EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 687-712 ISSN 1307-5543 – ejpam.com Published by New York Business Global Degenerate Apostol-Frobenius-Type Poly-Genocchi Polynomials of Higher Order 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. This paper introduces another variation of poly-Genocchi polynomials by mixing the concept of modified degenerate polyexponential function, Apostol-Genocchi polynomials and Frobe- nius polynomials. These polynomials are called the degenerate Apostol-Frobenius-type poly- Genocchi polynomials with parameters a and b. Several identities and formulas are derived in- cluding recurrence relations, explicit formulas and certain differential identity. Moreover, some relations are established connecting these polynomials to degenerate Stirling numbers of the first and second kind, higher order degenerate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials. 2020 Mathematics Subject Classifications: 11B68, 11B73, 05A15 Key Words and Phrases: Poly-Genocchi polynomials, degenerate exponential function, polyex- ponential function, polylogarithm, Frobenius polynomials, Appell polynomials, Euler polynomials, Bernoulli polynomials 1. Introduction There are many ways of constructing a generalization of certain special function, poly- nomial or number. One of these is by mixing it with the concept of some other known functions and polynomials. For instance, multiplying the generating function of the Genoc- chi numbers Gn (see [12]) ∞∑ n=0 Gn tn n! = 2t et + 1 , |t| < π, and its variations with exponential polynomials yields the Genocchi polynomials, the Genocchi polynomials of higher order, the Apostol-Genocchi polynomials, and Apostol- Genocchi polynomials of higher order, which are respectively defined as follows: ∞∑ n=0 Gn(x) tn n! = 2t et + 1 ext, |t| < π, (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4694 Email addresses: rcorcino@yahoo.com (R. Corcino), corcinoc@cnu.edu.ph (C. Corcino) https://www.ejpam.com 687 © 2023 EJPAM All rights reserved. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 688 ∞∑ n=0 G(k) n (x) tn n! = ( 2t et + 1 )k ext, (2) ∞∑ n=0 Gn(x, λ) tn n! = 2t λet + 1 ext, (3) ∞∑ n=0 G(k) n (x, λ) tn n! = ( 2t λet + 1 )k ext, (4) where |t| < π when λ = 1 and |t| < log(−λ) when λ ̸= 1, λ ∈ C (see [1–3, 5, 6, 40]). The first two polynomials are well-studied and two of the most recent studies are the works of Corcino-Corcino [14, 15] on asymptotic approximations, while the last two were given asymptotic approximation and Fourier series expansion in [13, 16, 20]. Also, incorporating the concept of 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, (5) (see [4, 6, 7, 19, 21, 24, 32, 43, 44] for other interesting studies related to these polynomials). Moreover, mixing the Genocchi numbers with the concept of polylogarithm Lik(z) Lik(z) = ∞∑ n=0 zn nk , k ∈ Z, (6) yields the poly-Genocchi polynomials, which are defined as follows ∞∑ n=0 G(k) n (x) xn n! = 2Lik(1− et) et + 1 ext. (7) Furthermore, with a slight modification of the generating function, another generalization, denoted by G (k) n,2(x), was defined by Kim et al. [9, 25, 34] as follows ∞∑ n=0 G (k) n,2(x) xn n! = Lik(1− e−2t) et + 1 ext. (8) These polynomials are called modified poly-Genocchi polynomials. Note that, when k = 1, equations (7) and (8) give the Genocchi polynomials in (1). That is, G(1) n (x) = G (1) n,2(x) = Gn(x). Kim et. al [25] obtained several properties of these polynomials. By introducing additional three parameters a, b, and c, Kurt [33] defined two forms of generalized poly-Genocchi polynomials as follows 2Lik(1− (ab)−t) a−t + bt ext = ∞∑ n=0 G(k) n (x; a, b, c) xn n! (9) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 689 2Lik(1− (ab)−2t) a−t + bt ext = ∞∑ n=0 G (k) n,2(x; a, b, c) xn n! . (10) These were motivated by the generalizations introduced in (7) and (8), respectively. Note that, when x = 0, (7) reduces to 2Lik(1− et) et + 1 = ∞∑ n=0 G(k) n xn n! , (11) where G (k) n are called the poly-Genocchi numbers. It is worth-mentioning that, using multi-polylogarithm, the generalized poly-Genocchi polynomials in (9) and (10) have been extended further in [39]. Recently, a new variation of poly-Genocchi polynomials with pa- rameters a, b and c was defined in [17] by mixing the definitions of polylogarithm, Apostol- Genocchi polynomials and Frobenius polynomials, namely, the Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a, b and c. More precisely, the said polynomials, denoted by Ĝ(k,α) n (x;λ, ρ, u, a, b), are defined as coefficients of the following generating function ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( Lik,ρ(1− (ab)−(1−u)t) λbt − ua−t )α cxt. (12) It is worth-mentioning the following interesting applications of Genocchi numbers, Genocchi polynomials and poly-Genocchi polynomials: (i) The Genocchi numbers were used to bound the number of finite languages over a two-letter alphabet accepted by a deterministic finite automation (DFA) with n states (see [18]); (ii) Certain wavelets method was constructed using Genocchi polynomials which has been used to obtain a numerical solution for the classical and time-fractional Rosenau- Hyman equation arising in the formation of patterns in liquid drops (see [37]); (iii) The orthogonal version of poly-Genocchi polynomials coincides with the shifted Leg- endre polynomials. Also, the poly-Genocchi polynomials were used to solve the frac- tional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme (see [8]). The degenerate exponential function, denoted by exλ(t), was defined in [10, 11]) exλ(t) = (1 + λt)x/λ = ∞∑ n=0 (x)n,λ tn n! , λ ∈ R+ ∪ {0}, (13) where eλ(t) = e1λ(t) = (1 + λt)1/λ and (x)0,λ = 1, (x)n,λ = x(x− λ)(x− 2λ) . . . (x− (n− 1)λ), n ≥ 1. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 690 It can easily be seen that ex+y λ (t) = (1 + λt)(x+y)/λ = (1 + λt)(x/λ)+(y/λ) = (1 + λt)(x/λ)(1 + λt)(y/λ) = exλ(t)e y λ(t), (14) and d dx exλ(t) = log(1 + λt)1/λexλ(t). (15) The degenerate Bernoulli polynomials Bn,λ(x) and degenerate Euler polynomials En,λ(x) were defined by Carlitz [11] by means of the following generating functions t eλ(t) + 1 exλ(t) = t (1 + λt)1/λ + 1 (1 + λt)x/λ = ∞∑ n=0 Bn,λ(x) tn n! 2 eλ(t) + 1 exλ(t) = 2 (1 + λt)1/λ + 1 (1 + λt)x/λ = ∞∑ n=0 En,λ(x) tn n! . Parallel to these, Lim [36] defined the degenerate Genocchi polynomials as follows 2t eλ(t) + 1 exλ(t) = 2t (1 + λt)1/λ + 1 (1 + λt)x/λ = ∞∑ n=0 Gn,λ(x) tn n! . (16) Kim and Kim [42] introduced the degenerate Frobenius-Euler polynomials as coefficients of the following generating function 1− u eλ(t)− u exλ(t) = ∞∑ n=0 hn,λ(x|u) tn n! (17) and Kim et al. [26] derived formulas that express any polynomial in terms of hn,λ(x|u). In their separate paper, Kim and Kim [27] defined the generalized degenerate Euler-Genocchi polynomials, denoted by A (r) n,λ(x), as coefficients of the following generating function 2tr eλ(t) + 1 exλ(t) = ∞∑ n=0 A (r) n,λ(x) tn n! , which reduce to the degenerate Genocchi polynomials in (26) when r = 1. That is, A (1) n,λ(x) = Gn,λ(x). The degenerate Stirling numbers of the first and second kind, denoted by S1,ρ(n, k) and S2,ρ(n, k), were defined in ([23, 29, 31]) (logρ(1 + t))k k! = ∞∑ n=0 S1,ρ(n, k) tn n! , (eρ(t)− 1)k k! = ∞∑ n=0 S2,ρ(n, k) tn n! , (18) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 691 where logρ(eρ(t)) = eρ(logρ(t)) = t. (19) When ρ → 0, lim ρ→0 S1,ρ(n, k) = S1(n, k), lim ρ→0 S2,ρ(n, k) = S2(n, k) where S1(n, k) and S2(n, k) are the classical Stirling numbers of the first and second kind. Also, the degenerate Bernoulli polynomials of the second kind are defined by (1 + t)x logρ(1 + t) t = ∞∑ n=0 bn,ρ(x) tn n! . (20) The degenerate Stirling numbers of the second kind appeared in the probability distribu- tion of the random variable given as the sum of a finite number of random variables with degenerate zero-truncated Poisson distributions and a random variable with degenerate Poisson distribution, all having the same parameter (see [31]). The polyexponential functions are defined by the following generating functions [22, 23, 28, 30] Eik(x) = ∞∑ n=1 xn nk(n− 1)! , k ∈ Z. (21) For k = 1, Ei1(x) = ex − 1. The modified degenerate polyexponential function are given by ([22, 23, 28, 30]) Eik,ρ(x) = ∞∑ n=1 (1)n,ρx n nk(n− 1)! , ρ ∈ R. (22) Note that Ei1,ρ(x) = ∞∑ n=1 (1)n,ρ xn n! = eρ(x)− 1, ρ ∈ R. (23) Also, we have d dx Eik,ρ(logρ(1 + x)) = (1 + x)ρ−1 logρ(1 + x) Eik−1,ρ(logρ(1 + x)), (24) which implies Eik,ρ(logρ(1 + x)) = ∫ x 0 (1 + t)ρ−1 logρ(1 + t) ∫ y 0 . . . (1 + x)ρ−1 logρ(1 + x) ∫ y 0 (1 + x)ρ−1 logρ(1 + x) xdx . . . dx = ∞∑ m=0 ∑ m1+m2+...