EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5957 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Gould-Hopper-based bivariate Fubini polynomials Raimah G. Bago1,∗, Normalah S. Abdulcarim1 1 Department of Mathematics, College of Natural Sciences and Mathematics, Mindanao State University Main Campus, 9700 Marawi City, Philippines Abstract. In this paper, we incorporate the bivariate Fubini polynomials with Gould-Hopper polynomials to introduce new polynomials called Gould-Hopper-based bivariate Fubini polynomials by modifying the classical generating function of the bivariate Fubini polynomials. Also, properties such as addition formula, explicit formula, implicit formula, recurrence formula and symmetric identities are obtained. Moreover, some relations between the Gould-Hopper-based bivariate Fubini polynomials and some of the other special polynomials and numbers, such as the 2-variable Gould- Hopper polynomials and the Stirling numbers of the second kind were investigated. 2020 Mathematics Subject Classifications: 05A15, 26A24, 26A36 Key Words and Phrases: Gould-Hopper polynomials, Fubini numbers and polynomials, bivari- ate Fubini polynomials, generating functions, derivatives, integration 1. Introduction In 1975, Tanny introduced the classical Fubini polynomials Fn(y) which are defined in [1] by: Fn(y) = n∑ k=1 k!S(n, k)yk, (1) where S(n, k) is the Stirling numbers of the second kind. These polynomials can be generated by 1 1− y(et − 1) = ∞∑ n=0 Fn(y) tn n! . (2) Note that when setting y = 1, gives Fn(1) = Fn, the classical Fubini number. One calls these numbers the Fubini numbers, ordered Bell numbers, or geometric numbers. In ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5957 Email addresses: bago.rg36@s.msumain.edu.ph (R. G. Bago), normalah.abdulcarim@msumain.edu.ph (N. S. Abdulcarim) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 2 of 25 2017, Kargin defined the bivariate Fubini polynomials Fn(x, y) (see[2]), by the following generating function ∞∑ n=0 Fn(x, y) tn n! = ext 1− y(et − 1) . (3) Another well-researched special polynomial are the Hermite polynomials introduced by French mathematician Charles Hermite in the mid-19th century, which are deemed to be the most beneficial orthogonal special functions during the classical period. The Hermite polynomials Hn(x) defined in [3], are given by e−t2+2tx = ∑ n≥0 Hn(x) tn n! . (4) Later on, it was then generalized by Appell and de Fériet [4] as follows: Hn(x, y) = n! ∞∑ r=0 yrxn−2r r!(n− 2r)! , (5) which are now well-known as the 2-variable Hermite Kamṕe de Fériet polynomials. These polynomials exponentially defined by the following generating function: ∞∑ n=0 Hn(x, y) tn n! = ext+yt2 . (6) Then in 2019, another type of Hermite polynomials involving Fubini polynomials was introduced by Khan et al [5], which was the Hermite-Fubini polynomials. They defined the 3-variable Hermite-Fubini polynomials by means of the following generating function ext+yt2 1− z(et − 1) = ∞∑ n=0 HFn(x, y; z) tn n! . For y = 0 in the above equation, they obtain the bivariate Fubini polynomials. They also investigate some properties of these polynomials and then establish summation formulas and derive symmetric identities. Recently, researchers have studied and investigated another type of special polynomial called Gould-Hopper polynomials. These polynomials sometimes also called higher-order Hermite or Kamṕe de Fériet polynomials. In 1962, Gould and Hopper [6] defined the Gould-Hopper polynomials H (j) n (x, y) by means of the following generating function ext+ytj = ∞∑ n=0 H(j) n (x, y) tn n! . (7) These polynomials are represented by the series: H(j) n (x, y) = n! [ n j ]∑ k=0 xn−jkyk (n− jk)!k! . (8) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 3 of 25 When setting j = 2, H (2) n (x, y) = Hn(x, y), where Hn(x, y) is the 2-variable Hermite Kamṕe de Fériet polynomials. In this paper, the authors introduced the Gould-Hopper-based bivariate Fubini polyno- mials which we define in parallel to the definition of 3-variable Hermite-Fubini polynomials described in [5]. We use j instead of 2 making it a generalization of Hermite-Fubini poly- nomials and investigate some of its properties. 