EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1471-1489 ISSN 1307-5543 – ejpam.com Published by New York Business Global Higher Order Bivariate Bell-Based Apostol-Frobenius-Type Poly-Genocchi Polynomials with Parameters a and b Roberto B. Corcino1,2,∗, Cristina B. Corcino1,2 1 Research Institute for Computational Mathematics and Physics, Cebu Normal University, 6000 Cebu City, Philippines 2 Mathematics Department, Cebu Normal University, 6000 Cebu City, Philippines Abstract. In this paper, we unveil a novel category of Frobenius-Genocchi polynomials, grounded in the Bell numbers and Apostol-type functions. Our exploration delves into a comprehensive examination of these polynomials, elucidating various properties. Employing diverse analytical methods and leveraging generating functions for Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order, we derive explicit and implicit summation formulas, complemented by their symmetric identities. 2020 Mathematics Subject Classifications: 05A15, 11B68, 11B73, 26C05, 33B10 Key Words and Phrases: Genocchi polynomials, Bell polynomials, Apostol-Frobenius-Type poly-Genocchi polynomials, Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials, Stir- ling numbers 1. Introduction Similar to the Bernoulli and Euler numbers [1], the Genocchi numbers, denoted as Gn, are established by means of the subsequent generating function: ∞∑ n=0 Gn tn n! = 2t et + 1 , |t| < π. Some novel identities involving these numbers can be found in [5, 19, 27]. These numbers have undergone diverse generalizations, often achieved by combining them with the prin- ciples of well-known polynomials. A specific example is the integration with exponential ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5193 Email addresses: rcorcino@yahoo.com (R. Corcino), corcinoc@cnu.edu.ph (C. Corcino) https://www.ejpam.com 1471 © 2024 EJPAM All rights reserved. 1472 polynomials, leading to the formation of Genocchi polynomials and higher-order Genocchi polynomials (see [18]), outlined as follows: ∞∑ n=0 Gn(x) tn n! = 2t et + 1 ext, |t| < π, (1.1) ∞∑ n=0 G(k) n (x) tn n! = ( 2t et + 1 )k ext. (1.2) Mixing with the Apostol polynomials yields the Apostol-Genocchi polynomials, and Apostol- Genocchi polynomials of higher order, which are respectively defined as follows: ∞∑ n=0 Gn(x, λ) tn n! = 2t λet + 1 ext, (1.3) ∞∑ n=0 G(k) n (x, λ) tn n! = ( 2t λet + 1 )k ext, (1.4) where |t| < π when λ = 1 and |t| < log(−λ) when λ ̸= 1, λ ∈ C. Also, mixing with Frobenius polynomials yields the so-called Frobenius-Genocchi polynomials, which are given by ∞∑ n=0 GF n (x;u) tn n! = (1− u)t et − u ext, (1.5) and further gives ∞∑ n=0 GF n (x;u, λ) tn n! = (1− u)t λet − u ext, (1.6) the Apostol-Frobenius-Genocchi polynomials by mixing with Apostol-Genocchi polynomi- als (see [6, 15–17, 22, 23, 28, 30, 32, 33]). Further generalization and other variation of Frobenius-Genocchi polynomials, specifically, the generalized Apostol-Frobenius-Genocchi polynomials and Frobenius-Euler-Genocchi polynomials, are introduced in [34] and [3] re- spectively, and defined as follows: ∞∑ n=0 Hr n(x;u, a, b, c, λ, ) tn n! = ( (at − u)t λbt − u )r cxt, (1.7) ∞∑ n=0 Ar n(x;u) tn n! = (1− u)tr et − u ext. (1.8) It is worth-mentioning that (1.7) is parallel to the generalized Apostol type Frobenius- Euler polynomials of Kurt and Simsek [24]. Moreover, mixing the Genocchi numbers with the concept of polylogarithm Lik(z) [9] Lik(z) = ∞∑ n=0 zn nk , k ∈ Z, (1.9) 1473 yields the poly-Genocchi polynomials, which are defined as follows ∞∑ n=0 G(k) n (x) xn n! = 2Lik(1− et) et + 1 ext, (1.10) such that when k = 1, Li1(1− et) = ln(1− (1− et)) = ln(et) = t