3 − 56αµ (1− µ)2 + 4αµ 1− µ − 7αµ2 + 7α2µ2 + 144αµ2 (1− µ)4 − 216αµ2 (1− µ)3 + 84αµ2 (1− µ)2 − 6αµ2 1− µ − 96αµ3 (1− µ)4 + 144αµ3 (1− µ)3 − 56αµ3 (1− µ)2 + 4αµ3 1− µ + 24αµ4 (1− µ)4 − 36αµ4 (1− µ)3 + 14αµ4 (1− µ)2 − αµ4 1− µ + ρ + 4αρ+ 6α2ρ+ 4α3ρ+ 10αµρ+ 12α2µρ+ 4αµ2ρ + 7ρ2 + 12αρ2 + 6α2ρ2 + 6αµρ2 + 6ρ3 + 4αρ3 + ρ4 + 4α3σ + 4αµσ + 12α2µσ + 4αµ2σ + 4ρσ + 12αρσ + 12α2ρσ + 12αµρσ + 12ρ2σ + 12αρ2σ + 4ρ3σ + 6α2σ2 + 6αµσ2 + 6ρσ2 + 12αρσ2 + 6ρ2σ2 + 4ασ3 + 4ρσ3 + σ4. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 14 of 19 We obtain the zeros of the Bell-based Frobenius-type Eulerian polynomials of order α, namely the solutions of BelA (α) ε (σ, ρ;µ), by making use of technology, given in Figure 1 for 1 ≤ ε ≤ 20: -40 -20 0 20 40 -40 -20 0 20 40 Re(σ) Im(σ) -40 -20 0 20 40 -40 -20 0 20 40 Re(σ) Im(σ) -40 -20 0 20 40 -40 -20 0 20 40 Re(σ) Im(σ) -40 -20 0 20 40 -40 -20 0 20 40 Re(σ) Im(σ) Figure 1: Zeros of BelA(α) ε (σ, ρ;µ) Figure 1 (top-left) shows the zeros of BelA (5) ε (σ, 5; 3) for 1 ≤ ε ≤ 20. Figure 1 (top-right) shows the zeros of BelA (5) ε (σ,−5; 3) for 1 ≤ ε ≤ 20. Figure 1 (bottom-left) shows the zeros of BelA (5) ε (σ, 5;−3) for 1 ≤ ε ≤ 20. Figure 1 (bottom-right) shows the zeros of BelA (5) ε (σ,−5;−3) for 1 ≤ ε ≤ 20. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 15 of 19 Stacks of zeros of the generalized Bell-based Frobenius-type Eulerian polynomials BelA (α) ε (σ, ρ;µ) of order α for 1 ≤ ε ≤ 20, creating a 3D structure, are given in Figure 2: Figure 2: Solutions of BelA(α) ε (σ, ρ;µ) = 0 Figure 2 (top-left) shows the zeros of BelA (5) ε (σ, 5; 3) for 1 ≤ ε ≤ 20. Figure 2 (top-right) shows the zeros of BelA (5) ε (σ,−5; 3) for 1 ≤ ε ≤ 20. Figure 2 (bottom-left) shows the zeros of BelA (5) ε (σ, 5;−3) for 1 ≤ ε ≤ 20. Figure 2 (bottom-right) shows the zeros of BelA (5) ε (σ,−5;−3) for 1 ≤ ε ≤ 20. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 16 of 19 Plots of real zeros of the generalized Bell-based Frobenius-type Eulerian polynomials BelA (α) ε (σ, ρ;µ) of order α for 1 ≤ ε ≤ 20 are provided by Figure 3: Figure 3: Real zeros of BelA(α) ε (σ, ρ;µ) Figure 3 (top-left) shows the real zeros of BelA (5) ε (σ, 5; 3) for 1 ≤ ε ≤ 20. Figure 3 (top- right) shows the real zeros of BelA (5) ε (σ,−5; 3) for 1 ≤ ε ≤ 20. Figure 3 (bottom-left) shows the real zeros of BelA (5) ε (σ, 5;−3) for 1 ≤ ε ≤ 20. Figure 3 (bottom-right) shows the real zeros of BelA (5) ε (σ,−5;−3) for 1 ≤ ε ≤ 20. M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 17 of 19 Lastly, we compute approximate solutions of BelA (5) ε (σ, 5; 3) = 0 in Table 1: Table 1. Approximate solutions of BelA (5) ε (σ, 5; 3) = 0 degree ε σ 1 −10.000 2 −10.0000− 4.4721i, −10.0000 + 4.4721i 3 −11.063, −9.4684− 7.8005i, −9.4684 + 7.8005i 4 −11.3040− 3.4430i, −11.3040 + 3.4430i, −8.6960− 10.5616i, −8.6960 + 10.5616i 5 −12.134, −11.1517− 6.3553i, −11.1517 + 6.3553i, −7.781− 12.967i, −7.781 + 12.967i 6 −12.4670− 2.9563i, −12.4670 + 2.9563i, −10.7625− 8.9281i, −10.7625 + 8.9281i, −6.770− 15.121i, −6.770 + 15.121i 7 −13.208, −12.4943− 5.5954i, −12.4943 + 5.5954i, −10.213− 11.259i, −10.213 + 11.259i, −5.689− 17.086i, −5.689 + 17.086i 8 −13.5902− 2.6625i, −13.5902 + 2.6625i, −12.312− 8.006i, −12.312 + 8.006i, −9.547− 13.404i, −9.547 + 13.404i, −4.552− 18.901i, −4.552 + 18.901i 9 −14.284, −13.723− 5.110i, −13.723 + 5.110i, −11.975− 10.240i, −11.975 + 10.240i, −8.791− 15.403i, −8.791 + 15.403i, −3.369− 20.595i, −3.369 + 20.595i 6. Conclusion Hermite-based special polynomials have been studied for a long time, and many mathemati- cians and scientists have improved their properties and applications; [2, 14] and the references cited therein. In recent years, by the similar motivation mentioned above, Bell-based special polynomials such as Bell-based Bernoulli polynomials in [8], Bell-based Bernoulli polynomials of the first kind in [7], and Bell-based Frobenius-type sine-(and cosine-)Eulerian polynomials in [15] have been defined, and diverse properties, applications, and relations have been derived. As a link in this working chain, we have worked on Bell-based Frobenius-type Eulerian polynomials of order α, and we then have acquired some relations, formulas, and derivative properties. Also, we have examined some implicit summation formulas and symmetric identities for Bell-based Frobenius-type Eulerian polynomials of order α. Lastly, we have shown the stack of zeros and surface representations of Bell-based Frobenius-type Eulerian polynomials for several specific parameters with specific values. Future research can explore several directions, including the development of q-analogues and degenerate forms of the proposed polynomials to study associated q-difference equations and limiting behaviors in [22, 23]. Investigating orthogonality conditions and suitable weight functions will help identify inner product spaces where these polynomials are orthogonal. Their M. Sharma et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6660 18 of 19 application in interpolation, approximation theory, and spectral methods also warrants atten- tion, particularly in solving differential or integral equations. Potential uses in mathematical physics and engineering such as quantum systems and signal analysis highlight their applied significance. 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