EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 587-607 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global A Unification of the Generalized Multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi Polynomials of Higher Order Nestor G. Acala Mathematics Department, College of Natural Sciences and Mathematics, Mindanao State University, Marawi City, Lanao Del Sur, Philippines Abstract. Most unifications of the classical or generalized Bernoulli, Euler, and Genocchi polyno- mials involve unifying any two or all of the three special types of polynomials (see, [1, 4, 9, 18, 19, 21, 24–26, 30, 31]). In this paper, we introduce a new class of multiparameter Fubini-type gener- alized polynomials that unifies four families of higher order generalized Apostol-type polynomials such as the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Fubini polynomials. Moreover, we obtain an explicit formula of these unified generalized polynomials in terms of the Gaussian hypergeometric function, and establish several symmetry identities. 2020 Mathematics Subject Classifications: 11B68, 11B73, 33C05, 05A10, 11B83 Key Words and Phrases: Fubini polynomials, Bernoulli polynomials, Euler polynomials, Genoc- chi polynomials, Apostol-type polynomials, Ordered Bell polynomials, Gauss hypergeometric func- tion, Generalized Bernoulli polynomials, Generalized Euler polynomials, Generalized Genocchi polynomials, Generalized Fubini polynomials 1. Introduction In recent years, extensive researches on various families of numbers and polynomials such as the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Fubini numbers and polynomials, and also their generalizations and unifications (see, for instance the recent works of [1, 3, 4, 10, 11, 17, 25, 26, 28, 31]) have become popular due to the abundance of their applications in many branches of mathematics such as in p-adic analytic number theory, umbral calculus, special functions and mathematical analysis, numerical analysis, combinatorics and other related fields. This motivates the author to obtain and explore a new unification of some of the recent generalizations of these special types of polynomials. In this section, we present some of the known generalizations of Bernoulli, Euler, Genocchi, and Fubini polynomials of higher order. Throughout this paper, we use the DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3757 Email address: nestor.acala@gmail.com (N. G. Acala) https://www.ejpam.com 587 c© 2020 EJPAM All rights reserved. N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 588 usual notations N,Z,R, and C for the sets of natural numbers, integers, real numbers, and complex numbers respectively. Also, we let N0 := N ∪ {0}, Z− := {−1,−2,−3, · · · }, and Z−0 := Z− ∪ {0}. The bivariate Fubini polynomials of order α is defined through the generating function ext [1− y(et − 1)]α = ∞∑ n=0 F (α) n (x, y) tn n! (see [12, 13, 15, 17]). (1) When α = 1, F (1) n (x, y) := Fn(x, y), the two-variable Fubini polynomials given by ext 1− y(et − 1) = ∞∑ n=0 Fn(x, y) tn n! (see [10, 11, 16]). Moreover, setting x = 0 in (1), we obtain Fαn (0, y) := F (α) n (y) and F (α) n (1) := F (α) n where F (α) n (y) and F (α) n are called the higher order Fubini polynomials and the higher order Fubini numbers respectively (see [6, 14]). For α = 1 F (1) n (0, y) := Fn(y) and F (1) n (1) := Fn, where Fn(y) are the classical Fubini polynomials or the ordered Bell polynomials, and Fn are the classical Fubini numbers or the ordered Bell numbers (see [2, 27]). The classical Bernoulli polynomials Bn(x), Euler polynomials En(x), and Genocchi polynomials