+mk−1=m ( m m1, . . . ,mk−1 ) × bm1,ρ(ρ− 1) m1 + 1 bm2,ρ(ρ− 1) m1 +m2 + 1 . . . bmk−1,ρ(ρ− 1) m1 + . . .+mk−1 + 1 xm+1 m! . (25) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 692 The degenerate poly-Euler polynomials were defined in [35] by means of the following generating function 2 Eik,ρ(logρ(1 + t)) teρ(t) + 1 exρ(t) = ∞∑ n=0 E(k) n,ρ(x) tn n! , (26) where k ∈ Z. In this paper, a new variation of poly-Genocchi polynomials is constructed by mixing the concepts of modified degenerate polyexponential function, Apostol-Genocchi polyno- mials and Frobenius polynomials. These polynomials are called the degenerate Apostol- Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b. Some special cases of these polynomials are enumerated and some identities that contain a num- ber of relations of this new variation with some Genocchi-type polynomials are provided. Finally, some connections of these degenerate Apostol-Frobenius-type poly-Genocchi poly- nomials to degenerate Stirling numbers of the first and second kind, higher order degener- ate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials are discussed. 2. Definition and Some Explicit Formulas Analogous to the definition of degenerate poly-Euler polynomials in (26), the desired variation of Apostol-type poly-Genocchi polynomials can be constructed by introducing the parameter u to incorporate the concept of Frobenius polynomials as well as the pa- rameters a and b. The following contains the formal definition of the desired polynomials. Definition 2.1. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b, denoted by Ĝ(k,α) n (x;λ, ρ, u, a, b), are defined as coefficients of the following generating function: ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α exρ(t), (27) where |t| < √ (ln(λ u))2+4π2 | ln a+ln b| . When α = 1, (27) yields ∞∑ n=0 Ĝ(k) n (x;λ, ρ, u, a, b) tn n! = Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t exρ(t). (28) where Ĝ(k) n (x;λ, ρ, u, a, b) = Ĝ(k,1) n (x;λ, ρ, u, a, b) denotes the degenerate Apostol-Frobenius- type poly-Genocchi polynomials with parameters a and b. Now, if x = (1− u)t ln ab, then (25) yields Eik,ρ(logρ(1 + (1− u)t ln ab)) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 693 = t ∞∑ m=0 ((1− u) ln ab)m+1 ∑ m1+m2+...+mk−1=m ( m m1, . . . ,mk−1 ) × bm1,ρ(ρ− 1) m1 + 1 bm2,ρ(ρ− 1) m1 +m2 + 1 . . . bmk−1,ρ(ρ− 1) m1 + . . .+mk−1 + 1 tm m! . Also, exρ(t) λbt − ua−t = ∞∑ m=0 m∑ j=0 ∞∑ n=0 ( m j ) (x)j,ρ (u λ )n (−n log ab)m−j t m m! . With Bm1,m2,...,mk−1 (m, ρ− 1) = ((1− u) ln ab)m+1 ∑ m1+m2+...+mk−1=m ( m m1, . . . ,mk−1 ) × bm1,ρ(ρ− 1) m1 + 1 bm2,ρ(ρ− 1) m1 +m2 + 1 . . . bmk−1,ρ(ρ− 1) m1 + . . .+mk−1 + 1 , (29) we have ∞∑ m=0 Ĝ(k) m (x;λ, ρ, u, a, b) tm m! = Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t exρ(t) = t ∞∑ m=0 m∑ i=0 m−i∑ j=0 ∞∑ n=0 Bm1,m2,...,mk−1 (i, ρ− 1) ( m− i j )( m i ) (x)j,ρ (u λ )n (−n log ab)m−i−j t m m! . Note that the right-hand side of the preceding equation has no constant term. Hence, when m = 0, Ĝ(k) 0 (x;λ, ρ, u, a, b) = 0. Moreover, ∞∑ m=0 1 m Ĝ(k) m (x;λ, ρ, u, a, b) tm−1 (m− 1)! = ∞∑ m=0 m∑ i=0 m−i∑ j=0 ∞∑ n=0 Bm1,m2,...,mk−1 (i, ρ− 1) ( m− i j )( m i ) (x)j,ρ (u λ )n (−n log ab)m−i−j t m m! . By comparing the coefficients of tm m! , we obtain the following explicit formula: 1 m+ 1 Ĝ(k) m+1(x;λ, ρ, u, a, b) = m∑ i=0 m−i∑ j=0 ∞∑ n=0 Bm1,m2,...,mk−1 (i, ρ− 1) ( m− i j )( m i ) (x)j,ρ (u λ )n (−n log ab)m−i−j . R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 694 Furthermore, using the arithmetic-geometric series formula in [12, p.245], we can write ∞∑ n=0 (u λ )n nm−i−j = Am−i−j ( u λ )( 1− u λ )m−i−j+1 , where An(u) is the Eulerian polynomial An(u) = n∑ k=0 A(n, k)uk (30) with A(n, k), the Eulerian number, satisfying A(n, k) = (n− k + 1)A(n− 1, k − 1) + kA(n− 1, k). Thus, 1 m+ 1 Ĝ(k) m+1(x;λ, ρ, u, a, b) = m∑ i=0 m−i∑ j=0 Bm1,m2,...,mk−1 (i, ρ− 1)Am−i−j (u λ )(m− i j )( m i ) (x)j,ρ(− log ab)m−i−j( 1− u λ )m−i−j+1 . To state formally this result, we have the following theorem. Theorem 2.2. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials with pa- rameters a and b are equal to 1 m+ 1 Ĝ(k) m+1(x;λ, ρ, u, a, b) = m∑ i=0 m−i∑ j=0 Bm1,m2,...,mk−1 (i, ρ− 1)Am−i−j (u λ )(m− i j )( m i ) (x)j,ρ(− log ab)m−i−j( 1− u λ )m−i−j+1 , where Ĝ(k,α) 0 (x;λ, ρ, u, a, b) = 0, An(u) is the Eulerian polynomial and Bm1,m2,...,mk−1 (i, ρ− 1) satisfies (29). Now, let us extend the explicit formula to higher order degenerate Apostol-Frobenius- type poly-Genocchi polynomials with parameters a and b. First, we have to find the expansion of the following function: Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t = t ∞∑ m=0  m∑ j=0 ∞∑ n=0 1 m! ( m j ) Bm1,m2,...,mk−1 (j, ρ− 1) (u λ )n (−n log ab)m−j  tm. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 695 Raising this to power α gives( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α = tα ∞∑ m=0 ∑ k1+k2+...+kα=m α∏ i=1  ki∑ j=0 ∞∑ n=0 1 ki! ( ki j ) Bm1,m2,...,mk−1 (j, ρ− 1) (u λ )n (−n log ab)ki−j  tm. Hence, ∞∑ m=0 Ĝ(k,α) m (x;λ, ρ, u, a, b) tm m! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α exρ(t) = tα ∞∑ m=0 m∑ q=0 ( m q ) (x)q,ρ(m− q)! × ∑ k1+k2+...+kα=m−q α∏ i=1  ki∑ j=0 ∞∑ n=0 1 ki! ( ki j ) Bm1,m2,...,mk−1 (j, ρ− 1) (u λ )n (−n log ab)ki−j  tm m! . Note that, when 0 ≤ m ≤ α − 1, Ĝ(k,α) m (x;λ, ρ, u, a, b) = 0. Now, we can further rewrite the preceding equation as follows: ∞∑ m=−α Ĝ(k,α) m+α(x;λ, ρ, u, a, b) tm (m+ α)! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α exρ(t) = ∞∑ m=0 m∑ q=0 ( m q ) (x)q,ρ(m− q)! × ∑ k1+k2+...+kα=m−q α∏ i=1  ki∑ j=0 ∞∑ n=0 1 ki! ( ki j ) Bm1,m2,...,mk−1 (j, ρ− 1) (u λ )n (−n log ab)ki−j  tm m! . Comparing the coefficients of tm m! and using the arithmetic-geometric formula yield the following explicit formula. Theorem 2.3. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b are equal to 1 (m+ α)α Ĝ(k,α) m+α(x;λ, ρ, u, a, b) = m∑ q=0 ( m q ) (x)q,ρ(m− q)! ∑ k1+k2+...+kα=m−q α∏ i=1 ki∑ j=0 1 ki! ( ki j ) × Bm1,m2,...,mk−1 (j, ρ− 1)Aki−j (u λ ) (− log ab)ki−j( 1− u λ )ki−j+1 . where Ĝ(k,α) m (x;λ, ρ, u, a, b) = 0 for m = 0, 1, . . . , α − 1, An(u) is the Eulerian polynomial defined in (30) and Bm1,m2,...,mk−1 (i, ρ− 1) satisfies (29). R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 696 Remark 2.4. It can easily be seen that, when α = 1, the explicit formula in Theorem 2.3 reduces to that in Theorem 2.2. 3. Relation With Some Genocchi-type Polynomials By giving special values to the parameters involved, Ĝ(k,α) n (x;λ, ρ, u, a, b) reduces to some interesting Genocchi-type polynomials. (i) Using (23), when k = 1, (27) yields ∞∑ n=0 Ĝ(α) n (x;λ, ρ, u, a, b) tn n! = ( (1− u)t ln ab λbt − ua−t )α exρ(t), (31) where the polynomials Ĝ(α) n (x;λ, ρ, u, a, b) = Ĝ(1,α) n (x;λ, ρ, u, a, b) are called the de- generate Apostol-Frobenius-type Genocchi polynomials of higher order with param- eters a, b and c. When α = 1, (31) yields ∞∑ n=0 Ĝ(1) n (x;λ, ρ, u, a, b) tn n! = (1− u)t ln ab λbt − ua−t exρ(t), (32) where the polynomials Ĝ(1) n (x;λ, ρ, u, a, b) are called the degenerate Apostol-Frobenius- type Genocchi polynomials with parameters a and b. (ii) When x = 0, equation (27) reduces to ∞∑ n=0 Ĝ(k,α) n (λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α , (33) the degenerate Apostol-Frobenius-type poly-Genocchi numbers with parameters a and b. (iii) When a = 1, b = e, (27) will reduce to ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u) tn n! = ( Eik,ρ(logρ(1 + (1− u)t)) λet − u )α exρ(t), (34) and call Ĝ(k,α) n (x;λ, ρ, u), the degenerate Apostol-Frobenius-type poly-Genocchi poly- nomials of higher order. When x = 0, we get ∞∑ n=0 Ĝ(k,α) n (λ, ρ, u) tn n! = ( Eik,ρ(logρ(1 + (1− u)t) λet − u )α , (35) the degenerate Apostol-Frobenius-type Genocchi numbers of higher order. