# A Note onq-Fubini-Appell Polynomials and Related Properties

Authors: Abdulghani Muhyi; Serkan Araci
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The present article is aimed at introducing and investigating a new class ofq-hybrid special polynomials, namely, q-Fubini-Appell polynomials. The generating functions, series representations, and certain other significant relations and identities of this class are established. Some members of q-Fubini-Appell polynomial family are investigated, and some properties of these members are obtained. Further, the class of 3-variable q-Fubini-Appell polynomials is also introduced, and some formulae related to this class are obtained. In addition, the determinant representations for these classes are established.


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## Body

## 1. Introduction

Theq-calculus subject has gained prominence and numerous popularity during the last three decades or so (see [1–4]). The contemporaneous interest in this subject is due to the fact that q-series has popped in such diverse fields as quantum groups, statistical mechanics, and transcendental number theory. The notations and definitions related to q-calculus used in this article are taken from [2] (see also [5, 6]).Theq-analogues of a number ℓ∈ℂ and the factorial function are, respectively, specified by
(1)ℓq=1−qℓ1−q,q∈ℂ\1,and
(2)κq!=∏l=1κlq=1q2q3q⋯κq,0q!=1,κ∈ℕ,q∈ℂ\0,1.Theq-binomial coefficient κlq is specified by
(3)κlq=κq!lq!κ−lq!,l=0,1,2,⋯,κ;κ∈ℕ0.Theq-analogue of u⊕vκ is specified as
(4)u⊕vqκ=∑l=0κκlqqκ−l2ulvκ−l.Theq-derivative of a function f at a point τ∈ℂ\0 is given as
(5)Dqfτ=fτ−fqττ−qτ,0<∣q∣<1.The functions(6)eqτ=∑κ=0∞τκκq!,0<∣q∣<1,∣τ∣<1−q−1,(7)Eqτ=∑κ=0∞qκ2τκκq!,0<q<1,τ∈ℂ,are called q-exponential functions and satisfy the following identities:
(8)Dqeqτ=eqτ,DqEqτ=Eqqτ,eqτEq−τ=Eqτeq−τ=1.The Fubini polynomials (FP)Fκw [7] (also known as geometric polynomials) are defined as
(9)11−weτ−1=∑κ=0∞Fκwτκκ!,together with the geometric series
(10)11−wFmw1−w=∑l=0∞lmwl,∣w∣<1.Recently, Duran et al. [8] introduced the q-analogue of the FP Fκw, denoted by Fκ,qw and defined by means of the generating function
(11)11−weqτ−1=∑κ=0∞Fκ,qwτκκq!.Forw=1, the q-Fubini polynomials (q-FP) Fκ,qw reduce to the q-Fubini numbers Fκ,q1≔Fκ,q, that is
(12)12−eqτ=∑κ=0∞Fκ,qτκκq!.Further, we recall the 3-variableq-Fubini polynomials (3Vq-FP) Fκ,qu,v,w [8] which are given as
(13)11−weqτ−1equτEqvτ=∑κ=0∞Fκ,qu,v,wτκκq!.Substantial properties of Fubini numbers and polynomials and theirq-analogue have been studied and investigated by many researchers (see [7–9] and the references cited therein). Further, these numbers and polynomials have enormous applications in analytic number theory, physics, and the other related areas.The class of theq-special polynomials such as q-Fubini polynomials, q-Appell polynomials, and certain members belonging to the family of q-Appell polynomials such as q-Bernoulli polynomials and q-Euler polynomials is an expanding field in mathematics [3, 7, 8, 10, 11].The class ofq-Appell polynomial sequences Aκ,qwκ=0∞ was established and investigated by Al-Salam [1]. These polynomials are defined by means of the generating function
(14)Aqτeqwτ=∑κ=0∞Aκ,qwτκκq!,where
(15)Aqτ=∑κ=0∞Aκ,qτκκq!,Aqτ≠0;A0,q=1,is an analytic function at τ=0 and Aκ,q≔Aκ,q0 denotes the q-Appell numbers.Certain significant members belonging toq-Appell polynomials class are obtained based on suitable selection for the function Aqτ as
(1)
IfAqτ=τ/eqτ−1, the q-AP Aκ,qw reduce to the q-Bernoulli polynomials (q-BP) Bκ,qw (see [12, 13]), that is(16)Aκ,qw≔Bκ,qw,where Bκ,qw are defined by
(17)τeqτ−1eqwτ=∑κ=0∞Bκ,qwτκκq!,and Bκ,q given by
(18)Bκ,q≔Bκ,q0,denotes the q-Bernoulli numbers.
(2)
IfAqτ=2/eqτ+1, the q-AP Aκ,qw reduce to the q-Euler polynomials (q-EP) Eκ,qw (see [13, 14]), that is(19)Aκ,qw≔Eκ,qw,where Eκ,qw are defined by
(20)2eqτ+1eqwτ=∑κ=0∞Eκ,qwτκκq!,and Eκ,q given by
(21)Eκ,q≔Eκ,q0,denotes the q-Euler numbers.Also, we recall the family of the numbers denoted byS2,qκ,l and defined by
(22)eqτ−1llq!=∑κ=l∞S2,qκ,lτκκq!.In recent years, many authors have shown their interest to introduce and study new families ofq-special polynomials, especially the hybrid type (see [15–17] and the references therein).The work in this article is summarized as follows: in Section2, the replacement technique is used to introduce the class of q-Fubini-Appell polynomials by combining the polynomials, q-Fubini polynomials and q-Appell polynomials. In Section 3, the 3-variable q-Fubini-Appell polynomials are introduced which are considered as a generalization of the q-Fubini-Appell polynomials. The generating relations, series representations, and some other useful properties related to these polynomials are established. In Section 4, the determinant representations of these two classes are defined. Further, certain members belonging to these polynomial families are considered, and the corresponding results are also derived.