and so (1.10) gives (1.1). Furthermore, with a slight modification of the generating function, another generalization, denoted by G (k) n,2(x), was defined by Kim et al. [31] as follows ∞∑ n=0 G (k) n,2(x) xn n! = Lik(1− e−2t) et + 1 ext. (1.11) These polynomials are called modified poly-Genocchi polynomials. Note that, when k = 1, equations (1.10) and (1.11) give the Genocchi polynomials in (1.1). That is, G(1) n (x) = G (1) n,2(x) = Gn(x). Kim et. al [31] obtained several properties of these polynomials. The higher order Apostol-Type poly-Genocchi polynomials G(k,α) n (x;λ, a, b, c) and Apostol- Frobenius-Type poly-Genocchi polynomials G(k,α) n (x;λ, u, a, b, c) and are respectively de- fined by (see [11, 12]) ∞∑ n=0 G(k,α) n (x;λ, a, b, c) tn n! = ( Lik(1− (ab)−2t) a−t + λbt )α cxt, (1.12) ∞∑ n=0 G(k,α) n (x;λ, u, a, b, c) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )α cxt. (1.13) Further extension and variation of these polynomials can be found in [13, 14]. The Bell polynomials, represented as Bn(x), are defined as polynomials with coeffi- cients corresponding to the Stirling numbers of the second kind. To be more precise, Bn(x) = n∑ k=0 S(n, k)xk, (1.14) where S(n, k) denotes the Stirling numbers of the second kind. These numbers adhere to the following exponential generating function ∞∑ n=0 S(n, j) tn n! = (et − 1)j j! , (1.15) (see [10, 26]). By mixing the concept of Bell polynomials with partially degenerate Bernoulli polynomials of the first kind defined by the generating function log(1 + λt)1/λ et − 1 ext = ∞∑ n=0 Bn,λ(x) tn n! , 1474 the partially degenerate Bell-Bernoulli polynomials of the first kind are defined in [20] by the generating function log(1 + λt)1/λ et − 1 ext+y(et−1) = ∞∑ n=0 BelBn,λ(x, y) tn n! . Further generalization is introduced in [20] by incorporating the concept of Dirichlet char- acter with conductor d. Also, by mixing the concept of Bell polynomials, Alam et al. [2] developed generating functions for new families of special polynomials, including two parametric types of Bell- based Bernoulli and Euler polynomials, defined as follows: ∞∑ n=0 BellBr n(ξ + iη, x;u, λ) tn n! = ( 1 et − 1 )r e(ξ+iη)teζ(e x−1), ∞∑ n=0 BellHr n(ξ + iη, x;u, λ) tn n! = ( 2 et + 1 )r e(ξ+iη)teζ(e x−1). They investigated fundamental properties of these generating functions and used them, along with certain identities, to present relations among trigonometric functions, two parametric types of Bell-based Bernoulli and Euler polynomials, and Stirling numbers. They also derived computational formulae for these polynomials. By applying a partial derivative operator to these generating functions, they obtained various derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers. In separate papers, Alam et al. [4] and Ayed et al. [8] introduced a novel class of Bell- based Apostol-type Frobenius-Euler polynomials and Bell-based Apostol-type Frobenius- Genocchi polynomials, respectively. These are defined as follows: ∞∑ n=0 BellHr n(x;u, λ) tn n! = ( 1− u λet − u )r eζ(e x−1), (1.16) ∞∑ n=0 BellGr n(ξ + iη, x;u, λ) tn n! = ( (1− u)t λet − u )r e(ξ+iη)teζ(e x−1). (1.17) Their research explored various properties of these polynomials and numbers, deriving summation formulas in terms of Apostol-type Bernoulli, Euler, and Genocchi polynomi- als [4]. They established numerous identities using diverse analytical methods and the generating function technique, and introduced parametric variations that unveiled specific polynomial identities [4]. Additionally, they investigated various formulas and proper- ties, including differentiation rules, addition formulas, relations, and summation formulas. Moreover, they