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 697 (iv) When ρ → 0, equation (27) reduces to ∞∑ n=0 Ĝ(k,α) n (x;λ, 0, u, a, b) tn n! = ( Eik,0(log0(1 + (1− u)t ln ab)) λbt − ua−t )α ex0(t) ∞∑ n=0 Ĝ(k,α) n (x;λ, u, a, b) tn n! = ( Eik(log(1 + (1− u)t ln ab)) λbt − ua−t )α ext, (36) where the polynomials G(k,α) n (x;λ, u, a, b) are the type 2 Apostol-Frobenius-type poly- Genocchi polynomials of higher order with parameters a, b and c with c = e in [17]. (v) When λ = 1, (34) gives ∞∑ n=0 Ĝ(k,α) n (x;u, 1, e) tn n! = ( Eik,ρ(logρ(1 + (1− u)t)) et − u )α ext. (37) which is the higher order version of equation (8) and are called the higher order poly-Genocchi polynomials. We may use Ĝ(k,α) n (x;u) to denote Ĝ(k,α) n (x;u, 1, e). (vi) When k = 1, a = 1 and b = e, (36) gives ∞∑ n=0 Ĝ(1,α) n (x;λ, u) tn n! = ( (1− u)t λet − u )α ext, (38) and when λ = 1, (38) gives ∞∑ n=0 Ĝ(1,α) n (x; 1, u) tn n! = ( (1− u)t et − u )α ext, where Ĝ(1,α) n (x;λ, u) = Ĝ(α) n (x;λ, u) and Ĝ(1,α) n (x; 1, u) = Ĝ(α) n (x;u) are called the degenerate Apostol-Frobenius-type Genocchi polynomials and Frobenius-Genocchi polynomials of higher order in (4) and (2), respectively. Furthermore, when α = 1, we have ∞∑ n=0 Ĝn(x;λ, u) tn n! = (1− u)t λet − u ext, (39) and ∞∑ n=0 Ĝn(x;u) tn n! = (1− u)t et − u ext, where Ĝn(x;λ, u) and Ĝn(x;u) are called the degenerate Apostol-Frobenius-type Genocchi polynomials and Frobenius-Genocchi polynomials in (4) and (2), respec- tively. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 698 Now, let us consider some some relations of Ĝ(k,α) n (x;λ, ρ, u, a, b) with other Genocchi- type polynomials. First is to establish a kind of addition formula for Ĝ(k,α) n (x;λ, ρ, u, a, b) expressing them as polynomials in x. Theorem 3.1. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b satisfy the relation Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ m=0 ( n m ) Ĝ(k,α) n−m(λ, ρ, u, a, b)(x)m,ρ. (40) Moreover, the expression of Ĝ(k,α) n (x;λ, ρ, u, a, b) as polynomial in x is given by Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ j=0 Ĝ(k,α) n,j,w̃(λ, ρ, u, a, b)x j , (41) where Ĝ(k,α) n,j,w̃(λ, ρ, u, a, b) = n∑ m=j ( n m ) Ĝ(k,α) n−m(λ, ρ, u, a, b)w̃ρ(m, j), Proof. Using (33), we can write (27) as follows: ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α exρ(t) = ( ∞∑ n=0 Ĝ(k,α) n (λ, ρ, u, a, b) tn n! )( ∞∑ n=0 (x)n,ρ tn n! ) = ∞∑ n=0 ( n∑ m=0 ( n m ) Ĝ(k,α) n−m(λ, ρ, u, a, b)(x)m,ρ ) tn n! Comparing the coefficients of tn n! yields (40). To prove (41), we first recall that the r- Whitney numbers of the first kind, denoted by wm,r(n, k) were defined by Mező [38] by means of the following horizontal generating function: mn(x)n = n∑ j=0 wm,r(n, j)(mx+ r)j , (42) where (x)n = x(x− 1)(x− 2) . . . (x−n+1). Replacing x with x/m and letting r = 0 yield (x)n,m = n∑ j=0 w̃m(n, j)xj , R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 699 where w̃m(n, j) = wm,0(n, j), a certain of generalization of Stirling numbers of the first kind, i.e. S1(n, j) = w̃0(n, j). Using (42), equation (40) can further be written as polyno- mial in x Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ j=0  n∑ m=j ( n m ) Ĝ(k,α) n−m(λ, ρ, u, a, b)w̃ρ(m, j) xj , with coefficients Ĝ(k,α) n,j,w̃(λ, ρ, u, a, b) = n∑ m=j ( n m ) Ĝ(k,α) n−m(λ, ρ, u, a, b)w̃ρ(m, j), the convolution of Ĝ(k,α) n (λ, ρ, u, a, b) and w̃ρ(n, k). The relation in (41) is useful in constructing the orthogonal version of Ĝ(k,α) n (x;λ, ρ, u, a, b) using Gram-Schmidt process. It can easily be verified from Theorem 2.3 that Ĝ(k,α) m (x;λ, ρ, u, a, b) = 0,m = 0, 1, . . . , α− 1, Ĝ(k,α) α (x;λ, ρ, u, a, b) = 1. Let ϕα(x), ϕα+1(x), . . . , ϕs(x) be the orthogonal version of of poly-Genocchi polynomials obtained from Gram-Schmidt process in which the polynomial is orthogonal with respect to the inner product < f, g >= ∫ 1 0 w(x)f(x)g(x)dx. Then ϕs+1 = Ĝ(k,α) s+1 (x;λ, ρ, u, a, b)− s∑ i=1 λiϕi(x) (43) satisfies < ϕs+1, ϕj >= ∫ 1 0 w(x)ϕs+1(x)ϕjdx = 0, j = 0, 1, . . . , s with λj = < Ĝ(k,α) s+1 (x;λ, ρ, u, a, b), ϕj > < ϕj , ϕj > . Clearly, ϕi(x) = 0 when 0 ≤ i ≤ α− 1 and ϕα(x) = 1. Then, ϕα+1(x) = Ĝ(k,α) α+1 (x;λ, ρ, u, a, b)− α∑ i=1 λiϕi(x) = α+1∑ j=0 Ĝ(k,α) α+1,j,w̃(λ, ρ, u, a, b)x j − ∫ 1 0 ∑α+1 j=0 Ĝ(k,α) α+1,j,w̃(λ, ρ, u, a, b)x jϕα(x)dx∫ 1 0 (ϕα(x))2dx ϕα(x) = α+1∑ j=0 