Gn(x) together with their natural higher order generalizations B (α n (x), E (α n (x), and G (α n (x) are usually defined by means of the generating functions (see [1, 3, 4, 29])( t et − 1 )α ext = ∞∑ n=0 B(α) n (x) tn n! (|t| < 2π, α ∈ C) , ( 2 et + 1 )α ext = ∞∑ n=0 E(α) n (x) tn n! (|t| < π, α ∈ C) , ( 2t et + 1 )α ext = ∞∑ n=0 G(α) n (x) tn n! (|t| < π, α ∈ C) . Hence, B(1) n (x) := Bn(x) E(1) n (x) := En(x) and G(1) n (x) := Gn(x). The classical Bernoulli numbers Bn, Euler numbers En, and Genocchi numbers Gn are obtained by setting further x = 0. That is Bn(0) := Bn En(0) := En and Gn(0) := Gn. N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 589 In [9], Karande and Thakare obtained a class of polynomials Dn(x;u, k) unifying the classical Bernoulli, Euler, and Genocchi polynomials through the generating function: 21−ktk et − u ext = ∞∑ n=0 Dn(x;u, k) tn n! (u ∈ R− {0}, k ∈ Z), in which Dn(x; 1, 1) = Bn(x), Dn(x;−1, 0) = En(x), and 2Dn(0;−1, 1) = Gn. The higher order Apostol-Bernoulli polynomials B (α) n (x;λ), higher order Apostol-Euler polynomials E (α) n (x;λ), and higher order Apostol-Genocchi polynomials G (α) n (x;λ) (see [18, 20–24, 32, 33]) are defined through the generating functions:( t λet − 1 )(α) ext = ∞∑ n=0 B(α) n (x;λ) (|t+ lnλ| < 2π; 1α = 1, α ∈ C), ( 2 λet + 1 )(α) ext = ∞∑ n=0 E(α) n (x;λ) (|t+ lnλ| < π; 1α = 1, α ∈ C), ( 2t λet + 1 )(α) ext = ∞∑ n=0 G(α) n (x;λ) (|t+ lnλ| < π; 1α = 1, α ∈ C). For λ = 1, B(α) n (x; 1) = Bα n (x), E(α) n (x, 1) = E(α) n (x), and ; G(α) n (x, 1) = G(α) n (x), where Bα n (x), Eαn (x), ang Gαn(x) are the Bernoulli, Euler and Genocchi polynomials of order α, respectively. Further setting α = 1, each reduces to its classical kind. In [26], Ozden et al. introduced a more general unification of Apostol-type Bernoulli, Euler and Genocchi polynomials via the generating function: 21−ktk βbet − ab ext = ∞∑ n=0 Yn,β(x; k, a, b) tn n! (|t+ b ln(β/a)| < 2π; k ∈ N; a, b ∈ R+;β ∈ C). This was further extended by Ozarslan [25] to higher order type of polynomials through this generating function: ( 21−ktk βbet−ab )α ext = ∞∑ n=0 P (α) n,β (x; k, a, b) tn n! (|t+ b ln(β/a)| < 2π; k ∈ N; a, b ∈ R+;α, β ∈ C). (2) Clearly, P (1) n,λ(x; k, a, b) = Yn,β(x; k, a, b), P (α) n,λ (x; 1, 1, 1) = B(α) n (x;λ), N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 590 P (α) n,λ (x; 0,−1, 1) = E(α) n (x;λ), P (α) n,λ 2 (x; 1,−1 2 , 1) = G(α) n (x;λ). Finally, more generalized higher order Apostol-type polynomials of parameters a, b, c are defined via the generating functions (see [1, 5, 7, 8, 28, 30, 31]):( t λbt − at )α cxt = ∞∑ n=0 B(α) n (x; a, b, c;λ) (|t ln(b/a) + lnλ| < 2π; a 6= b; 1α = 1, α ∈ C), ( 2 λbt + at )α cxt = ∞∑ n=0 E(α) n (x; a, b, c;λ) (|t ln(b/a) + lnλ| < π; 1α = 1, α ∈ C), ( 2t λbt + at )α cxt = ∞∑ n=0 G(α) n (x; a, b, c;λ) (|t ln(b/a) + lnλ| < π; 1α = 1, α ∈ C). In the next section, we try to unify all the previously mentioned special polynomials using a more generalized generating function. 2. A new class of unified generalized polynomials of higher order Motivated by the generating relations (1), (2), and the definitions of the higher order Aposotol-type polynomials of parameters a, b, c, we consider the following unification of the generalized special types of polynomials mentioned in the previous section. Definition 1. Let a, b, c > 0, we define a unified form of generalized polynomials F (α) n,k(x, y; a, b, c;λ) by means of the generating function a−ttk 1− y ( λ ( b a )t − 1 ) α cxt = ∞∑ n=0 F (α) n,k(x, y; a, b, c;λ) tn n! , (3) (∣∣∣∣t ln ( b a ) + ln ( λy y + 1 )∣∣∣∣ < 2π;α ∈ C; a, b, c ∈ R+;x, y ∈ R, k ∈ N0; 1α := 1 ) . Setting a = 1, b = e and c = e in (3), we obtain new generalized Fubini-type polyno- mials F (α) n,k (x, y;λ) given by the following generating function( tk 1− y (λet − 1) )α ext = ∞∑ n=0 F (α) n,k (x, y;λ) tn n! . (4) Taking k = 0, F (α) n,0 (x, y;λ) := F (α) n (x, y;λ), where( 1 1− y (λet − 1) )α ext = ∞∑ n=0 F (α) n (x, y;λ) tn n! . (5) We call F (α) n (x, y;λ) as the bivariate Apostol-Fubini polynomials of order α. Setting λ = 1 in (5), we get the two-variable Fubini polynomials of higher order F (α) n (x, y) given in (1). N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 591 Remark 1. Setting y = −1 2 and k = 0 in (3), we obtain F (α) n,0 ( x,−1 2 ; a, b, c;λ ) = E(α) n (x; a, b, c;λ). Remark 2. Setting y = −1 2 and k = 1 in (3), we have F (α) n,1 ( x,−1 2 ; a, b, c;λ ) = G(α) n (x; a, b, c;λ). Remark 3. Setting y = −2 and k = 1, and replacing λ by λ 2 in (3), we get F (α) n,1 ( x,−2; a, b, c; λ 2 ) = B(α) n (x; a, b, c;λ). Remark 4. Setting a = 1, b = c = e, α = 1, y = −(2k−1u+1), and λ = 2k−1 2k−1u+1 (u 6= 0); we obtain F (1) n,k ( x,−(2k−1u+ 1); 1, e, e; 2k−1 2k−1u+ 1 ) = Dn(x;u, k). Remark 5. Let a, b, β be the parameters used in (2). Setting y = −(2k−1ab + 1) and λ = 2k−1βb 2k−1ab+1 in (4), we obtain, F (α) n,k ( x,−(2k−1ab + 1); 2k−1βb 2k−1ab + 1 ) = F (α) n,k ( x,−(2k−1ab + 1); 1, e, e; 2k−1βb 2k−1ab + 1 ) = P (α) n,β (x; k, a, b). Some of the basic properties and identities for F (α) n,k(x, y; a, b, c;λ) are given in the next theorems and corollaries. The following addition formulas are straighforward consequences of relation (3). Theorem 1. For α, β, λ ∈ C and x, z ∈ R, we have F (α) n,k(x+ z, y; a, b, c;λ) = n∑ j=0 ( n j ) (ln c)n−jF (α) j,k (x, y; a, b, c;λ)zn−j (z 6= 0) (6) = n∑ j=0 ( n j ) (ln c)n−jF (α) j,k (z, y; a, b, c;λ)xn−j (x 6= 0), (7) F (α) n,k(x+ z; y; a; b; c;λ) = n∑ j=0 ( n j ) F (β) j,k (z, y; a, b, c;λ)F (α−β) n−j,k (x, y; a, b, c;λ), (8) F (α+β) n,k (x; y; a; b; c;λ) = n∑ j=0 ( n j ) F (β) j,k (x, y; a, b, c;λ)F (α) n−j,k(x, y; a, b, c;λ), (9) F (α+β) n,k (x+ z; y; a; b; c;λ) = n∑ j=0 ( n j ) F (α) j,k (x, y; a, b, c;λ)F (β) n−j,k(z, y; a, b, c;λ) N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 592 = n∑ j=0 ( n j ) F (β) j,k (x, y; a, b, c;λ)F (α) n−j,k(z, y; a, b, c;λ). Replacing x by x − z in (6), (7), and (8); and replacing α by α − β in (9), we obtain the next results. Corollary 1. For α, β, λ ∈ C and x, z ∈ R, we have F (α) n,k(x, y; a, b, c;λ) = n∑ j=0 ( n j ) (ln c)n−jF (α) j,k (x− z, y; a, b, c;λ)zn−j (z 6= 0), = n∑ j=0 ( n j ) (ln c)n−jF (α) j,k (z, y; a, b, c;λ)(x− z)n−j . F (α) n,k(x, y; a, b, c;λ) = n∑ j=0 ( n j ) F (α) j,k (z, y; a, b, c;λ)F (α) n−j,k(x− z, y; a, b, c;λ) (z 6= x), F (α) n,k(x, y; a, b, c;λ) = n∑ j=0 ( n j ) F (β) j,k (x, y; a, b, c;λ)F (α−β) n−j,k (x, y; a, b, c;λ) Setting z = (p− 1)x in (6), we obtain a multiplication formula for F (α) n,k(x, y; a, b, c;λ). Corollary 2. Let p 6= 1 and x 6= 0. Then F (α) n,k(px, y; a, b, c;λ) = n∑ j=0 ( n j ) [(p− 1)x ln c]n−j F (α) j,k (x, y; a, b, c;λ). Theorem 2. Let α and λ be arbitrary real or complex parameters. Then F (α) n,k(−x, y; a, b, c;λ) = (−1)kα+nF (α) n,k ( x, y; 1 a , 1 b , c;λ ) , F (α) n,k(x+ α, y; a, b, c;λ) = F (α) n,k ( x, y; a c , b c , c;λ ) , F (α) n,k(α− x, y; a, b, c;λ) = F (α) n,k ( −x, y; a c , b c , c;λ ) , = (−1)kα+nF (α) n,k ( x, y; c a , c b , c;λ ) . Basic differential and integral identities of F (α) n,k(x, y; a, b, c;λ) are given in the next the- orem. N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 593 Theorem 3. Let m, l ∈ N0. Then for any real numbers u and v, we have ∂m ∂xm F (l) n,k(x, y; a, b, c;λ) = n! (n−m)! (ln c)mF (l) n−m,k(x, y; a, b, c;λ), (10)∫ v u F (l) n,k(x, y; a, b, c;λ)dx = 1 (n+ 1) ln c [ F (l) n+1,k(v, y; a, b, c;λ)− F (l) n+1,k(u, y; a, b, c;λ) ] . (11) Expression (10) follows from standard arguments and induction. Moreover, (11) follows from integrating both sides of (10) with respect to x (when m = 1). 