## 2.q-Fubini-Appell Polynomials

Theq-Fubini-Appell polynomials are established by means of the generating function and series representation. To achieve this, we prove the following results:Theorem 1.
Theq-Fubini-Appell polynomials (q-FAP) FAκ,qw are defined by means of the following generating function:
(23)Aqτ1−weqτ−1=∑κ=0∞FAκ,qwτκκq!.Proof.
Utilizing equation (14), based on expanding the function eqwτ, then replacing the powers of w, i.e., w0,w,w2,⋯,wκ by the corresponding polynomials F0,qw,F1,qw,⋯,Fκ,qw and thereafter summing up the terms in the left-hand side of the resulting equation, we obtain that
(24)Aqτ∑κ=0∞Fκ,qwτκκq!=∑κ=0∞Aκ,qF1,qwτκκq!.
Now, denoting the resultantq-FAP in the right hand side of the above equation by FAκ,qw and utilizing equation (11) yield the assertion in equation (23).Remark 2.
Takingw=1, the q-FAP FAκ,qw reduce to q-Fubini-Appell numbers (q-FAN) FAκ,q. Therefore, in view of equation (23), we have
(25)Aqτ2−eqτ=∑κ=0∞FAκ,qτκκq!.Corollary 3.
TakingAqτ=τ/eqτ−1 in equation (23), we get the following generating function of the q-Fubini-Bernoulli polynomials (q-FBP) FBκ,qw.(26)τeqτ−11−weqτ−1=∑κ=0∞FBκ,qwτκκq!.Corollary 4.
TakingAqτ=2/eqτ+1 in equation (23), we get the following generating function of the q-Fubini-Euler polynomials (q-FEP) FEκ,qw(27)2eqτ+11−weqτ−1=∑κ=0∞FEκ,qwτκκq!.Theorem 5.
The following series representation for theq-FAP FAκ,qw holds true:
(28)FAκ,qw=∑l=0κκlqAl,qFκ−l,qw.Proof.
In view of equations (11) and (15) and equation (23), we have
(29)∑κ=0∞FAκ,qwτκκq!=Aqτ1−weqτ−1=∑κ=0∞Aκ,qτκκq!∑κ=0∞Fκ,qwτκκq!=∑κ=0∞∑l=0κκlqAl,qFκ−l,qwτκκq!,which on comparing the coefficients of τκ/κq! yield assertion in equation (28).Theorem 6.
Forn∈ℕ0, the following series representation for the q-FAP FAκ,qw holds true:
(30)FAκ,qw=∑l=0κ∑σ=0lκlqσq!wσAκ−l,qS2,ql,σ.Proof.
In view of equations (15), (22), and (23), we can write
(31)∑κ=0∞FAκ,qwτκκq!=Aqτ1−weqτ−1=∑κ=0∞Aκ,qτκκq!∑σ=0∞wσeqτ−1σ=∑κ=0∞Aκ,qτκκq!∑σ=0∞wσσq!∑l=σ∞S2,ql,στllq!=∑κ=0∞Aκ,qτκκq!∑l=0∞∑σ=0lwσσq!S2,ql,στllq!=∑κ=0∞∑l=0κκlqAκ−l,q∑σ=0lwσσq!S2,ql,στκκq!,which on comparing the coefficients of τκ/κq! yield assertion in equation (30).Corollary 7.
TakingAqτ=τ/eqτ−1 in equations (28) and (30), we get the following series representations of the q-FBP FBκ,qw(32)FBκ,qw=∑l=0κκlqBl,qFκ−l,qw,FBκ,qw=∑l=0κ∑σ=0lκlqσq!wσBκ−l,qS2,ql,σ,n∈ℕ0.Corollary 8.
TakingAqτ=2/eqτ+1 in equations (28) and (30), we get the following series representations of the q-FEP FEκ,qw:
(33)FEκ,qw=∑l=0κκlqEl,qFκ−l,qw,FEκ,qw=∑l=0κ∑σ=0lκlqσq!wσEκ−l,qS2,ql,σ,n∈ℕ0.Theorem 9.
The following formula for theq-FAP FAκ,qw holds true:
(34)ddwFAκ,qw=∑l=0κκlqFAκ−l,qwFl,q1,0,w−FAl,qwFκ−l,qw.Proof.
Utilizing equation (23), we have
(35)ddw∑κ=0∞FAκ,qwτκκq!=ddwAqτ1−weqτ−1Aqτ1−weqτ−1=Aqτeqτ−11−weqτ−12=Aqτeqτ1−weqτ−12−Aqτ1−weqτ−12=∑κ=0∞∑l=0κκlqFAκ−l,qwFl,q1,0,wτκκq!−∑κ=0∞∑l=0κκlqFAl,qwFκ−l,qwτhhq!,which on equating the coefficients of the like powers of τ yields the assertion in equation (34).Corollary 10.
TakingAqτ=τ/eqτ−1 in equations (34), we get the formula satisfied by the q-FBP FBκ,qw as
(36)ddwFBκ,qw=∑l=0κκlqFBκ−l,qwFl,q1,0,w−FBl,qwFκ−l,qw.Corollary 11.
TakingAqτ=2/eqτ+1 in equations (34), we get the formula satisfied by the q-FEP FEκ,qw as
(37)ddwFEκ,qw=∑l=0κκlqFEκ−l,qwFl,q1,0,w−FEl,qwFκ−l,qw.