2. Preliminaries Definition 1. [3] The n falling factorial of order k, denoted by (n)k, is defined as (n)k = k∏ i=1 (n− i+ 1) = n! (n− k)! = n(n− 1)...(n− k + 1), if k ≥ 1; (n)0 = 1. Theorem 1. [7](Newton’s Binomial Theorem) For all real numbers r, (1 + x)r = ∞∑ i=0 ( r i ) xi, where ( r i ) = { 1 , if i = 0, r(r−1)...(r−i+1) i! , if i > 0. Remark 1. The following series arises in the Binomial theorem where r is any positive real number, (x+ y)−r = ∞∑ i=0 (−1)i ( r + i− 1 i ) x−r−iyi, (|y| < |x|). Example 1. Consider the expression [ 1− y(et − 1) ]−1 . By applying Remark 1, we have [ 1− y(et − 1) ]−1 = ∞∑ i=0 (−1)i ( 1 + i− 1 i ) (1)−1−i[−y(et − 1)]i = ∞∑ i=0 yi(et − 1)i. R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 4 of 25 Definition 2. [8] Geometric series are series of the form a+ ar + ar2 + · · ·+ arn−1 + · · · = ∞∑ n=1 arn−r (9) in which a and r are fixed real numbers and a ̸= 0. The series can also be written as∑∞ n=0 ar n. The ratio r can be positive or negative. Example 2. For positive integers a and b. Consider the geometric series a−1∑ n=0 ebtn. (10) Note that ebt a−1∑ n=0 ebtn − a−1∑ n=0 ebtn =(ebt) [ (1) + (1)(ebt) + (1)(ebt)2 + · · ·+ (1)(ebt)a−1 ] − [ (1) + (1)(ebt) + (1)(ebt)2 + · · ·+ (1)(ebt)a−1 ] ebt a−1∑ n=0 ebtn − a−1∑ n=0 ebtn = [ ebt + (ebt)2 + (ebt)3 + · · ·+ (ebt)a ] − [ 1 + ebt + (ebt)2 + · · ·+ (ebt)a−1 ] a−1∑ n=0 ebtn ( ebt − 1 ) =(ebt)a − 1. So, a−1∑ n=0 ebtn = (ebt)a − 1 ebt − 1 . (11) Theorem 2. [9] The following formula holds ∞∑ m,n=0 f(m+ n) xm m! yn n! = ∞∑ N=0 f(N) (x+ y)N N ! . (12) Theorem 3. [10] Let A(t) = ∑∞ n=0 an tn n! and B(t) = ∑∞ n=0 bn tn n! be the exponential gen- erating function for the sequence (an) and (bn), respectively. Then A(t)B(t) is given by( ∞∑ n=0 an tn n! )( ∞∑ n=0 bn tn n! ) = ∞∑ n=0 n∑ k=0 ( n k ) akbn−k tn n! . (13) Theorem 4. [3] The Stirling numbers of the second kind S(n, k) have a vertical generating function given by: 1 k! (et − 1)k = n∑ k=0 S(n, k) tn n! . (14) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 5 of 25 Definition 3. [1] The classical Fubini polynomials or geometric polynomials Fn(y) are defined by Fn(y) = n∑ k=0 S(n, k)k!yk, (15) where S(n, k) are the Stirling numbers of the second kind. Setting y = 1, we obtain the nth Fubini number Fn, defined by Fn = n∑ k=0 S(n, k)k!. (16) Theorem 5. [1] The Fubini polynomials satisfy the following generating function ∞∑ n=0 Fn(y) tn n! = 1 1− y(et − 1) . (17) Specializing to the case y = 1 yields ∞∑ n=0 Fn tn n! = 1 2− et . (18) Definition 4. [2] The bivariate Fubini polynomials Fn(x; y) are defined by the fol- lowing generating function ext 1− y(et − 1) = ∞∑ n=0 Fn(x; y) tn n! . (19) Theorem 6. [2] The bivariate Fubini polynomials satisfy the following yFn(x+ 1, y) = (1 + y)Fn(x, y)− xn. (20) Definition 5. [11] The Gould-Hopper polynomials H (j) n (x, y) are defined by the fol- lowing generating function ext+ytj = ∞∑ n=0 H(j) n (x, y) tn n! . (21) Theorem 7. [11] The Gould-Hopper polynomials H (j) n (x, y) satisfy the following generat- ing function H(j) n (x, y) = n! [ n j ]∑ r=0 yrxn−jr r!(n− jr)! . (22) Setting j = 2 gives H (2) n (x, y) = Hn(x, y), where Hn(x, y) is the 2-variable Hermite Kamṕe de Fériet polynomials. R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 6 of 25 3. Gould-Hopper-based bivariate Fubini polynomials Definition 6. Let j ∈ N, j ≥ 2. The Gould-Hopper-based bivariate Fubini poly- nomials, denoted by HF (j) n (x, y; z), are defined by the following generating function ∞∑ n=0 HF (j) n (x, y; z) tn n! = ext+ztj 1− y(et − 1) , (23) where |t| < 2π, if y y+1 = 1, and |t| < ∣∣∣ln( y y+1 )∣∣∣, if y y+1 ̸= 1. Observe that when j = 2, ∞∑ n=0 HF (2) n (x, y; z) tn n! = ext+zt2 1− y(et − 1) = ∞∑ n=0 HFn(x, y; z) tn n! , (24) which are the 3-variable Hermite-Fubini polynomials. Furthermore, when z = 0, ∞∑ n=0 HF (j) n (x, y; 0) tn n! = ext 1− y(et − 1) = ∞∑ n=0 Fn(x, y) tn n! , (25) which are the bivariate Fubini polynomials. When x = z = 0, ∞∑ n=0 HF (j) n (0, y; 0) tn n! = 1 1− y(et − 1) = ∞∑ n=0 Fn(y) tn n! , (26) which are the classical Fubini polynomials. And when x = z = 0, y = 1, HF (j) n (0, 1; 0) = 1 2− et = ∞∑ n=0 Fn tn n! , which is the nth Fubini