identified the first few zero values of the Apostol-type Frobenius-Genocchi polynomials and provided graphical representations of these zero values [7, 8]. It is note- worthy that an alternative method for introducing Bell-based Frobenius-Euler polynomials has been established in [21]. This variation, known as Bell-based Frobenius-type Eulerian 1475 polynomials, is defined as follows: ∞∑ n=0 BellAr n(ξ, ζ|u) tn n! = ( 1− u et(u−1) − u )r eξteζ(e x−1) In line with the polynomial exploration in [12], it is equally compelling to investigate Bell-based Apostol-Frobenius-type poly-Genocchi polynomials. 2. Higher Order Bivariate Bell-Based Apostol-Frobenius-Type Poly-Genocchi Polynomials In this section, we introduce higher-order bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials, aligning them with the Bell-based Apostol-type Frobenius- Euler polynomials as defined by Alam et al. [4] and the generalized Apostol-Frobenius- Type poly-Genocchi polynomials by Khan [34]. The following definition formally presents the higher-order Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials. Definition 2.1. The bivariate Bell-based Apostol-Frobenius-type Poly-Genocchi polyno- mials of higher order with parameters a and b, denoted by BG (r) n (x, y;u, λ, a, b) are defined by ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1). (2.1) When x = 0, the Bell-based Apostol-Frobenius-type poly-Genocchi polynomials of higher order BG (r) n,k(y;u, λ) are defined by ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ey(e t−1) (2.2) Remark 2.2. Using the fact that Li1(z) = − ln(1− z), we get Li1(1− (ab)−(1−u)t) = − ln(1− (1− (ab)−(1−u)t)) = (1− u)t ln ab. Hence, when k = 1, the Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order BG (r) n,k(y;u, λ) in (2.2) can further be reduced to ∞∑ n=0 BG (r) n (x, y;u, λ, a, b) tn n! = ( (1− u)t ln ab λbt − ua−t )r ext+y(et−1), (2.3) the higher order bivariate Bell-based Apostol-Frobenius-Type Genocchi numbers with pa- rameters a and b. 1476 Remark 2.3. When y = 1, the Bell-based Apostol-Frobenius-Type poly-Genocchi polyno- mials of higher order BG (r) n,k(y;u, λ) in (2.2) can further be reduced to ∞∑ n=0 BG (r) n,k(1;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ee t−1, the Bell-based Apostol-Frobenius-Type poly-Genocchi numbers of higher order. Remark 2.4. The Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order BG (r) n,k(y;u, λ) in (2.2) can further be reduced as follows: (i) When k = 1, a = 1, b = e, ∞∑ n=0 BG (r) n (x, y;u, λ) tn n! = ( (1− u)t λet − u )r ext+y(et−1), (2.4) the Higher Order Bivariate Bell-based Apostol-Frobenius-Type Genocchi polynomials, where BGn(x, y;u, λ) = BG (1) n (x, y;u, λ, 1, e). (ii) When r = 1, (2.4) gives ∞∑ n=0 BGn(x, y;u, λ) tn n! = (1− u)t λet − u ext+y(et−1), (2.5) the Bivariate Bell-based Apostol-Frobenius-Type Genocchi polynomials, where BGn(x, y;u, λ) = BG (1) n (x, y;u, λ, 1, e). Remark 2.5. When r = 0, the Bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order BG (r) n,k(y;u, λ) in (2.1) can be reduced to ∞∑ n=0 BG (0) n (x, y;u, λ) tn n! = ext+y(et−1), Bn(x, y) = BG (0) n (x, y;u, λ) (2.6) the Bivariate Bell Polynomial. The following theorem contains the first identity for the bivariate Bell-based Apostol- Frobenius-Type poly-Genocchi polynomials of higher order expressed in terms Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order and the Bell polyomi- als. Theorem 2.6. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order with parameters a and b, BG (r) n,k(x, y;u, λ, a, b), are equal to BG (r) n,k(x, y;u, λ, a, b) = n∑ k=0 ( n k ) G (r) k (x;u, λ, a, b)Bn−k(y). (2.7) 1477 Proof. Using Definition 2.1, we have ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1) = {( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext } ey(e t−1) = ( ∞∑ n=0 G (r) n,k(x;u, λ, a, b) tn n! )( ∞∑ n=0 Bn(y) tn n! ) = ∞∑ n=0 { n∑ k=0 ( n k ) G (r) k (x;u, λ, a, b)Bn−k(y) } tn n! . Comparing the coefficients of tn n! yields the desired identity in (2.7). The next theorem expresses the bivariate Bell-based Apostol-Frobenius-Type poly- Genocchi polynomials of higher order as polynomial in x with Bell-based Apostol-Frobenius- Type poly-Genocchi polynomials of higher order as the coefficients. Theorem 2.7. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order BG (r) n,k(x, y;u, λ, a, b) are equal to BG (r) n,k(x, y;u, λ, a, b) = n∑ j=0 ( n k ) BG (r) n−j,k(y;u, λ, a, b)x j . (2.8) Proof. Using Definition 2.1, we have ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ey(e t−1)ext = ( ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! )( ∞∑ n=0 (xt)n n! ) = ∞∑ n=0  n∑ j=0 ( n j ) BG (r) j,k(y;u, λ, a, b)x n−j  tn n! Comparing the coefficients of tn n! yields BG (r) n,k(x, y;u, λ, a, b) = n∑ j=0 ( n j ) BG (r) j,k(y;u, λ, a, b)x n−j , which is equivalent to the desired identity in (2.8). 1478 The next theorem contains the addition formula for bivariate Bell-based Apostol- Frobenius-Type poly-Genocchi polynomials of higher order. Theorem 2.8. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order BG (r) n,k(x, y;u, λ, a, b) are equal to BG (r) n,k(x+ y, z;u, λ, a, b) = n∑ j=0 ( n j ) G (r) j,k(x;u, λ, a, b)Bn−j(y, z). (2.9) Proof. Using Definition 2.1, we have ∞∑ n=0 BG (r) n,k(x+ y, z;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r e(x+y)t+z(et−1) = {( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext } eyt+z(et−1) = ( ∞∑ n=0 G (r) n,k(x;u, λ, a, b) tn n! )( ∞∑ n=0 Bn(y, z) tn n! ) = ∞∑ n=0  n∑ j=0 ( n k ) G (r) j,k(x;u, λ, a, b)Bn−j(y, z)  tn n! . Comparing the coefficients of tn n! yields the desired identity in (2.9). 3. Implicit Summation Formula Within this section, we will derive different summation formulas for BG (r) n (x+y, z;u, λ), establishing implicit connections among the variables by considering them as arguments. The subsequent theorem encapsulates a particular expression of these summation formulas. Theorem 3.1. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order BG (r) n (x, y;u, λ) satisfy the following summation formula: BG (r1+r2) n,k (x1 + x2, y2 + y2;u, λ, a, b) = n∑ j=0 ( n j ) BG (r1) j,k (x1, y1;u, λ, a, b)BG (r2) n−j,k(x2, y2;u, λ). (3.1) Proof. Note that we can express the right hand side of (2.1) as follows:( Lik(1− (ab)−(1−u)t) λbt − ua−t )r1+r2 e(x1+x2)t+(y1+y2)(et−1) 1479 = {( Lik(1− (ab)−(1−u)t) λbt − ua−t )r1 ex1t+y1(et−1) } {( Lik(1− (ab)−(1−u)t) λbt − ua−t )r2 ex2t+y2(et−1) } . Applying (2.1) yields ∞∑ n=0 BG (r1+r2) n,k (x1 + x2, y2 + y2;u, λ, a, b) tn n! = ( ∞∑ n=0 BG (r1) n,k (x1, y1;u, λ, a, b) tn n! )( ∞∑ n=0 BG (r2) n,k (x2, y2;u, λ, a, b) tn n! ) = ∞∑ n=0 n∑ j=0 BG (r1) j,k (x1, y1;u, λ, a, b)BG (r2) n−j,k(x2, y2;u, λ, a, b) ( n j ) . By comparing the coefficients of tn n! , we obtain BG (r1+r2) n,k (x1 + x2, y2 + y2;u, λ, a, b) = n∑ j=0 ( n k ) BG (r1) j,k (x1, y1;u, λ, a, b)BG (r2) n−j,k(x2, y2;u, λ), which is exactly the desired summation formula in (3.5). Remark 3.2. When r1 = r,r2 = 0,x1 = x,x2 = 1,y1 = y, y2 = 0, the summation formula in (3.5) reduces to BG (r) n,k(x+ 1, y;u, λ, a, b) = n∑ j=0 ( n j ) BG (r) j,k(x, y;u, λ, a, b)Bn−k(1, 0) = n∑ j=0 ( n j ) BG (r) j,k(x, y;u, λ, a, b). (3.2) On the other hand, when y = 1, (2.9) gives BG (r) n,k(x+ 1, z;u, λ, a, b) = n∑ j=0 ( n j ) G (r) j,k(x;u, λ, a, b)Bn−j(1, z). (3.3) Replacing z with y in (3.3) and subtract it from (3.2) yields n∑ j=0 ( n j ) G (r) j,k(x;u, λ, a, b)Bn−j(1, z) = n∑ j=0 ( n j ) BG (r) j,k(x, y;u, λ, a, b). 