3. Explicit formulas involving the Gaussian hypergeometric function We now establish an explicit expression of F (r) n,k(x, y; a, b, c;λ) in terms of the Gaussian hypergeometric function 2F1(a, b; c; z) which is given by 2F1(a, b; c; z) := ∞∑ n=0 (a)n(b)n (c)n zn n! , where c /∈ Z0; |z| < 1; z = 1 and <(c − a − b) > 0; z = −1 and <(c − a − b) > −1. Here, (q)0 = 1, and (q)n = q(q + 1) · · · (q + n− 1) for n > 0. Theorem 4. For n, k, r ∈ N0 and y − λy 6= −1, we have F (r) n,k(x, y; a, b, c;λ) = (kr)! ( n kr ) n−kr∑ i=0 ( n− kr i )( r + i− 1 i ) [ −λy ln ( b a )]i (y + 1− λy)r+i × i∑ m=0 (−1)mmi ( i m )[ x ln c− r ln a+m ln ( b a )]n−kr−i 2F1 ( −n+ kr + i, i; 1 + i; m m+ x ln c−r ln a ln b−ln a ) . Proof : Note that the left hand side of (3) can be written as (y + 1− λy)−r [ 1− λy y + 1− λy ( et ln( b a) − 1 )]−r tkrex ln c−r ln a. Let Dt := d dt . Thus, F (r) n,k(x, y; a, b, c;λ) = (y + 1− λy)−r n∑ s=0 ( n s ) (x ln c− r ln a)n−s ×Ds t [ tkr ( 1− λy y + 1− λy [ et ln( b a) − 1 ])−r] t=0 = (y + 1− λy)−r n∑ s=0 ( n s ) (x ln c− r ln a)n−s(kr)! ( s kr ) N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 594 ×Ds−kr t [( 1− λy y + 1− λy [ et ln( b a) − 1 ])−r] t=0 . Using (A+ w)−r = ∞∑ i=0 ( r + i− 1 i ) A−r−i(−w)i, (|w| < |A|) and (et − 1)i = i! ∞∑ j=1 S(j, i) tn n! , where S(j, i) is the Stirling numbers of the second kind, we obtain F (r) n,k(x, y; a, b, c;λy) = n∑ s=kr ( n s ) (x ln c− r ln a)n−s(kr)! ( s kr ) × s−kr∑ i=0 ( r + i− 1 i ) (λy)i (y + 1− λy)r+i [ ln ( b a )]s−kr i!S(s− kr, i). Using the explicit formula S(j, i) = 1 i! i∑ m=0 (−1)i−m ( i m ) mj and the identity ( n s )( s kr ) = ( n kr )( n− kr n− s ) , we get F (r) n,k(x, y; a, b, c;λ) = n∑ s=kr ( n s ) (kr)! ( s kr ) s−kr∑ i=0 ( r + i− 1 i ) (λy)i (y + 1− λy)r+i × (x ln c− r ln a)n−s [ ln ( b a )]s−kr i∑ m=0 (−1)i−m ( i m ) ms−kr. = (kr)! ( n kr ) n−kr∑ i=0 n−i−kr∑ s=0 ( n− kr n− s− kr − i )( r + i− 1 i )[ ln ( b a )]s+i (−λy)i (y + 1− λy)r+i (x ln c− r ln a)n−s−i−kr i∑ m=0 (−1)m ( i m ) ms+i. Using the identity (n− s− kr − i)! = (−1)s(n− kr − i)! (−n+ kr + i)s , N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 595 gives F (r) n,k(x, y; a, b, c;λ) = (kr)! ( n kr ) n−kr∑ i=0 ( n− kr i )( r + i− 1 i ) [ −λy ln ( b a )]i (y + 1− λy)r+i × i∑ m=0 (−1)m ( i m ) mi(x ln c− r ln a)n−i−kr 2F1 ( −n+ kr + i, 1; 1 + i; −m ln ( b a ) x ln c− r ln a ) . Finally, applying Pfaff-Kummer hypergeometric transformation 2F1(a, b; c; z) = (1− z)−a 2F1 ( a, c− b; c; z z − 1 ) ( c /∈ Z−0 ; | arg(1− z)| ≤ π − ε (0 < ε < π) ) , yields F (r) n,k(x, y; a, b, c;λ) = (kr)! ( n kr ) n−kr∑ i=0 ( n− kr i )( r + i− 1 i ) [ −λy ln ( b a )]i (y + 1− λy)r+i × i∑ m=0 (−1)m ( i m ) mi [ x ln c− r ln a+m ln ( b a )]n−kr−i 2F1 ( −n+ kr + i, i; 1 + i; m m+ x ln c−r ln a ln b−ln a ) . Setting y = −1 2 and k = 0 in Theorem 4, we obtain an explict expression for E (r) n (x; a, b, c;λ) (see Theorem 6 [31]). Corollary 3. For n, r ∈ N0 and λ 6= −1, we have E(r) n (x; a, b, c;λ) = 2r n∑ i=0 ( n i )( r + i− 1 i )[ λ ln ( b a )]i (λ+ 1)r+i × i∑ m=0 (−1)mmi ( i m )[ x ln c− r ln a+m ln ( b a )]n−i 2F1 ( −n+ i, i; 1 + i; m m+ x ln c−r ln a ln b−ln a ) . Setting y = −2 and k = 1, and replacing λ by λ 2 in Theorem 4, we obtain an explicit formula for B (r) n (x; a, b, c;λ) (see Theorem 6 [30]). Corollary 4. For n, r ∈ N0 and λ 6= 1, we have B(r) n (x; a, b, c;λ) = r! ( n r ) n−r∑ i=0 ( n− r i )( r + i− 1 i )[ λ ln ( b a )]i (λ− 1)r+i × i∑ m=0 (−1)mmi ( i m )[ x ln c− r ln a+m ln ( b a )]n−r−i 2F1 ( −n+ i, i; 1 + i; m m+ x ln c−r ln a ln b−ln a ) . N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 596 Setting y = −1 2 and k = 1 in Theorem 4, we obtain an explicit formula forG (r) n (x; a, b, c;λ) (see Theorem 9 [31]). Corollary 5. For n, r ∈ N0, and λ 6= −1, we have G(r) n (x; a, b, c;λ) = 2rr! ( n r ) n−r∑ i=0 ( n− r i )( r + i− 1 i )[ λ ln ( b a )]i (λ+ 1)r+i × i∑ m=0 (−1)mmi ( i m )[ x ln c− r ln a+m ln ( b a )]n−r−i 2F1 ( −n+ i, i; 1 + i; m m+ x ln c−r ln a ln b−ln a ) . Taking a = 1 and b = c = e Theorem 4, we get an explicit formula of F (r) n,k(x, y;λ). Corollary 6. For n, k, r ∈ N0 and y − λy 6= −1, we have F (r) n,k(x, y;λ) = (kr)! ( n kr ) n−kr∑ i=0 ( n− kr i )( r + i− 1 i ) (−λy)i (y + 1− λy)r+i × i∑ m=0 (−1)m ( i m ) mi(x+m)n−kr−i 2F1 ( −n+ kr + i, i; 1 + i; m m+ x ) . Setting y = −(2k−1ab + 1) and λ = 2k−1βb 2k−1ab+1 in Corollary 6, we obtain an explicit formula of P (r) n,β(x; a, b) (see Theorem 2.1 [25]). Corollary 7. For n, k, r ∈ N0, a, b ∈ R+, β 6= a, P (r) n,β(x; a, b) = 2(1−k)r(kr)! ( n kr ) n−kr∑ i=0 ( n− kr i )( r + i− 1 i ) βbi (βa − ab)r+i × i∑ m=0 (−1)m ( i m ) mi(x+m)n−kr−i 2F1 ( −n+ kr + i, i; 1 + i; m m+ x ) . 4. Symmetry Identities In this section, we derive and investigate some symmetry identities for F (r) n,k(x, y; a, b, c;λ). For each k ∈ N0, the sum of integer powers Sk(n) is defined by Sk(n) = n−1∑ j=0 jk and has the exponential generating function ∞∑ k=0 Sk(n) tk k! = ent − 1 et − 1 . N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 597 In [19], Lu and Srivastava defined the generalized sum of integer powers Sk(n;λ) through the generating function ∞∑ k=0 Sk(n;λ) tk k! = λent − 1 λet − 1 (λ ∈ C). Clearly, Sk(n; 1) = Sk(n). Definition 2. Let λ be any real or complex paramete and b > 0, we define a more generalized sum of integer powers Sk(n; b, λ) using the generating function ∞∑ k=0 Sk(n; b, λ) tk k! = λbnt − 1 λbt − 1 . Obviously, Sk(n; e, λ) = Sk(n;λ) and Sk(n; e, 1) = Sk(n). Now, we establish some symmetry identities involving these new class of unified gen- eralized polynomials. The techniques used in here are parallel to the methods in [25, 33]. Thus, we also include some results in [25] as corollaries. Theorem 5. For u, v,m ∈ N; n ∈ N0; and y 6= −1, we have n∑ r=0 ( n r ) un−rvr+kF (m) n−r,k(vx− v u logc a, y; a, b, c;λ) r∑ l=0 ( r l ) Sl ( u− 1, b a ; λy y + 1 ) F (m−1) r−l,k (uz, y; a, b, c;λ) = n∑ r=0 ( n r ) vn−rur+kF (m) n−r,k(ux− u v logc a, y; a, b, c;λ) r∑ l=0 ( r l ) Sl ( v − 1, b a ; λy y + 1 ) F (m−1) r−l,k (vz, y; a, b, c;λ). Proof: Let G(t) := t2km−ka−m(u+v)tcuvxt ( 1− y ( λ ( b a )uvt − 1 )) cuvzt( 1− y ( λ ( b a )ut − 1 ))m ( 1− y ( λ ( b a )vt − 1 ))m . Grouping factors and expanding G(t) into series, we obtain G(t) = 1 ukmvk(m−1)  a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cvx(ut)a−vt ×  λy y+1 ( b a )uvt − 1 λy y+1 ( b a )vt − 1  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m−1 cuz(vt) = 1 ukmvk(m−1) ∞∑ n=0 F (m) n,k ( vx− v u logc a, y; a, b, c;λ ) (ut)n n! N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 598 × ∞∑ n=0 Sn ( u− 1, b a ; λy y + 1 ) (vt)n n! · ∞∑ n=0 F (m−1) n,k (uz, y; a, b, c;λ) (vt)n n! = 1 (uv)km ∞∑ n=0 [ n∑ r=0 ( n r ) un−rvr+kF (m) n−r,k ( vx− v u logc a, y; a, b, c;λ ) × r∑ l=0 ( r l ) Sl ( u− 1, b a ; λy y + 1 ) F (m−1) r−l,k (uz, y; a, b, c;λ) ] tn n! . (12) Similarly, G(t) = 1 vkmuk(m−1)  (vt)k 1− y ( λ ( b a )vt − 1 ) m cux(vt)a−ut ×  λy y+1 ( b a )uvt − 1 λy y+1 ( b a )ut − 1  (ut)k 1− y ( λ ( b a )ut − 1 ) m−1 cvz(ut) = 1 vkmuk(m−1) ∞∑ n=0 F (m) n,k ( ux− u v logc a, y; a, b, c;λ ) (vt)n n! × ∞∑ n=0 Sn ( v − 1, b a ; λy y + 1 ) (ut)n n! · ∞∑ n=0 F (m−1) n,k (vz, y; a, b, c;λ) (ut)n n! = 1 (vu)km ∞∑ n=0 [ n∑ r=0 ( n r ) vn−rur+kF (m) n−r,k ( ux− u v logc a, y; a, b, c;λ ) × r∑ l=0 ( r l ) Sl ( v − 1, b a ; λy y + 1 ) F (m−1) r−l,k (vz, y; a, b, c;λ) ] tn n! . (13) Comparing the (12) and (13) yields the desired result. Setting y = −1 2 and k = 0 in Theorem 5, we have a symmetry identity for E (α) n (x; a, b, c;λ). Corollary 8. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) un−rvrE (m) n−r(vx− v u logc a; a, b, c;λ) r∑ l=0 ( r l ) Sl ( u− 1, b a ;−λ ) E (m−1) r−l (uz; a, b, c;λ) = n∑ r=0 ( n r ) vn−rurE (m) n−r(ux− u v logc a; a, b, c;λ) r∑ l=0 ( r l ) Sl ( v − 1, b a ;−λ ) E (m−1) r−l (vz; a, b, c;λ). Setting y = −2 and k = 1, and replacing λ by λ 2 in Theorem 5, we have a symmetry identity for B (α) n (x; a, b, c;λ). N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 599 Corollary 9. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) un−rvr+1B (m) n−r ( vx− v u logc a; a, b, c;λ ) r∑ l=0 ( r l ) Sl ( u− 1, b a ;λ ) B (m−1) r−l (uz; a, b, c;λ) = n∑ r=0 ( n r ) vn−rur+1B (m) n−r ( ux− u v logc a; a, b, c;λ ) r∑ l=0 ( r l ) Sl ( v − 1, b a ;λ ) B (m−1) r−l (vz; a, b, c;λ) . Setting y = −1 2 and k = 0 in Theorem 5, we have a symmetry identity forG (α) n (x; a, b, c;λ). Corollary 10. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) un−rvr+1G (m) n−r(vx− v u logc a; a, b, c;λ) r∑ l=0 ( r l ) Sl ( u− 1, b a ;−λ ) G (m−1) r−l (uz; a, b, c;λ) = n∑ r=0 ( n r ) vn−rur+1G (m) n−r(ux− u v logc a; a, b, c;λ) r∑ l=0 ( r l ) Sl ( v − 1, b a ;−λ ) G (m−1) r−l (vz; a, b, c;λ). Setting b = c = e and a = 1 in Theorem 5, we obtain a symmetry identity for the higher order generalized Fubini-type polynomials F (α) n,k (x, y;λ). Corollary 11. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) un−rvr+kF (m) n−r,k(vx, y;λ) r∑ l=0 ( r l ) Sl ( u− 1; λy y + 1 ) F (m−1) r−l,k (uz, y;λ) = n∑ r=0 ( n r ) vn−rur+kF (m) n−r,k(ux, y;λ) r∑ l=0 ( r l ) Sl ( v − 1; λy y + 1 ) F (m−1) r−l,k (vz, y;λ). Setting y = −(2k−1ab+1) and λ = 2k−1βb 2k−1 + 1 in Corollary 11, we get Theorem 3.1 of [25]. Corollary 12. For a, b ∈ R− {0};β ∈ C; u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) un−rvr+kP (m) n−r,β (vx; k, a, b) r∑ l=0 ( r l ) Sl ( u− 1; ( β a )b) P (m−1) r−l,β (uz; k, a, b) = n∑ r=0 ( n r ) vn−rur+kP (m) n−r,β (ux; k, a, b) r∑ l=0 ( r l ) Sl ( v − 1; ( β a )b) P (m−1) r−l,β (vz; k, a, b) . Theorem 6. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ v u i logc (b/a) , y; a, b, c;λ ) F (m) n−r,k ( uz + u v j logc (b/a) , y; a, b, c;λ ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ u v i logc (b/a) , y; a, b, c;λ ) F (m) n−r,k ( vz + v u j logc (b/a) , y; a, b, c;λ ) . N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 600 Proof: Consider H(t) = t2kma−t(u+v)mcuvxt ( −(λy)u ( b a )uvt + (y + 1)u )( −(λy)v ( b a )uvt + (y + 1)v ) cuvzt( 1− y ( λ ( b a )ut − 1 ))m+1 ( 1− y ( λ ( b a )vt − 1 ))m+1 . Expanding H(t) into series, we have H(t) = (y + 1)u−1(y + 1)v−1 (uv)km  a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cvx(ut)  ( λy y+1 )u ( b a )uvt − 1 λy y+1 ( b a )vt − 1  ×  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cuz(vt)  ( λy y+1 )v ( b a )uvt − 1 λy y+1 ( b a )ut − 1  . = (y + 1)u−1(y + 1)v−1 (uv)km u−1∑ i=0 ( λy y + 1 )i( b a )ivt a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cvx(ut) × v−1∑ j=0 ( λy y + 1 )j ( b a )jut a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cuz(vt) = (y + 1)u+v−2 (uv)km ∞∑ n=0  n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ v u i logc (b/a) , y; a, b, c;λ ) × F (m) n−r,k ( uz + u v j logc (b/a) , y; a, b, c;λ )] tn n! . (14) Similarly, H(t) = (y + 1)u−1(y + 1)v−1 (uv)km  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cux(vt)  ( λy y+1 )v ( b a )uvt − 1 λy y+1 ( b a )ut − 1  ×  a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cvz(ut)  ( λy y+1 )u ( b a )uvt − 1 λy y+1 ( b a )vt − 1  . = (y + 1)u−1(y + 1)v−1 (uv)km v−1∑ i=0 ( λy y + 1 )i( b a )iut a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cux(vt) × u−1∑ j=0 ( λy y + 1 )j ( b a )jvt a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cvz(ut) = (y + 1)u+v−2 (uv)km ∞∑ n=0  n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ u v i logc (b/a) , y; a, b, c;λ ) N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 601 × F (m) n−r,k ( vz + v u j logc (b/a) , y; a, b, c;λ )] tn n! . (15) Comparing (14) and (15), we get the desired result. Setting y = −1 2 and k = 0 in Theorem 6, we obtain the following corollary. Corollary 13. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (−λ)i+j urvn−rE(m) r ( vx+ v u i logc (b/a) ; a, b, c;λ ) E (m) n−r ( uz + u v j logc (b/a) ; a, b, c;λ ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (−λ)i+j vrun−rE(m) r ( ux+ u v i logc (b/a) ; a, b, c;λ ) E (m) n−r ( vz + v u j logc (b/a) ; a, b, c;λ ) . Setting y = −2, k = 0 and replacing λ by λ 2 in Theorem 6, we obtain the following corollary. Corollary 14. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (λ)i+j urvn−rB(m) r ( vx+ v u i logc (b/a) ; a, b, c;λ ) B (m) n−r ( uz + u v j logc (b/a) ; a, b, c;λ ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (λ)i+j vrun−rB(m) r ( ux+ u v i logc (b/a) ; a, b, c;λ ) B (m) n−r ( vz + v u j logc (b/a) ; a, b, c;λ ) . Setting y = −1 2 and k = 1 in Theorem 6, we obtain the following corollary. Corollary 15. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (−λ)i+j urvn−rG(m) r ( vx+ v u i logc (b/a) ; a, b, c;λ ) G (m) n−r ( uz + u v j logc (b/a) ; a, b, c;λ ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (−λ)i+j vrun−rG(m) r ( ux+ u v i logc (b/a) ; a, b, c;λ ) G (m) n−r ( vz + v u j logc (b/a) ; a, b, c;λ ) . Setting a = 1 and b = c = e in Theorem 6, we obtain another symmetry identity for the higher order bivariate Fubini-type polynomials F (α) n,k (x, y;λ). N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 602 Corollary 16. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ v u i, y;λ ) F (m) n−r,k ( uz + u v j, y;λ ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ u v i, y;λ ) F (m) n−r,k ( vz + v u j, y;λ ) . Taking y = −(2k−1ab+1) and λ = 2k−1βb 2k−1 + 1 in Corollary 16, we get Theorem 3.5 of [25]. Corollary 17. For a, b > 0;β ∈ C; u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( β a )b(i+j) urvn−rP (m) r,β ( vx+ v u i; k, a, b ) P (m) n−r,β ( uz + u v j; k, a, b ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( β a )b(i+j) vrun−rP (m) r,β ( ux+ u v i; k, a, b ) p (m) n−r,k ( vz + v u j; k, a, b ) . Theorem 7. For u, v,m ∈ N, n ∈ N0 and y 6= −1, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ ( i v u + j ) logc(b/a), y; a, b, c;λ ) F (m) n−r,k (uz, y; a, b, c;λ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ ( i u v + j ) logc(b/a), y; a, b, c;λ ) F (m) n−r,k (vz, y; a, b, c;λ) . Proof: Consider L(t) = t2kma−t(u+v)mcuvxt ( −(λy)u ( b a )uvt + (y + 1)u )( −(λy)v ( b a )uvt + (y + 1)v ) cuvzt( 1− y ( λ ( b a )ut − 1 ))m+1 ( 1− y ( λ ( b a )vt − 1 ))m+1 . Expanding L(t) into a series, we get L(t) = (y + 1)u−1(y + 1)v−1 (uv)km  a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m cuvxtt  ( λy y+1 )u ( b a )uvt − 1 λy y+1 ( b a )vt − 1  ×  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cuvzt  ( λy y+1 )v ( b a )uvt − 1 λy y+1 ( b a )ut − 1  . = (y + 1)u−1(y + 1)v−1 (uv)km u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j ( b a )(iv+ju)t cvx(ut)  a−ut(ut)k 1− y ( λ ( b a )ut − 1 ) m N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 603 ×  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cuz(vt) = (y + 1)u−1(y + 1)v−1 (uv)km u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j ∞∑ n=0 F (m) n,k ( vx+ ( i v u + j ) logc(b/a), y; a, b, c;λ ) (ut)n n!  × ∞∑ n=0 F (m) n,k (uz, y; a, b, c;λ) (vt)n n! = (y + 1)u+v−2 (uv)km ∞∑ n=0  n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ ( i v u + j ) logc(b/a), y; a, b, c;λ ) × F (m) n−r,k (uz, y; a, b, c;λ) ] tn n! . (16) Similarly, L(t) = (y + 1)u−1(y + 1)v−1 (uv)km  a−vt(vt)k 1− y ( λ ( b a )vt − 1 ) m cux(vt)  ( λy y+1 )v ( b a )uvt − 1 λy y+1 ( b a )ut − 1  ×  ( λy y+1 )u ( b a )uvt − 1 λy y+1 ( b a )vt − 1  a−ut(ut)k 1− y ( λ ( b a )vt − 1 ) m cvz(ut). = (y + 1)u+v−2 (uv)km ∞∑ n=0  n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ ( i u v + j ) logc(b/a), y; a, b, c;λ ) × F (m) n−r,k (vz, y; a, b, c;λ) ] tn n! . (17) Combining (16) and (17) gives the desired identity. Setting y = −1 2 and k = 0 in Theorem 7, we obtain the following corollary. Corollary 18. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (−λ)i+j urvn−rE(m) r ( vx+ ( i v u + j ) logc(b/a); a, b, c;λ ) E (m) n−r (uz; a, b, c;λ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (−λ)i+j vrun−rE(m) r ( ux+ ( i u v + j ) logc(b/a); a, b, c;λ ) E (m) n−r (vz; a, b, c;λ) . Setting y = −2, k = 1 and replacing λ by λ 2 in Theorem 7, we obtain the following corollary. N. G. Acala / Eur. J. Pure Appl. Math, 13 (3) (2020), 587-607 604 Corollary 19. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (λ)i+j urvn−rB(m) r ( vx+ ( i v u + j ) logc(b/a); a, b, c;λ ) B (m) n−r (uz; a, b, c;λ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (λ)i+j vrun−rB(m) r ( ux+ ( i u v + j ) logc(b/a); a, b, c;λ ) B (m) n−r (vz; a, b, c;λ) . Setting y = −1 2 and k = 1 in Theorem 7, we obtain the following corollary. Corollary 20. For u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 (−λ)i+j urvn−rG(m) r ( vx+ ( i v u + j ) logc(b/a); a, b, c;λ ) G (m) n−r (uz; a, b, c;λ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 (−λ)i+j vrun−rG(m) r ( ux+ ( i u v + j ) logc(b/a); a, b, c;λ ) G (m) n−r (vz; a, b, c;λ) . Setting a = 1 and b = c = e in Theorem 7, we obtain another symmetry identity for the polynomials F (α) n,k (x, y;λ). Corollary 21. For u, v,m ∈ N, n ∈ N0 and y 6= −1, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( λy y + 1 )i+j urvn−rF (m) r,k ( vx+ i v u + j, y;λ ) F (m) n−r,k (uz, y;λ) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( λy y + 1 )i+j vrun−rF (m) r,k ( ux+ i u v + j, y;λ ) F (m) n−r,k (vz, y;λ) . Taking y = −(2k−1ab+1) and λ = 2k−1βb 2k−1 + 1 in Corollary 21, we get Theorem 3.9 of [25]. Corollary 22. For a, b > 0;β ∈ C; u, v,m ∈ N and n ∈ N0, we have n∑ r=0 ( n r ) u−1∑ i=0 v−1∑ j=0 ( β a )b(i+j) urvn−rP (m) r,β ( vx+ i v u + j; k, a, b ) P (m) n−r,β (uz; k, a, b) = n∑ r=0 ( n r ) v−1∑ i=0 u−1∑ j=0 ( β a )b(i+j) vrun−rP (m) r,β ( ux+ i u v + j; k, a, b ) P (m) n−r,k (vz; k, a, b) . Acknowledgements The author greatly appreciates the anonymous reviewers for their valuable comments and suggestions for this paper. REFERENCES 605 References [1] S. Araci, W.A. Khan, M. Acikgoz, C. Ozel, and P. Kumam. A new generalization of Apostol-type Hermite-Genocchi polynomials and its application. 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