number. Theorem 8. For n ≥ 0, HF (j) n (x+ u, y; z + v) = n∑ r=0 ( n r ) HF (j) n−r(x, y; z)H (j) r (u, v). (27) Proof. By Definition 6, ∞∑ n=0 HF (j) n (x+ u, y; z + v) tn n! = e(x+u)t+(z+v)tj 1− y(et − 1) = ext+ztj 1− y(et − 1) · eut+vtj . Then again by applying Definition 6 and Definition 5 to the right-hand side of the above equation, we have R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 7 of 25 ∞∑ n=0 HF (j) n (x+ u, y; z + v) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn n! · ∞∑ r=0 H(j) r (u, v) tr r! . (28) Moreover, applying Theorem 3 to the right-hand side of equation (28) we get ∞∑ n=0 HF (j) n (x+ u, y; z + v) tn n! = ∞∑ n=0 ( n∑ r=0 ( n r ) HF (j) n−r(x, y; z)H (j) r (u, v) ) tn n! . (29) Comparing the coefficients of tn n! yield (27). Remark 2. Setting j = 2 in Theorem 8, the following formula involving Hermite-Fubini polynomials holds: HFn(x+ u, y; z + v) = n∑ r=0 ( n r ) HFn−r(x, y; z)Hr(u, v). Setting z = 0 in Theorem 8, a relation between the Gould-Hopper-based bivariate Fubini polynomials, bivariate Fubini polynomials and Gould-Hopper polynomials were established in the following corollary. Corollary 1. For n ≥ 0, the following equation holds: HF (j) n (x+ u, y; v) = n∑ r=0 ( n r ) Fn−r(x, y)H (j) r (u, v). Setting j = 2 and z = 0 in Theorem 8, a relation between the 3-variable-Hermite Fubini polynomials, bivariate Fubini polynomials and 2-variable Hermite polynomials will be established in the following corollary. Corollary 2. For n ≥ 0, the following equation holds: HFn(x+ u, y; v) = n∑ r=0 ( n r ) Fn−r(x, y)Hr(u, v). Theorem 9. For n ≥ 0, HF (j) n (x+ u, y; z) = n∑ m=0 ( n m ) xn−m HF (j) m (u, y; z). Proof. By Definition 6, ∞∑ n=0 HF (j) n (x+ u, y; z) tn n! = eut+ztj 1− y(et − 1) · ext. R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 8 of 25 Then, again applying Definition 6 to the right-hand side of the above equation and ex- pressing ext in its series form, we have ∞∑ n=0 HF (j) n (x+ u, y; z) tn n! = ∞∑ m=0 HF (j) m (u, y; z) tm m! · ∞∑ n=0 xn tn n! . (30) Moreover, applying Theorem 3 to equation (30) we get ∞∑ n=0 HF (j) n (x+ u, y; z) tn n! = ∞∑ n=0 ( n∑ m=0 ( n m ) xn−m HF (j) m (u, y; z) ) tn n! . Comparing the coefficients of tn n! yield to the desired result. Setting j = 2 in Theorem 9, a formula involving 3-variable-Hermite Fubini polynomials will be established in the following corollary. Corollary 3. For n ≥ 0, the following equation holds: HFn(x+ u, y; z) = n∑ m=0 ( n m ) xn−m HFm(u, y; z). Setting z = 0 in Theorem 9, a formula involving bivariate Fubini polynomials will be established in the following corollary. Corollary 4. For n ≥ 0, the following equation holds: Fn(x+ u, y) = n∑ m=0 ( n m ) xn−mFm(u, y). Theorem 10. For n ≥ 0, the following formula for Gould-Hopper-based bivariate Fubini polynomials holds: yHF (j) n (x+ 1, y; z) = (1 + y)HF (j) n (x, y; z)−H(j) n (x, z). (31) Proof. Using Definition 6, ∞∑ n=0 yHF (j) n (x+ 1, y; z) tn n! = ye(x+1)t+ztj 1− y(et − 1) = ext+ztj − ext+ztj + yext+ztj − yext+ztj + ye(x+1)t+ztj 1− y(et − 1) = ext+ztj + yext+ztj − ext+ztj (1− yet + y) 1− y(et − 1) = ext+ztj 1− y(et − 1) + yext+ztj 1− y(et − 1) − ext+ztj . R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 9 of 25 Then, by using Definition 6 and Definition 5 to the right-hand side of the above equation we have ∞∑ n=0 yHF (j) n (x+ 1, y; z) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn n! + ∞∑ n=0 yHF (j) n (x, y; z) tn n! − ∞∑ n=0 H(j) n (x, z) tn n! = (1 + y) ∞∑ n=0 HF (j) n (x, y; z) tn n! − ∞∑ n=0 H(j) n (x, z) tn n! = ∞∑ n=0 [ (1 + y)HF (j) n (x, y; z)−H(j) n (x, z) ] tn n! . Thus, comparing the coefficients of tn n! in the above equation we get (31). When we set j = 2 in Theorem 10, a relationship between 3-variable Hermite-Fubini polynomials and 2-variable Hermite polynomials will be established in the following corol- lary. Corollary 5. For n ≥ 0, the following equation holds: yHFn(x+ 1, y; z) = (1 + y)HFn(x, y; z)−Hn(x, z). Remark 3. Setting z = 0, Theorem 10 reduces to Theorem 6. Theorem 11. For n ≥ 0, the following formula for Gould-Hopper-based bivariate Fubini polynomials holds: H(j) n (x, z) = HF (j) n (x, y; z)− yHF (j) n (x+ 1, y; z) + yHF (j) n (x, y; z). Proof. Note that ext+ztj = 1− y(et − 1) 1− y(et − 1) · ext+ztj = ext+ztj − yet(ext+ztj ) + y(ext+ztj ) 1− y(et − 1) = ext+ztj 1− y(et − 1) − y e(x+1)t+ztj 1− y(et − 1) + y ext+ztj 1− y(et − 1) . Then, applying Definition 5 to the left-hand side and Definition 6 to the right-hand side of the above equation we have ∞∑ n=0 H(j) n (x, z) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn n! − y ∞∑ n=0 HF (j) n (x+ 1, y; z) tn n! + y ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 [ HF (j) n (x, y; z)− yHF (j) n (x+ 1, y; z) + yHF (j) n (x, y; z) ] tn n! . Therefore, comparing the coefficients of tn n! yields to the desired result. R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 10 of 25 Theorem 12. For n ≥ 0 and y1 ̸= y2, the following formula for Gould-Hopper-based bivariate Fubini polynomials holds: n∑ k=0 ( n k ) HF (j) n−k(x1, y1; z1)HF (j) k (x2, y2; z2) = y2HF (j) n (x1 + x2, y2; z1 + z2)− y1HF (j) n (x1 + x2, y1; z1 + z2) y2 − y1 . Proof. Using Definition 6 ∞∑ n=0 HF (j) n (x1, y1; z1) tn n! ∞∑ k=0 HF (j) k (x2, y2; z2) tk k! = ex1t+z1tj 1− y1(et − 1) · ex2t+z2tj 1− y2(et − 1) . Then, applying Theorem 3 to the left-hand side of the above equation we get ∞∑ n=0 ( n∑ k=0 ( n k ) HF (j) n−k(x1, y1; z1)HF (j) k (x2, y2; z2) ) tn n! = ex1t+z1tj 1− y1(et − 1) · ex2t+z2tj 1− y2(et − 1) . (32) Note that the right-hand side of equation (32) can be expressed as ex1t+z1tj 1− y1(et − 1) · ex2t+z2tj 1− y2(et − 1) = e(x1+x2)t+(z1+z2)tj (1− y1(et − 1))(1− y2(et − 1)) · y2 − y1 y2 − y1 = y2e (x1+x2)t+(z1+z2)tj − y1e (x1+x2)t+(z1+z2)tj (1− y2(et − 1))(1− y1(et − 1))(y2 − y1) = ( y2e (x1+x2)t+(z1+z2)tj 1− y2(et − 1) − y1e (x1+x2)t+(z1+z2)tj 1− y1(et − 1) )( 1 y2 − y1 ) . (33) Thus, applying Definition 6 to the right-hand side of (33) we have( ∞∑ n=0 y2HF (j) n (x1 + x2, y2; z1 + z2) tn n! − ∞∑ n=0 y1HF (j) n (x1 + x2, y1; z1 + z2) tn n! )( 1 y2 − y1 ) = ∞∑ n=0 ( y2HF (j) n (x1 + x2, y2; z1 + z2)− y1HF (j) n (x1 + x2, y1; z1 + z2) y2 − y1 ) tn n! . So that, ex1t+z1tj 1− y1(et − 1) · ex2t+z2tj 1− y2(et − 1) = ∞∑ n=0 ( y2HF (j) n (x1 + x2, y2; z1 + z2)− y1HF (j) n (x1 + x2, y1; z1 + z2) y2 − y1 ) tn n! . (34) Moreover, equating the left-hand side of (32) and the right-hand side of (34) we get ∞∑ n=0 ( n∑ k=0 ( n k ) HF (j) n−k(x1, y1; z1)HF (j) k (x2, y2; z2) ) tn n! R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 11 of 25 = ∞∑ n=0 ( y2HF (j) n (x1 + x2, y2; z1 + z2)− y1HF (j) n (x1 + x2, y1; z1 + z2) y2 − y1 ) tn n! . Furthermore, comparing the coefficients of tn n! yields the desired result. In the next theorems, we derive some explicit formulae for Gould-Hopper-based bivari- ate Fubini polynomials. Theorem 13. For n ≥ 0, the following formula for Gould-Hopper-based bivariate Fubini polynomials holds: HF (j) n (x, y; z) = n∑ m=0 ( n m ) Fn−m(y)H(j) m (x, z). (35) Proof. By Definition 6, ∞∑ n=0 HF (j) n (x, y; z) tn n! = ext+ztj 1− y(et − 1) = 1 1− y(et − 1) ext+ztj . (36) Then by applying Theorem 5 and Definition 5 to the right-hand side of the equation (36), we get ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 Fn(y) tn n! ∞∑ m=0 H(j) m (x, z) tm m! . Moreover, applying Theorem 3 to the above equation we have ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 ( n∑ m=0 ( n m ) Fn−m(y)H(j) m (x, z) ) tn n! . Finally, comparing the coefficients of tn n! yields to the desired result. Remark 4. When j = 2 in Theorem 13, the following formula involving Hermite-Fubini polynomials holds: HFn(x, y; z) = n∑ m=0 ( n m ) Fn−m(y)Hm(x, z). Theorem 14. For n ≥ 0 and p, q ∈ R, the following summation formula for Gould- Hopper-based bivariate Fubini polynomials holds: HF (j) n (px, y; qz) = n∑ k=0 [ k j ]∑ r=0 (n)k r!(k − jr)! HF (j) n−k(x, y; z)(p− 1)k−jr(q − 1)rzrxk−jr. (37) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 12 of 25 Proof. By applying Definition 6 ∞∑ n=0 HF (j) n (px, y; qz) tn n! = epxt+qztj 1− y(et − 1) = ext+ztj 1− y(et − 1) · e(p−1)xt+(q−1)ztj . Then, applying Definition 6 again and Definition 5 to the right-hand side of the above equation we have ∞∑ n=0 HF (j) n (px, y; qz) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn n! · ∞∑ n=0 H(j) n ((p− 1)x, (q − 1)z) tn n! . (38) Note that, by applying Theorem 7 ∞∑ n=0 H(j) n ((p− 1)x, (q − 1)z) tn n! = ∞∑ n=0 n! [ n j ]∑ r=0 ((q − 1)z)r((p− 1)x)n−jr r!