1480 We recall the following series manipulation formula: ∞∑ N=0 f(N) (x+ y)N N ! = ∞∑ n=0 ∞∑ m=0 f(n+m) xn n! yn m! . (3.4) Applying (3.4) obtains( Lik(1− (ab)−(1−u)(t+v)) λbt+v − ua−(t+v) )r ey(e t+v−1) = e−x(t+v) ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) (t+ v)n n! = e−x(t+v) ∞∑ j=0 ∞∑ l=0 BG (r) j+l,k(x, y;u, λ, a, b) tj j! vl l! . Replacing x with z yields( Lik(1− (ab)−(1−u)(t+v)) λbt+v − ua−(t+v) )r ey(e t+v−1) = e−z(t+v) ∞∑ j=0 ∞∑ l=0 BG (r) j+l,k(z, y;u, λ, a, b) tj j! vl l! . Thus, by using (3.4) again, we have ∞∑ j=0 ∞∑ l=0 BG (r) j+l,k(x, y;u, λ, a, b) tj j! vl l! = e(x−z)(t+v) ∑ j,l≥0 BG (r) j+l,k(z, y;u, λ, a, b) tj j! vl l! = ( ∞∑ N=0 (x− z)N (t+ v)N N ! )∑ j,l≥0 BG (r) j+l,k(z, y;u, λ, a, b) tj j! vl l!  =  ∑ n,m≥0 (x− z)n+m tn n! vm m! ∑ j,l≥0 BG (r) j+l,k(z, y;u, λ, a, b) tj j! vl l!  = ∑ j,l≥0  j,l∑ n,m=0 ( j n )( l m ) (x− z)n+m BG (r) j+l,k(z, y;u, λ, a, b)  tj j! vl l! . Comparing the coefficients of tj j! vl l! completes the proof of the following theorem. 1481 Theorem 3.3. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order BG (r) n,k(x, y;u, λ, a, b) satisfy the following summation formula: BG (r) j+l,k(x, y;u, λ, a, b) = j,l∑ n,m=0 ( k n )( l m ) (x− z)n+m BGk+l−n−m(z, y;u, λ, a, b). (3.5) The next theorem gives the difference when the variable x in BG (r) n,k(x, y;u, λ, a, b) is shifted by 1. Theorem 3.4. For n ≥ 1, the difference BG (r) n,k(x+1, y;u, λ)−BG (r) n,k(x, y;u, λ, a, b) equals BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b) = n−1∑ j=0 ( n k ) BG (r) j,k(x, y;u, λ, a, b). (3.6) Proof. Using Definition 2.1, we have ∞∑ n=0 BG (r) n,k(x+ 1, y;u, λ, a, b) tn n! − ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r e(x+1)t+y(et−1) − ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1) = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1)(et − 1) = ( ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! )∑ n≥0 tn+1 (n+ 1)!  = ∞∑ n=0  n∑ j=0 ( n+ 1 j ) BG (r) j,k(x, y;u, λ, a, b)  tn n! . Thus, we have ∞∑ n=0 ( BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b) ) tn n! = ∞∑ n=0  n∑ j=0 ( n+ 1 j ) BG (r) j,k(x, y;u, λ, a, b)  tn+1 (n+ 1)! = ∞∑ n=1  n−1∑ j=0 ( n j ) BG (r) j,k(x, y;u, λ, a, b)  tn n! . This immediately gives BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b) = n−1∑ j=0 ( n j ) BG (r) j,k(x, y;u, λ, a, b). 1482 4. Connection with Second Kind Stirling Numbers and Bivariate Bell Polynomials In this section, we derive some formulas connecting BG (r) n (x, y;u, λ) with Stirling num- bers of the second kind given in (1.15) and bivariate Bell polynomials in (2.6). Theorem 4.1. The bivariate Bell-based Apostol-Frobenius-Type poly-Genocchi polynomi- als of higher order BG (r) n (x, y;u, λ) satisfy the following summation formula BG (r) n,k(x, y;u, λ) = n∑ i=0 i∑ j=0 ( n i ) (x)jS(i, j)BG (r) n−i,k(y;u, λ). (4.1) Proof. Using Definition 2.1, we have ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ey(e t−1)(1 + et − 1)x = ( ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! ) ∞∑ j=0 ( x n ) (et − 1)n  = ( ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! ) ∞∑ j=0 (x)j (et − 1)j j!  = ( ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! ) ∞∑ j=0 (x)j ∞∑ n=0 S(n, j) tn n!  = ( ∞∑ n=0 BG (r) n,k(y;u, λ, a, b) tn n! ) ∞∑ n=0  ∞∑ j=0 (x)jS(n, j)  tn n!  = ∞∑ n=0 n∑ i=0 ( n i ) ∞∑ j=0 (x)jS(i, j)BG (r) n−i,k(y;u, λ, a, b)  tn n! = ∞∑ n=0  n∑ i=0 ( n i ) ∞∑ j=0 (x)jS(i, j)BG (r) n−i,k(y;u, λ, a, b)  tn n! BG (r) n,k(x, y;u, λ, a, b) = n∑ i=0 ∞∑ j=0 ( n i ) (x)jS(i, j)BG (r) n−i,k(y;u, λ, a, b) = n∑ i=0 i∑ j=0 ( n i ) (x)jS(i, j)BG (r) n−i,k(y;u, λ, a, b). The subsequent theorem is another relation for BG (r) n,k(x, y;u, λ, a, b) in connection with Stirling numbers of the second kind. 1483 Theorem 4.2. The higher order Bivariate Bell-based Apostol-Frobenius-type poly-Genocchi polynomials with parameters a, b satisfy the relation, BG (r) n,k(x, y;u, λ, a, b) = n∑ j=0 ( n j ) (−1)rBG (r) n−j(x, y;u, λ, a, b)dj (4.2) where dj = ∑ n1+n2+...+nr=j r∏ i=1 cni ( j n1, n2, . . . , nr ) cj = j∑ m=0 (−1)m+j+1 ((1− u) ln ab)jm!S(j + 1,m+ 1) (j + 1)(m+ 1)k−1 . Proof. Now, (2.1) can be written as ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ext+y(et−1) (λbt − ua−t)r ( ∞∑ m=1 (1− e−(1−u)t ln ab)m mk )r = ext+y(et−1) (λbt − ua−t)r ( ∞∑ m=0 (1− e−(1−u)t ln ab)m+1 (m+ 1)k )r = ext+y(et−1) (λbt − ua−t)r ( ∞∑ m=0 m! (m+ 1)k−1 (1− e−(1−u)t ln ab)m+1 (m+ 1)! )r = ext+y(et−1) (λbt − ua−t)r  ∞∑ m=0 (−1)m+1m! (m+ 1)k−1 ∞∑ j=m+1 S(j,m+ 1) (−(1− u)t ln ab)j j! r = (−1)rext+y(et−1) ( (1− u)t ln ab λbt − ua−t )r  ∞∑ j=0 cj tj j! r , where cj = j∑ m=0 (−1)m+j+1 ((1− u) ln ab)jm!S(j + 1,m+ 1) (j + 1)(m+ 1)k−1 . Note that the power series (∑∞ j=0 cj tj j! )r can be expressed as ∞∑ j=0 cj tj j! r = ∞∑ n=0 dn tn n! , where dn = ∑ n1+n2+...+nr=n r∏ i=1 cni ( n n1, n2, . . . , nr ) , 1484 (see [10]). It follows that ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = (−1)r ( ∞∑ n=0 BG (r) n (x, y;u, λ, a, b) tn n! )( ∞∑ n=0 dn tn n! ) = ∞∑ n=0  n∑ j=0 ( n j ) (−1)rBG (r) n−j(x, y;u, λ, a, b)dj  tn n! Comparing the coefficients completes the proof of the theorem. The next theorem contains a relation that expresses the bivariate Bell polynomials in terms of bivariate Bell-based Apostol-Frobenius-type poly-Genocchi polynomials. Theorem 4.3. The following relation holds Bn(x, y) = λ BGn+1(x+ 1, y;u, λ)− u BGn+1(x, y;u, λ) (1− u)(n+ 1) . (4.3) Proof. Using equation (2.6), we have ∞∑ n=0 Bn(x, y) tn n! = ( λet − u (1− u)t )( (1− u) t λet − u ext+y(et−1) ) = 1 (1− u)t ( λ ( (1− u) t λet − u e(x+1)t+y(et−1) ) − u ( (1− u) t λet − u ext+y(et−1) )) = 1 (1− u) ( λ ∞∑ n=0 BGn(x+ 1, y;u, λ) tn−1 n! − u ∞∑ n=0 BG (r) n (x, y;u, λ) tn−1 n! ) = 1 1− u ( λ ∞∑ n=−1 BGn(x+ 1, y;u, λ) tn (n+ 1)! − u ∞∑ n=−1 BG (r) n (x, y;u, λ) tn (n+ 1)! ) = ∞∑ n=−1 ( λ BGn+1(x+ 1, y;u, λ)− u BGn+1(x, y;u, λ) (1− u)(n+ 1) ) tn n! Comparing the coefficients of tn n! yields Bn(x, y) = λ BGn+1(x+ 1, y;u, λ)− u BGn+1(x, y;u, λ) (1− u)(n+ 1) . 5. Derivative Formulas The derivative formulas for special polynomials play a crucial role in various areas of mathematics, physics, engineering, and other scientific disciplines. For instance, derivative formulas allow for the analysis of the behavior and properties of special polynomials in terms of their rates of change. This is fundamental in calculus and mathematical analysis 1485 for understanding functions and their behavior. They are also essential for manipulat- ing generating functions, which represent sequences of coefficients of special polynomials. These functions are widely used in combinatorics, number theory, and discrete mathemat- ics for counting and enumerative purposes. The following theorem contains the derivative formula for BG (r) n,k(x, y;u, λ) with respect to the variable x. Theorem 5.1. The following derivative formula holds ∂ ∂x BG (r) n,k(x, y;u, λ) = nBG (r) n−1,k(x, y;u, λ). (5.1) Proof. Using Definition 2.1, we have ∂ ∂x ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ∂ ∂x ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1) ∞∑ n=0 ∂ ∂x BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1) t = t ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn n! = ∞∑ n=0 BG (r) n,k(x, y;u, λ, a, b) tn+1 