Ĝ(k,α) α+1,j,w̃(λ, ρ, u, a, b) ( xj − 1 j + 1 ) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 700 = Ĝ(k,α) α+1,0,w̃(λ, ρ, u, a, b) ( x− 1 2 ) + Ĝ(k,α) α+1,1,w̃(λ, ρ, u, a, b) ( x2 − 1 3 ) = 2 ( x− 1 2 ) . ϕ3(x) = Ĝ(k,α) 3 (x;λ, ρ, u, a, b)− 2∑ i=1 λiϕi(x) = 3∑ j=0 Ĝ(k,α) 3,j,w̃ (λ, ρ, u, a, b)x j − ∫ 1 0 ∑3 j=0 Ĝ (k,α) 3,j,w̃ (λ, ρ, u, a, b)x jϕ2(x)dx∫ 1 0 (ϕ2(x))2dx ϕ2(x) − ∫ 1 0 ∑3 j=0 Ĝ (k,α) 3,j,w̃ (λ, ρ, u, a, b)x jϕ1(x)dx∫ 1 0 (ϕ1(x))2dx ϕ1(x) = 3∑ j=0 Ĝ(k,α) 3,j,w̃ (λ, ρ, u, a, b) ( xj − 3j(2x− 1) (j + 2)(j + 1) − 1 j + 1 ) = Ĝ(k,α) 3,1,w̃(λ, ρ, u, a, b) ( x− 2x− 1 2 − 1 2 ) + Ĝ(k,α) 3,2,w̃(λ, ρ, u, a, b) ( x2 − 2x− 1 2 − 1 3 ) = Ĝ(k,α) 3,2,w̃(λ, ρ, u, a, b) ( x2 − x+ 1 6 ) = 3x2 − 3x+ 1 2 ϕ4(x) = Ĝ(k,α) 4 (x;λ, ρ, u, a, b)− 3∑ i=1 λiϕi(x) = 4∑ j=0 Ĝ(k,α) 4,j,w̃ (λ, ρ, u, a, b)x j − ∫ 1 0 ∑4 j=0 Ĝ (k,α) 4,j,w̃ (λ, ρ, u, a, b)x jϕ3(x)dx∫ 1 0 (ϕ3(x))2dx ϕ3(x) − ∫ 2 0 ∑4 j=0 Ĝ (k,α) 4,j,w̃ (λ, ρ, u, a, b)x jϕ2(x)dx∫ 1 0 (ϕ2(x))2dx ϕ2(x)− ∫ 2 0 ∑4 j=0 Ĝ (k,α) 4,j,w̃ (λ, ρ, u, a, b)x jϕ1(x)dx∫ 1 0 (ϕ1(x))2dx ϕ1(x) = 4∑ j=0 Ĝ(k,α) 4,j,w̃ (λ, ρ, u, a, b) ( xj − 9(20)j(j − 1) ( x2 − x+ 1 6 ) 6(j + 3)(j + 2)(j + 1) − 3j(2x− 1) (j + 2)(j + 1) − 1 j + 1 ) = Ĝ(k,α) 4,3,w̃(λ, ρ, u, a, b) ( x3 − 3 2 x2 + 3 5 x− 1 20 ) = 4x3 − 6x2 + 12 5 x− 1 5 The next identity gives the relation between Ĝ(k,α) n (x;λ, ρ, u, a, b) and Ĝ(k,α) n (x;λ, ρ, u). R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 701 Theorem 3.2. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b satisfy the relation, Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ m=0 ( n m ) (ln a)m(ln ab)n−mĜ(k,α) n−m ( x ln ab ;λ, ρ, u ) . (44) Proof. Using (34), we can rewrite (27) as follows ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) a−t(λ(ab)t − u) )α ex ln ab ρ ln ab(t ln ab) = aαt ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λet ln ab − u )α ex ln ab ρ ln ab(t ln ab) = ( ∞∑ n=0 (ln ab)nĜ(k,α) n ( x ln ab ;λ, ρ, u ) tn n! )( ∞∑ n=0 (t ln a)n n! ) = ∞∑ n=0 n∑ m=0 ( n m ) (ln a)m(ln ab)n−mĜ(k,α) n−m ( x ln ab ;λ, ρ, u ) tn n! Comparing the coefficients of tn n! , we obtain the desired result. The next result is another form of addition formula for Ĝ(k,α) n (x;λ, ρ, u, a, b). Theorem 3.3. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a and b satisfy the relation Ĝ(k,α) n (x+ y;λ, ρ, u, a, b) = n∑ m=0 ( n m ) Ĝ(k,α) n−m(x;λ, ρ, u, a, b)(y)m,ρ. Proof. We can write (27) as follows: ∞∑ n=0 Ĝ(k,α) n (x+ y;λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α ex+y ρ (t) = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α exρ(t)e y ρ(t) = ( ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! )( ∞∑ n=0 (y)n,ρ tn n! ) = ∞∑ n=0 ( n∑ m=0 ( n m ) Ĝ(k,α) n−m(x;λ, ρ, u, a, b)(y)m,ρ ) tn n! . Comparing the coefficients of tn n! completes the proof of the theorem. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 702 4. Differential and Integral Formulas In the following theorem, certain differential equation will be established containing the degenerate Apostol-type poly-Genocchi polynomials of higher order with parameters a and b. Here, we consider Ĝ(k,α) n (x;λ, ρ, u, a, b) as polynomial in x. Theorem 4.1. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials with pa- rameters a and b satisfy the relation, d dx Ĝ(k,α) n+1 (x;λ, ρ, u, a, b) = n∑ j=0 ( n j ) (−1)n−j n− j + 1 ρn−jĜ(k,α) j (x;λ, ρ, u, a, b). (45) Proof. Applying the first derivative to equation (27) with respect to x and using (15) yield ∞∑ n=0 d dx Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( Eik,ρ(logρ(1 + (1− u)t ln ab)) (λbt − ua−t) )α exρ(t) log(1 + ρt)1/ρ = t ( ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! )( ∞∑ n=0 (−1)n 1 n+ 1 (ρt)n n! ) , ∞∑ n=0 d dx Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = t ∞∑ n=0 n∑ j=0 ( n j ) Ĝ(k,α) j (x;λ, ρ, u, a, b) (−1)n−j n− j + 1 ρn−j t n n! . It follows that ∞∑ n=0 1 n+ 1 d dx Ĝ(k,α) n+1 (x;λ, ρ, u, a, b) tn n! = ∞∑ n=0 n∑ j=0 ( n j ) Ĝ(k,α) j (x;λ, ρ, u, a, b) (−1)n−j n− j + 1 ρn−j t n n! . Comparing the coefficients of tn n! yields the desired differential identity. Remark 4.2. When ρ → 0, equation (45) reduces to the following differential identity d dx