(n− jr)! tn n! = ∞∑ n=0 [ n j ]∑ r=0 n!((q − 1)z)r((p− 1)x)n−jr r!(n− jr)! tn n! . Substituting this to equation (38), gives ∞∑ n=0 HF (j) n (px, y; qz) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn n! · ∞∑ n=0 [ n j ]∑ r=0 n!((q − 1)z)r((p− 1)x)n−jr r!(n− jr)! tn n! . Thus, by applying Theorem 3 to the right-hand side of the above equation we get ∞∑ n=0 HF (j) n (px, y; qz) tn n! = ∞∑ n=0  n∑ k=0 [ k j ]∑ r=0 ( n k ) HF (j) n−k(x, y; z) k!((q − 1)z)r((p− 1)x)k−jr r!(k − jr)!  tn n! = ∞∑ n=0  n∑ k=0 [ k j ]∑ r=0 n! k!(n− k)! HF (j) n−k(x, y; z) k!((q − 1)z)r((p− 1)x)k−jr r!(k − jr)!  tn n! = ∞∑ n=0  n∑ k=0 [ k j ]∑ r=0 n! (n− k)! HF (j) n−k(x, y; z) ((q − 1)z)r((p− 1)x)k−jr r!(k − jr)!  tn n! = ∞∑ n=0  n∑ k=0 [ k j ]∑ r=0 (n)kHF (j) n−k(x, y; z) ((q − 1)z)r((p− 1)x)k−jr r!(k − jr)!  tn n! R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 13 of 25 = ∞∑ n=0  n∑ k=0 [ k j ]∑ r=0 (n)k r!(k − jr)! HF (j) n−k(x, y; z)(p− 1)k−jr(q − 1)rzrxk−jr  tn n! . Comparing the coefficients of tn n! yields (37). Setting p = 1 and q = 1 in Theorem 14, the following corollary involving Gould- Hopper-based bivariate Fubini polynomials holds. Corollary 6. For n ≥ 0, the following equation holds: HF (j) n (x, y; z) = n∑ k=0 [ k j ]∑ r=0 (n)k r!(k − jr)! HF (j) n−k(x, y; z)z rxk−jr. In the subsequent theorems, we give the symmetric identities for Gould-Hopper-based bivariate Fubini polynomials. Theorem 15. For integers a, b and n ≥ 0, the following symmetric identity for Gould- Hopper-based bivariate Fubini polynomials holds: n∑ r=0 ( n r ) bran−r HF (j) n−r(bx, y; b jz)HF (j) r (ax, y; ajz) = n∑ r=0 ( n r ) arbn−r HF (j) n−r(ax, y; a jz)HF (j) r (bx, y; bjz). (39) Proof. Consider the function A(t) = (eabxt+ajbjztj )2 (1− y(eat − 1))(1− y(ebt − 1)) = ebx(at)+bjz(at)j (1− y(eat − 1)) · eax(bt)+ajz(bt)j (1− y(ebt − 1)) . (40) Then, by applying Definition 6 to equation (40) we have A(t) = ∞∑ n=0 HF (j) n (bx, y; bjz) (at)n n! ∞∑ n=0 HF (j) n (ax, y; ajz) (bt)n n! = ∞∑ n=0 anHF (j) n (bx, y; bjz) tn n! ∞∑ n=0 bnHF (j) n (ax, y; ajz) tn n! . Thus, applying the Theorem 3 to the equation above we get A(t) = ∞∑ n=0 ( n∑ r=0 ( n r ) bran−r HF (j) n−r(bx, y; b jz) · HF (j) r (ax, y; ajz) ) tn n! . (41) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 14 of 25 Note that, A(t) can also be expressed as follows A(t) = eax(bt)+ajz(bt)j (1− y(ebt − 1)) · ebx(at)+bjz(at)j (1− y(eat − 1)) . (42) Applying Definition 6 to equation (42), gives A(t) = ∞∑ n=0 bnHF (j) n (ax, y; ajz) tn n! ∞∑ n=0 anHF (j) n (bx, y; bjz) tn n! . We then apply Theorem 3 to the equation above and obtain A(t) = ∞∑ n=0 ( n∑ r=0 ( n r ) bn−r HF (j) n−r(ax, y; a jz) · arHF (j) r (bx, y; bjz) ) tn n! = ∞∑ n=0 ( n∑ r=0 ( n r ) arbn−r HF (j) n−r(ax, y; a jz) · HF (j) r (bx, y; bjz) ) tn n! . (43) Furthermore, equating equations (41) and (43) yields ∞∑ n=0 ( n∑ r=0 ( n r ) bran−r HF (j) n−r(bx, y; b jz)HF (j) r (ax, y; ajz) ) tn n! = ∞∑ n=0 ( n∑ r=0 ( n r ) arbn−r HF (j) n−r(ax, y; a jz)HF (j) r (bx, y; bjz) ) tn n! . Finally, comparing the coefficients of tn n! we get (39). Theorem 16. For positive integers a, b and n ≥ 0, the following symmetric identity for Gould-Hopper-based bivariate Fubini polynomials holds: n∑ k=0 ( n k ) a−1∑ i=0 b−1∑ l=0 bkan−k HF (j) n−k(bx+ b a i+ l, y; bjz)Fk(au, y) = n∑ k=0 ( n k ) a−1∑ l=0 b−1∑ i=0 akbn−k HF (j) n−k(ax+ a b i+ l, y; ajz)Fk(bu, y). (44) Proof. Consider the following function: B(t) = eab(x+u)t+ajbjztj (eabt − 1)2 (1− y(eat − 1))(1− y(ebt − 1))(eat − 1)(ebt − 1) = eabxt+ajbjztj 1− y(eat − 1) · (e bt)a − 1 ebt − 1 · (e at)b − 1 eat − 1 · eabut 1− y(ebt − 1) . (45) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 15 of 25 By using Example 2, a−1∑ i=0 ebti = (ebt)a − 1 ebt − 1 , (46) and b−1∑ l=0 eatl = (eat)b − 1 eat − 1 . (47) Substituting equations (46) and (47) to equation (45), we get B(t) = ebx(at)+bjz(at)j 1− y(eat − 1) · a−1∑ i=0 ebti · b−1∑ l=0 eatl · eau(bt) 1− y(ebt − 1) = ebx(at)+bjz(at)j 1− y(eat − 1) · a−1∑ i=0 b−1∑ l=0 e( b a i+l)at · eau(bt) 1− y(ebt − 1) = ∑a−1 i=0 ∑b−1 l=0 e (bx+ b a i+l)at+bjz(at)j 1− y(eat − 1) · eau(bt) 1− y(ebt − 1) . (48) Then, by applying Definition 6 and Definition 4 to equation (48) we have B(t) = ∞∑ n=0 a−1∑ i=0 b−1∑ l=0 anHF (j) n (bx+ b a i+ l, y; bjz) tn n! · ∞∑ n=0 bnFn(au, y) tn n! . (49) Thus, by applying Theorem 3 to equation (49) we get B(t) = ∞∑ n=0 ( n∑ k=0 ( n k ) a−1∑ i=0 b−1∑ l=0 an−k HF (j) n−k(bx+ b a i+ l, y; bjz) · bkFk(au, y) ) tn n! = ∞∑ n=0 ( n∑ k=0 ( n k ) a−1∑ i=0 b−1∑ l=0 bkan−k HF (j) n−k(bx+ b a i+ l, y; bjz)Fk(au, y) ) tn n! . (50) Moreover, note that B(t) can also be expressed as follows B(t) = eab(x+u)t+ajbjztj (eabt − 1)2 (1− y(eat − 1))(1− y(ebt − 1))(eat − 1)(ebt − 1) = eabxt+ajbjztj 1− y(ebt − 1) · (e at)b − 1 eat − 1 · (e bt)a − 1 ebt − 1 · eabut 1− y(eat − 1) . (51) Again, by using Example 2 b−1∑ i=0 eati = (eat)b − 1 eat − 1 , (52) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 16 of 25 and a−1∑ l=0 ebtl = (ebt)a − 1 ebt − 1 . (53) Substituting equations (52) and (53) to equation (51), we get B(t) = eax(bt)+ajz(bt)j 1− y(ebt − 1) · b−1∑ i=0 eati · a−1∑ l=0 ebtl · ebu(at) 1− y(eat − 1) = ∑b−1 i=0 ∑a−1 l=0 e( a b i+l)bteax(bt)+ajz(bt)j 1− y(ebt − 1) · ebu(at) 1− y(eat − 1) = ∑b−1 i=0 ∑a−1 l=0 e(ax+ a b i+l)bt+ajz(bt)j 1− y(ebt − 1) · ebu(at) 1− y(eat − 1) . (54) Applying Definition 6 and Definition 4 to (54), gives B(t) = ∞∑ n=0 a−1∑ l=0 b−1∑ i=0 bnHF (j) n (ax+ a b i+ l, y; ajz) tn n! · ∞∑ n=0 anFn(bu, y) tn n! . (55) We then apply the Theorem 3 to equation (55) and obtain B(t) = ∞∑ n=0 ( n∑ k=0 ( n k ) a−1∑ l=0 b−1∑ i=0 akbn−k HF (j) n−k(ax+ a b i+ l, y; ajz)Fk(bu, y) ) tn n! . (56) Furthermore, equating (50) and (56) yields ∞∑ n=0 ( n∑ k=0 ( n k ) a−1∑ i=0 b−1∑ l=0 bkan−k HF (j) n−k(bx+ b a i+ l, y; bjz)Fk(au, y) ) tn n! = ∞∑ n=0 ( n∑ k=0 ( n k ) a−1∑ l=0 b−1∑ i=0 akbn−k HF (j) n−k(ax+ a b i+ l, y; ajz)Fk(bu, y) ) tn n! . (57) Finally, comparing the coefficients of tn n! we obtain (44). The following theorem is the recurrence relation for Gould-Hopper-based bivariate Fubini polynomials in the coming theorem. Theorem 17. The Gould-Hopper-based bivariate Fubini polynomials satisfy the following recurrence relation: HF (j) n+1(x, y; z) = n∑ m=0 ( n m ) yHF (j) n (x+ 1, y; z)Fn−m(y) + xHF (j) n (x, y; z) + jz(n)j+1HF (j) n−j+1(x, y; z). R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 17 of 25 Proof. Note that ext+ztj 1− y(et − 1) = ∞∑ n=0 HF (j) n (x, y; z) tn n! . Differentiating both sides with respect to t, we have d dt [ ext+ztj 1− y(et − 1) ] = d dt [ ∞∑ n=0 HF (j) n (x, y; z) tn n! ] = ∞∑ n=0 d dt [ HF (j) n (x, y; z) tn n! ] . (58) Now, for n ≥ 1, d dt [ HF (j) n (x, y; z) tn n! ] = nHF (j) n (x, y; z) tn−1 n! . (59) By substituting equation (59) to equation (58), we get d dt [ ext+ztj 1− y(et − 1) ] = ∞∑ n=1 nHF (j) n (x, y; z) tn−1 n! = ∞∑ n=0 HF (j) n+1(x, y; z) tn n! . (60) Note that ext+ztj 1− y(et − 1) = ext+ztj (1− y(et − 1))−1. Thus, using derivative of a product, we have d dt [ ext+ztj 1− y(et − 1) ] = d dt [ ext+ztj (1− y(et − 1))−1 ] = ext+ztj d dt [ (1− y(et − 1))−1 ] + (1− y(et − 1))−1 ( ext+ztj (x+ jztj−1) ) . (61) Observe that d dt [ (1− y(et − 1))−1 ] = −1(1− y(et − 1))−2(−yet) = (1− y(et − 1))−1(1− y(et − 1))−1(yet). (62) Thus, by substituting equation (62) to equation (61) gives d dt [ ext+ztj 1− y(et − 1) ] =ext+ztj (1− y(et − 1))−1(1− y(et − 1))−1(yet) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 18 of 25 + (1− y(et − 1))−1 ( ext+ztj (x+ jztj−1) ) =y [ e(x+1)t+ztj (1− y(et − 1))−1 ] (1− y(et − 1))−1 + x [ ext+ztj (1− y(et − 1))−1 ] + jztj−1 [ ext+ztj (1− y(et − 1))−1 ] . (63) Moreover, applying Definition 6 and Theorem 5 to the right-hand side of equation (63) we have d dt [ ext+ztj 1− y(et − 1) ] =y ∞∑ n=0 HF (j) n (x+ 1, y; z) tn n! ∞∑ n=0 Fn(y) tn n! + x ∞∑ n=0 HF (j) n (x, y; z) tn n! + jztj−1 ∞∑ n=0 HF (j) n (x, y; z) tn n! . Hence, applying Theorem 3 to the right-hand side of the above equation gives d dt [ ext+ztj 1− y(et − 1) ] =y ∞∑ n=0 n∑ m=0 ( n m ) HF (j) n (x+ 1, y; z)Fn−m(y) tn n! + x ∞∑ n=0 HF (j) n (x, y; z) tn n! + jztj−1 ∞∑ n=0 HF (j) n (x, y; z) tn n! . (64) But note that, jztj−1 ∞∑ n=0 HF (j) n (x, y; z) tn n! = jz ∞∑ n=0 HF (j) n (x, y; z) tn+j−1 n! = jz ∞∑ n=0 (n+ j − 1) . . . (n+ 1)HF (j) n (x, y; z) tn+j−1 (n+ j − 1)! = jz ∞∑ n=0 (n)j+1HF (j) n−j+1(x, y; z) tn n! = jz ∞∑ n=0 (n)j+1HF (j) n−j+1(x, y; z) tn n! . (65) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 19 of 25 Substituting equation (65) to equation (64) yields d dt [ ext+ztj 1− y(et − 1) ] =y ∞∑ n=0 n∑ m=0 ( n m ) HF (j) n (x+ 1, y; z)Fn−m(y) tn n! + x ∞∑ n=0 HF (j) n (x, y; z) tn n! + jz(n)j+1 ∞∑ n=0 HF (j) n−j+1(x, y; z) tn n! = ∞∑ n=0 n∑ m=0 ( n m ) yHF (j) n (x+ 1, y; z)Fn−m(y) + xHF (j) n (x, y; z) + jz(n)j+1HF (j) n−j+1(x, y; z) tn n! . (66) We then equate equation (60) and equation (66) and obtain ∞∑ n=0 HF (j) n+1(x, y; z) tn n! = ∞∑ n=0 n∑ m=0 ( n m ) yHF (j) n (x+ 1, y; z)Fn−m(y) + xHF (j) n (x, y; z) + jz(n)j+1HF (j) n−j+1(x, y; z) tn n! . Finally, comparing the coefficients of tn n! yields to the desired result. In the next theorem, we introduced the partial derivative formulae for Gould-Hopper- based bivariate Fubini polynomials. Theorem 18. For n ≥ 0, the following partial derivatives for the Gould-Hopper-based bivariate Fubini polynomials holds: i. ∂ ∂x HF (j) n+1(x, y; z) = (n+ 1)HF (j) n (x, y; z), (67) ii. ∂ ∂z HF (j) n+j(x, y; z) = (n+ j)j HF (j) n (x, y; z). (68) Proof. Note that ∞∑ n=0 HF (j) n (x, y; z) tn n! = ext+ztj (1− y(et − 1))−1. (69) Differentiating both sides with respect to x, we have ∂ ∂x [ ∞∑ n=0 HF (j) n (x, y; z) tn n! ] = ∂ ∂x [ ext+ztj (1− y(et − 1))−1 ] = text+ztj (1− y(et − 1))−1. R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 20 of 25 But note that, ∂ ∂x [ ∞∑ n=0 HF (j) n (x, y; z) tn n! ] = ∞∑ n=0 ∂ ∂x HF (j) n (x, y; z) tn n! . Thus, ∞∑ n=0 ∂ ∂x HF (j) n (x, y; z) tn n! = text+ztj (1− y(et − 1))−1. (70) By applying Definition 6 to the right-hand side of equation (70) we get ∞∑ n=0 ∂ ∂x HF (j) n (x, y; z) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn+1 n! . Differentiating both sides of the above equation with respect to t, we have ∞∑ n=1 ∂ ∂x HF (j) n (x, y; z) tn−1 (n− 1)! = ∞∑ n=0 (n+ 1)HF (j) n (x, y; z) tn n! . (71) Reindexing the left-hand side of equation (71), we have ∞∑ n=0 ∂ ∂x HF (j) n+1(x, y; z) tn n! = ∞∑ n=0 (n+ 1)HF (j) n (x, y; z) tn n! . Comparing the coefficients of tn n! yields to (67). Moreover, we will prove (68) analogously. That is, differentiating both side of equation (69) with respect to z gives us ∂ ∂z [ ∞∑ n=0 HF (j) n (x, y; z) tn n! ] = ∂ ∂z [ ext+ztj (1− y(et − 1))−1 ] = tjext+ztj (1− y(et − 1))−1. But note that, ∂ ∂z [ ∞∑ n=0 HF (j) n (x, y; z) tn n! ] = ∞∑ n=0 ∂ ∂z HF (j) n (x, y; z) tn n! . Hence, ∞∑ n=0 ∂ ∂z HF (j) n (x, y; z) tn n! = tjext+ztj (1− y(et − 1))−1. (72) Then, by applying Definition 6 to the right-hand side of equation (72) we get ∞∑ n=0 ∂ ∂z HF (j) n (x, y; z) tn n! = ∞∑ n=0 HF (j) n (x, y; z) tn+j n! . R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 21 of 25 Differentiating both sides of the above equation with respect to t, we have ∞∑ n=1 ∂ ∂z HF (j) n (x, y; z) tn−1 (n− 1)! = ∞∑ n=0 (n+ j)HF (j) n (x, y; z) tn+j−1 n! . (73) Reindexing the left-hand side of equation (73), we have ∞∑ n=0 ∂ ∂z HF (j) n+1(x, y; z) tn n! = ∞∑ n=0 (n+ j)HF (j) n (x, y; z) tn+j−1 n! . Again, differentiating both sides of the above equation with respect to t, we have ∞∑ n=1 ∂ ∂z HF (j) n+1(x, y; z) tn−1 (n− 1)! = ∞∑ n=0 (n+ j)(n+ j − 1)HF (j) n (x, y; z) tn+j−2 n! . (74) Reindexing the left-hand side of equation (74), we have ∞∑ n=0 ∂ ∂z HF (j) n+2(x, y; z) tn n! = ∞∑ n=0 (n+ j)(n+ j − 1)HF (j) n (x, y; z) tn+j−2 n! . Continue doing this, until we arrive with ∞∑ n=1 ∂ ∂z HF (j) n+j−1(x, y; z) tn−1 (n− 1)! = ∞∑ n=0 (n+ j)(n+ j − 1) . . . (n+ 1)HF (j) n (x, y; z) tn n! . (75) Reindexing the left-hand side of equation (75), we have ∞∑ n=0 ∂ ∂z HF (j) n+j(x, y; z) tn n! = ∞∑ n=0 (n+ j)(n+ j − 1) . . . (n+ 1)HF (j) n (x, y; z) tn n! = ∞∑ n=0 (n+ j)j HF (j) n (x, y; z) tn n! . Finally, comparing the coefficients of tn n! yields to (68). The subsequent theorem will established the integral formulae for Gould-Hopper-based bivariate Fubini polynomials. Theorem 19. For n ≥ 0, the following formulae for Gould-Hopper-based bivariate Fubini polynomials holds: i. ∫ HF (j) n (x, y; z)dx = 1 (n+ 1) HF (j) n+1(x, y; z), (76) ii. ∫ HF (j) n (x, y; z)dz = 1 (n+ j)j HF (j) n+j(x, y; z). (77) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 22 of 25 Proof. From (67), it follows that ∂ ∂x 1 (n+ 1) HF (j) n+1(x, y; z) = HF (j) n (x, y; z). (78) Integrating both sides of (78) with respect to x gives 1 (n+ 1) HF (j) n+1(x, y; z) = ∫ HF (j) n (x, y; z)dx. Thus, we obtain (76). Moreover, we will prove (77) analogously. By using (68) we obtain ∂ ∂z 1 (n+ j)j HF (j) n+j(x, y; z) = HF (j) n (x, y; z). (79) Integrating both sides of (79) with respect to z gives 1 (n+ j)j HF (j) n+j(x, y; z) = ∫ HF (j) n (x, y; z)dz. The next theorem will derive the implicit formula for Gould-Hopper-based bivariate Fubini polynomials. Theorem 20. For n ≥ 0, the following implicit summation formula for Gould-Hopper- based bivariate Fubini polynomials holds: HF (j) q+l(w, y; z) = q,l∑ p,r=0 ( q p )( l r ) (w − x)p+r HF (j) q+l−p−r(x, y; z). (80) Proof. Replacing t by t+ u in Definition 6, we have ∞∑ n=0 HF (j) n (x, y; z) (t+ u)n n! = ex(t+u)+z(t+u)j 1− y(et+u − 1) . (81) Applying Theorem 2 to the left-hand side of equation (81), we get ∞∑ n=0 HF (j) n (x, y; z) (t+ u)n n! = ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! . (82) Then, equating (81) and (82) gives ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ex(t+u)+z(t+u)j 1− y(et+u − 1) . (83) R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 23 of 25 Multiplying both sides of equation (83) by 1 ex(t+u) , we have e−x(t+u) ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ez(t+u)j 1− y(et+u − 1) . (84) Replacing x by w in equation (84), we get e−w(t+u) ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! = ez(t+u)j 1− y(et+u − 1) . (85) Hence, equating (84) and (85) and multiplying both sides by 1 e−w(t+u) we have e−x(t+u) ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = e−w(t+u) ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! e−x(t+u)ew(t+u) ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! e−x(t+u)+w(t+u) ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! e(w−x)(t+u) ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! . (86) Note that, expressing e(w−x)(t+u) in its exponential form and applying Theorem 2 gives e(w−x)(t+u) = ∞∑ n=0 (w − x)n (t+ u)n n! = ∞∑ q,l=0 (w − x)q+l t q q! ul l! . (87) Now, substituting (87) to (86) we find ∞∑ q,l=0 (w − x)q+l t q q! ul l! ∞∑ q,l=0 HF (j) q+l(x, y; z) tq q! ul l! = ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! . (88) Thus, applying Theorem 3 to the left-hand side of equation (88) we get ∞∑ q,l=0  q,l∑ p,r=0 ( q p )( l r ) (w − x)p+r HF (j) q+l−p−r(x, y; z)  tq q! ul l! R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. Math, 18 (2) (2025), 5957 24 of 25 = ∞∑ q,l=0 HF (j) q+l(w, y; z) tq q! ul l! . Comparing the coefficients of tq q! ul l! in the above equation we get (80). Setting l = 0 in Theorem 20, it will deduce to the following corollary. Corollary 7. For n ≥ 0, the following formula for Gould-Hopper-based bivariate Fubini polynomials holds: HF (j) q (w, y; z) = q∑ p=0 ( q p ) (w − x)pHF (j) q−p(x, y; z). In the following theorem, a relationship of Gould-Hopper-based bivariate Fubini poly- nomials with Gould-Hopper polynomials and Stirling number of the second kind will be introduced. Theorem 21. For n ≥ 0, the following relationship holds: HF (j) n (x, y; z) = n∑ m=0 ( n m ) H (j) n−m(x, z) m∑ k=0 k!ykS(m, k). (89) Proof. By Definition 6, we have ∞∑ n=0 HF (j) n (x, y; z) tn n! = ext+ztj 1 1− y(et − 1) . (90) Then, applying Definition 5 and Theorem 5 to the right-hand side of equation (90) gives us ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 H(j) n (x, z) tn n! ∞∑ m=0 Fm(y) tm m! . (91) Hence, applying Definition 3 to the right-hand side of equation (91) we have ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 H(j) n (x, z) tn n! ∞∑ m=0 m∑ k=0 S(m, k)k!yk tm m! = ∞∑ n=0 H(j) n (x, z) tn n! ∞∑ m=0 m∑ k=0 k!ykS(m, k) tm m! . (92) Thus, applying Theorem 3 to the right-hand side of equation (92), we obtain ∞∑ n=0 HF (j) n (x, y; z) tn n! = ∞∑ n=0 ( n∑ m=0 ( n m ) H (j) n−m(x, z) m∑ k=0 k!ykS(m, k) ) tn n! . (93) Comparing the coefficients of tn n! yields (89). R. G. Bago, N. S. Abdulcarim / Eur. J. Pure Appl. 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