n! = ∞∑ n=1 nBG (r) n−1,k(x, y;u, λ, a, b) tn n! . Comparing the coefficients of tn n! yields ∂ ∂x BG (r) n,k(x, y;u, λ, a, b) = nBG (r) n−1,k(x, y;u, λ, a, b). Remark 5.2. This relation shows that BG (r) n,k(x, y;u, λ, a, b) is an Apell Polynomial (see [25, 29]). Belonging to the category of Appell polynomials, the polynomials BG (r) n,k(x, y;u, λ, a, b) are expected to demonstrate the following characteristics: BG (r) n,k(x, y;u, λ, a, b) = n∑ j=0 ( n j ) cjx n−j BG (r) n,k(x, y;u, λ, a, b) =  n∑ j=0 cj j! Dj xn for some scalar ck ̸= 0. Clearly, using (2.8), cj = BG (r) j,k(y;u, λ, a, b). Hence, BG (r) n,k(x, y;u, λ, a, b) =  n∑ j=0 BG (r) j,k(y;u, λ, a, b) j! Dj xn. (5.2) REFERENCES 1486 The next theorem contains the derivative formula for BG (r) n,k(x, y;u, λ, a, b) with respect to the variable y. Theorem 5.3. The following derivative formula holds ∂ ∂y BG (r) n,k(x, y;u, λ, a, b) = BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b). (5.3) Proof. Using Definition 2.1, we have ∞∑ n=0 ∂ ∂y BG (r) n,k(x, y;u, λ, a, b) tn n! = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1)(et − 1) = ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r e(x+1)t+y(et−1) − ( Lik(1− (ab)−(1−u)t) λbt − ua−t )r ext+y(et−1) = ∞∑ n=0 BG (r) n,k(x+ 1, y;u, λ) tn n! − ∞∑ n=0 BG (r) n,k(x, y;u, λ) tn n! = ∞∑ n=0 { BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b) } tn n! . Comparing the coefficients of tn n! yields ∂ ∂y BG (r) n,k(x, y;u, λ, a, b) = BG (r) n,k(x+ 1, y;u, λ, a, b)− BG (r) n,k(x, y;u, λ, a, b). Acknowledgement . This research has been funded by Cebu Normal University (CNU) through its Center for Research and Development (CRD). References [1] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions, volume 48. Dover, New York, 1970. [2] N. Alam, W. A. Khan, S. Obeidat, G. Muhiuddin, N.S. Diab, H.N. Zaidi, A. Altaleb, and L. Bachioua. A note on bell-based bernoulli and euler polynomials of complex variable. Computer Modelling in Engineering and Sciences, 153(1):187–209, 2023. REFERENCES 1487 [3] N. Alam, W.A. Khan, C. Kizilates, S. Obeidat, C.S. Ryoo, and N.S. Diab. Some explicit properties of frobenius-euler-genocchi polynomials with applications in com- puter modeling. Symmetry, 15(1358):1–20, 2023. [4] N. Alam, W.A. Khan, and C.S. Ryoo. A note on bell-based apostol-type frobenius- euler polynomials of complex variable with its certain applications. Mathematics, 2022(12):2109, 10. [5] S. Araci. Novel identities involving genocchi numbers and polynomials arising from application of umbral calculus. Appl. Math. Comput., 233:599–607., 2014. [6] S. Araci, E. Sen, and M. Acikgoz. Theorems on genocchi polynomials of higher order arising from genocchi basis. Taiwanese J. Math. Math. Sci., 18(2):473–482, 2014. [7] A. Ayed, W.A. Khan, and C.S. Ryoo. Certain properties on bell-based apostol-type frobenius-genocchi polynomials and its applications. Advanced Mathematical Models and Applications, 1(8):92–107, 2023. [8] A. Ayed, W.A. Khan, and C.S. Ryoo. Certain properties on bell based apostol- frobenius-genocchi polynomials of complex variables. Journal of Mathematics and Computer Science, 33(3):326–338, 2024. [9] A. Bayad and Y. Hamahata. Polylogarithms and poly-bernoulli polynomials. Kyushu J. Math, 65:15–24, 2011. [10] L. Comtet. Advanced Combinatorics. Reidel, Dordrecht, The Netherlands., 1974. [11] C. Corcino and R. Corcino. Higher order apostol-type poly-genocchi polynomials with parameters a, b and c. Communication of the Korean Mathematical Society, 36(3):423–445, 2021. [12] R. Corcino and C. Corcino. Higher order apostol-frobenius-type poly-genocchi poly- nomials with parameters a, b and