Ĝ(k,α) n+1 (x;λ, u, a, b) = (n+ 1)Ĝ(k,α) n (x;λ, u, a, b), (46) where Ĝ(k,α) n (x;λ, u, a, b) is the type 2 Apostol-Frobenius-type poly-Genocchi polynomials in (36). Equation (46) was used to classify Ĝ(k,α) n (x;λ, u, a, b) as an Appell polynomial. R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 703 To derive the integral formula GIn = ∫ Ĝ(k,α) n (x;λ, u, a, b)dx, we need to consider the following lemma. Lemma 4.3. If an = n∑ j=0 (−1)n−j ( n j ) 1 n− j + 1 bj , then bn − nan−1 = n∑ j=0 (−1)n−j ( n j ) bj . (47) Proof. Using the fact that( n+ 1 j ) 1 n+ 1 = ( n j ) 1 n− j + 1 , we have −(n+ 1)an = n∑ j=0 (−1)n+1−j ( n+ 1 j ) 1 n− j + 1 bj , which is equivalent to (47). To compute the value of the integral GIn, we can use the following recurrence relation. Theorem 4.4. The integral GIn satisfies the following recurrence relation GIn = − ρn n+ 1  n−1∑ j=0 ( n+ 1 j ) 1 ρj GIj − n+1∑ j=0 ( n+ 1 j ) j ρj−1 Ĝ(k,α) j (x;λ, ρ, u, a, b)  . Proof. Integrating both sides of (45) gives 1 ρn Ĝ(k,α) n+1 (x;λ, ρ, u, a, b) = n∑ j=0 (−1)n−j ( n j ) 1 n− j + 1 1 ρj ∫ Ĝ(k,α) j (x;λ, ρ, u, a, b)dx. Applying Lemma 4.3 yields 1 ρn GIn − n 1 ρn−1 Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ j=0 (−1)n−j ( n j ) 1 ρj GIj . Using the inversion formula an = n∑ j=0 (−1)n−j ( n j ) bj ⇐⇒ bn = n∑ j=0 ( n j ) aj , R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 704 we have 1 ρn GIn = n∑ j=0 ( n j ){ 1 ρj GIj − j 1 ρj−1 Ĝ(k,α) j (x;λ, ρ, u, a, b) } . Thus, n−1∑ j=0 ( n j ) 1 ρj GIj = n∑ j=0 ( n j ) j ρj−1 Ĝ(k,α) j (x;λ, ρ, u, a, b). n 1 ρn−1 GIn−1 = − n−2∑ j=0 ( n j ) 1 ρj GIj + n∑ j=0 ( n j ) j ρj−1 Ĝ(k,α) j (x;λ, ρ, u, a, b). This completes the proof of the theorem. Remark 4.5. With the aid of the explicit formula in Theorem 2.3, one can easily compute the value of GIn recursively. 5. Connections with Some Special Polynomials Here, some connections of the higher order degenerate Apostol-type poly-Genocchi polynomials Ĝ(k,α) n (x;λ, ρ, u, a, b) with other well-known special numbers and polynomials will be established. Carlitz [10, 11] defined the degenerate Bernoulli polynomials, denoted by βn,ρ(x), as follows: t eρ(t)− 1 exρ(t) = ∞∑ n=0 βn,ρ(x) tn n! . (48) One may extend this to higher order degenerate Bernoulli polynomials, which can be defined as follows ( t eρ(t)− 1 )s exρ(t) = ∞∑ n=0 β(s) n,ρ(x) tn n! . (49) The degenerate Frobenius-Euler polynomials of higher order, denoted by H (s) n,ρ(x;µ), are defined in [42] as follows( 1− µ eρ(t)− µ )s exρ(t) = ∞∑ n=0 H(s) n,ρ(x;µ) tn n! . (50) When s = 1, w = 0, (50) gives E (s) n (x;µ, λ), the Apostol-type Frobenius-Euler polynomi- als in [41]. Now, if λ = 0, we can define the Frobenius-Euler polynomials, denoted by E (s) n,H(x;µ), as follows: ( 1− µ et − µ )s ext = ∞∑ n=0 E(s) n (x;µ) tn n! . (51) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 705 The following theorem contains an identity that relates the degenerate Apostol-Frobenius- type poly-Genocchi polynomials of higher order with parameters a, b and c to the degen- erate Stirling numbers of the first kind S1,ρ(n, k) in (18). Here, it is important to note that if (c0, c1, . . . , cj , . . .) is any sequence of numbers and l is a positive integer, then ∞∑ j=0 cj tj j! l = l∏ i=1 ( ∞∑ ni=0 cni ni! tni ) = ∞∑ n=0 { ∑ n1+n2+...+nα=n l∏ i=1 cni ( n n1, n2, . . . , nα )} tn n! . (52) (see [12]). Now, we are ready to introduce the following theorem. Theorem 5.1. The degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a, b and c satisfies the relation, Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ j=0 ( n j ) (ln ab)n−jĜ(α) n−j ( x ln ab ;λ, ρ, u ) dj (53) where dj = ∑ n1+n2+...