c. Journal of Inequalities and Special Functions, 12(3):54–72, 2021. [13] R. Corcino and C. Corcino. Generalized laguerre-apostol-frobenius-type poly- genocchi polynomials of higher order with parameters a,b and c. European Journal of Pure and Applied Mathematics, 15(4):1549–1565, 2022. [14] R. Corcino and C. Corcino. Degenerate apostol-frobenius-type poly-genocchi polyno- mials of higher order with parameters a and b. European Journal of Pure and Applied Mathematics, 16(2):687–712, 2023. [15] R. Corcino, C. Corcino, K. Casas, A. Elnar, and G. Maglasang. Construction of fourier series expansion of apostol-frobenius-type tangent and genocchi polynomials of higher-order. European Journal of Pure and Applied Mathematics, 16(2):1005– 1023, 2023. REFERENCES 1488 [16] Y. He. Some new results on products of the apostol-genocchi polynomials. J. Comput. Anal. Appl., 22(4):591–600, 2017. [17] Y. He, S. Araci, H.M. Srivastava, and M. Acikgoz. Some new identities for the apostol- bernoulli polynomials and the apostol-genocchi polynomials. Appl. Math. Comput., 262:31–41, 2015. [18] Y. He and T. Kim. General convolution identities of apostol-bernoulli, euler and genocchi polynomials. J. Nonlinear Sci. Appl., 9:4780–4797, 2016. [19] S. Hu, D. Kim, and M.S. Kim. New identities involving bernoulli, euler and genocchi numbers. Advances in Difference Equations, 2013:Article 74, 2013. [20] W. A. Khan, J. Younis, and M. Nadeem. Construction of partially degenerate bell- bernoulli polynomials of the first kind. Analysis, 43(3):171–184, 2022. [21] W.A. Khan, M.A. Alatawi, and U. Duran. Applications, and properties of bivariate bell-based frobenius-type eulerian polynomials. Journal of Function Spaces, 2023:Ar- ticle ID 5205867, 10 pages, 2023. [22] D.S. Kim, D.V. Dolgy, T. Kim, and S.H. Rim. Some formula for the product of two bernoulli and euler polynomials. Abstract and Applied Analysis, 2012:Article ID 784307, 15 pages., 2012. [23] D.S. Kim and T. Kim. Some new identities of frobenius-euler numbers and polyno- mials. J. Inequal. Appl., 2012:Article ID 307, 2012. [24] B. Kurt and Y. Symsek. On the generalized apostol type frobenius-euler polynomials. Advances in Difference Equation, 2013(1):1–9, 2013. [25] D.W. Lee. On multiple appell polynomials. Proc. Amer. Math. Soc., 139:2133–2141, 2011. [26] I. Mezo and R. Corcino. The estimation of the zeros of the bell and r-bell polynomials. Applied Mathematics and Computations, 250:727–732., 2015. [27] Araci S., Acikgoz M., and Sen E. Some new formulae for genocchi numbers and polynomials involving benoulli and euler polynomials. Int. J. Math. Sci., 2014:Article ID 760613, 2014. [28] Araci S., Khan W.A., Acikgoz M., Ozel C., and Kumam P. A new generalization of apostol type hermite-genocchi polynomials and its applications. Springerplus, 5:Art. ID 860, 2016. [29] J. Shohat. The relation of the classical orthogonal polynomials to the polynomials of appell. Amer. J. Math., 58:453–464, 1936. REFERENCES 1489 [30] Kim T. Some identities for the bernoulli, the euler and the genocchi numbers and polynomials. Adv. Stud. Contemp. Math., 20(1):23–28, 2010. [31] Kim T., Y.S. Jang, and J.J. Seo. A note on poly-genocchi numbers and polynomials. Applied Mathematical Sciences, 8:4775–4781, 2014. [32] Kim T., Rim. S.H., Dolgy D.V., and Lee S.H. Some identities of genocchi polynomials arising from genocchi basis. J. Ineq. Appl., 2013:Article ID 43, 2013. [33] G. Thomas, M. Weir, J. Hass, and F. Giordano. Thomas’ Calculus. Pearson Educa- tion, Inc., 11th edn. edition, 2005. [34] Khan W.A. and D. Srivastava. On the generalized apostol-frobenius-type poly- genocchi polynomials. Filomat, 33(7):1967–1977, 2019.