+nα=j α∏ i=1 cni ( j n1, n2, . . . , nα ) cj = j∑ m=0 (−1)m+j+1 ((1− u) ln ab)j(1)m+1,ρS1,ρ(j + 1,m+ 1) (j + 1)(m+ 1)k−1 . Proof. Now, (27) can be written as ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = exρ(t) (λbt − ua−t)α ( ∞∑ m=1 (1)m,ρ mk−1 (logρ(1 + (1− u)t ln ab))m m! )α = exρ(t) (λbt − ua−t)α ( ∞∑ m=0 (1)m+1,ρ (m+ 1)k−1 (logρ(1 + (1− u)t ln ab))m+1 (m+ 1)! )α = exρ(t) (λbt − ua−t)α  ∞∑ m=0 (1)m+1,ρ (m+ 1)k−1 ∞∑ j=m+1 S1,ρ(j,m+ 1) ((1− u)t ln ab)j j! α = exρ(t) (λbt − ua−t)α  ∞∑ m=0 (1)m+1,ρ (m+ 1)k−1 ∞∑ j=m S1,ρ(j + 1,m+ 1) ((1− u)t ln ab)j+1 (j + 1)! α R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 706 = exρ(t) (λbt − ua−t)α  ∞∑ j=0 j∑ m=0 (1)m+1,ρ (m+ 1)k−1 S1,ρ(j + 1,m+ 1) ((1− u)t ln ab)j+1 (j + 1)! α = exρ(t) ( (1− u)t ln ab λbt − ua−t )α  ∞∑ j=0 cj tj j! α , where cj = j∑ m=0 (−1)m+j+1 ((1− u) ln ab)j(1)m+1,ρS1,ρ(j + 1,m+ 1) (j + 1)(m+ 1)k−1 . Using (31), we get ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( ∞∑ n=0 Ĝ(α) n (x;λ, ρ, u, a, b) tn n! ) ∞∑ j=0 cj tj j! α . Note that, using (52), (∑∞ j=0 cj tj j! )α can be expressed as ∞∑ j=0 cj tj j! α = ∞∑ n=0 dn tn n! , where dn = ∑ n1+n2+...+nα=n α∏ i=1 cni ( n n1, n2, . . . , nα ) . It follows that ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ∞∑ n=0  n∑ j=0 ( n j ) Ĝ(α) n−j(x;λ, ρ, u, a, b)dj  tn n! . Comparing the coefficients and using equation (44) complete the proof of the theorem. Remark 5.2. When α = 1, dj = cj . The identities in the following theorem are derived using the fact that the polynomials Ĝ(k,α) n (x;λ, ρ, u, a, b, ) with parameters a and b satisfy the relation in (27). Theorem 5.3. The degenerate Apostol-type poly-Genocchi polynomials of higher order with parameters a, b satisfy the following explicit formulas: Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ l=0 n−l∑ m=0 ( n l ) S2,ρ(l + s, s) ( n−l m )( l+s s )β(s) m,ρ(x)Ĝ (k,α) n−l−m(λ, ρ, u, a, b), (54) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 707 Ĝ(k,α) n (x;λ, ρ, u, a, b) = n∑ m=0 ( n m ) (1− µ)s s∑ j=0 ( s j ) (−µ)s−jĜ(k,α) n−m(λ, ρ, u, j, a, b)E(s) n (x ln c;µ). (55) Proof. Using (49), (27) may be expressed as ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( (eρ(t)− 1)s s! )( tsexρ(t) (eρ(t)− 1)s )( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α s! ts = ( ∞∑ n=0 S2,ρ(n+ s, s) tn+s (n+ s)! )( ∞∑ m=0 β(s) m,ρ(x) tm m! )( ∞∑ n=0 Ĝ(k,α) n (λ, ρ, u, a, b) tm m! ) s! ts = ( ∞∑ n=0 S2,ρ(n+ s, s) tn+s (n+ s)! )( ∞∑ n=0 n∑ m=0 ( n m ) β(s) m,ρ(x ln c)Ĝ (k,α) n−m(λ, ρ, u, a, b) tn n! ) s! ts = ( ∞∑ n=0 n∑ l=0 S2,ρ(l + s, s) tl+s (l + s)! n−l∑ m=0 ( n− l m ) β(s) m,ρ(x)Ĝ (k,α) n−l−m(λ, ρ, u, a, b) tn−l (n− l)! ) s! ts . This can further be written as ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( ∞∑ l=0 ∞∑ n=l n−l∑ m=0 { l + s s } l!s! (l + s)! ( n− l m ) B(s) m (x ln c)Ĝ(k,α) n−l−m(x;λ, ρ, u, a, b) n! (n− l)!l! tn n! ) = ∞∑ n=0 ( n∑ l=0 n−l∑ m=0 ( n l ) S2,ρ(l + s, s) ( n−l m )( l+s s )β(s) m,ρ(x)Ĝ (k,α) n−l−m(λ, ρ, u, a, b) ) tn n! . Comparing the coefficients of tn n! gives (54). Now, to prove relation (55), (27) may be expressed as ∞∑ n=0 Ĝ(k,α) n (x;λ, ρ, u, a, b) tn n! = ( (1− µ)s (eρ(t)− µ)s exρ(t) )( (eρ(t)− µ)s (1− µ)s )( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α = 1 (1− µ)s ( ∞∑ n=0 H(s) n,ρ(x;µ) tn n! ) s∑ j=0 ( s j ) (−µ)s−j× × ( Eik,ρ(logρ(1 + (1− u)t ln ab)) λbt − ua−t )α (eρ(t)) j ) R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 687-712 708 = 1 (1− µ)s s∑ j=0 ( s j ) (−µ)s−j ( ∞∑ n=0 H(s) n,ρ(x ln c;µ) tn n! ) × × ( ∞∑ n=0 Ĝ(k,α) n (λ, ρ, u, j, a, b) tn n! ) = 1 (1− µ)s s∑ j=0 ( s j ) (−µ)s−j ∞∑ n=0 ( n∑ m=0 ( n m ) Ĝ(k,α) n−m(λ, ρ, u, j, a, b)× ×H(s) n,ρ(x ln c;µ) ) tn n! = ∞∑ n=0  n∑ m=0 ( n m ) (1− µ)s s∑ j=0 ( s j ) (−µ)s−jĜ(k,α) n−m(λ, ρ, u, j, a, b)× ×H(s) n,ρ(x ln c;µ) ) tn n! . Comparing the coefficients of tn n! gives (55). 6. Conclusion and Recommendations In this paper, a certain variation of poly-Genocchi polynomials, called the degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameter a, b and c was constructed using the concept of modified degenerate polylexponential function, Apostol-Genocchi polynomials and Frobenius polynomials. Some interesting properties and identities of these polynomials were explored parallel to those of the poly-Genocchi, poly-Euler and poly-Bernoulli polynomials. The paper was concluded by expressing these degenerate Apostol-Frobenius-type poly-Genocchi polynomials of higher order in terms of degenerate Stirling numbers of the first and second kind, higher order degenerate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials. 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