## 3. 3-Variableq-Fubini-Appell Polynomials

In this section, the class of 3-variableq-Fubini-Appell polynomials is established, which is a generalization of the class introduced in the previous section. The generating function, series representations, and other formulae for these polynomials are obtained.Theorem 12.
The 3-variableq-Fubini-Appell polynomials (3Vq-FAP) FAκ,qu,v,w are defined by means of the following generating function:
(38)Aqτ1−weqτ−1equτEqvτ=∑κ=0∞FAκ,qu,v,wτκκq!.Proof.
Utilizing equations (13) and (14) and following the same method as in the proof of Theorem 1, we can get the assertion in equation (38).Remark 13.
Settingw=0 in equation (38) gives the generating function of the 2-variable q-Appell polynomials (2Vq-AP) Aκ,qu,v [18], that is
(39)AqτequτEqvτ=∑κ=0∞Aκ,qu,vτκκq!.Corollary 14.
TakingAqτ=τ/eqτ−1 in equation (38), we get the generating function of the 3-variable q-Fubini-Bernoulli polynomials (3Vq-FBP) FBκ,qu,v,w as
(40)τeqτ−11−weqτ−1equτEqvτ=∑κ=0∞FBκ,qu,v,wτκκq!.Corollary 15.
TakingAqτ=2/eqτ+1 in equation (38), we get the generating function of the 3-variable q-Fubini-Euler polynomials (3Vq-FEP) FEκ,qu,v,w as
(41)2eqτ+11−weqτ−1equτEqvτ=∑κ=0∞FEκ,qu,v,wτκκq!.Theorem 16.
The 3Vq-FAPFAκ,qu,v,w are defined by the series
(42)FAκ,qu,v,w=∑l=0κκlqAl,qFκ−l,qu,v,w.Proof.
In view of equations (13), (15), and (38), we have
(43)∑κ=0∞FAκ,qu,v,wτκκq!=Aqτ1−weqτ−1equτEqvτ=∑κ=0∞Aκ,qτκκq!∑κ=0∞Fκ,qu,v,wτκκq!=∑κ=0∞∑l=0κκlqAl,qFκ−l,qu,v,wτκκq!,which on comparing the coefficients of τκ/κq! yield assertion in equation (42).Corollary 17.
TakingAqτ=τ/eqτ−1 in equation (42), we get the series representation of the 33Vq-FBP FBκ,qu,v,w as
(44)FBκ,qu,v,w=∑l=0κκlqBl,qFκ−l,qu,v,w.Corollary 18.
TakingAqτ=2/eqτ+1 in equation (42), we get the series representation of the 33Vq-FEP FEκ,qu,v,w as
(45)FEκ,qu,v,w=∑l=0κκlqEl,qFκ−l,qu,v,w.Suitably using equations (4), (6), (7), (11), and (23) in generating relation (38) and then making use of the Cauchy product rule in the resultant relations and thereafter comparing the identical powers of τ in both sides of the resultant expressions, we get the formulae given in the following theorem.Theorem 19.
The 3Vq-FAPFAκ,qu,v,w satisfy the following formulae
(46)FAκ,qu,v,w=∑l=0κκlqFAκ−1,qwu⊕vql,FAκ,qu,v,w=∑l=0κκlqqκ−l2FA1,qu,0,wvκ−l,FAκ,qu,v,w=∑l=0κκlqF1,qwFAκ−1,qu,v,0,FAκ,qu,v,w=∑l=0κκlqFA1,q0,v,wuκ−l.Applying theq-derivatives w.r.t. u and v to generating relation (38), we get the results given in the following theorem.Theorem 20.
The following identities for the 3Vq-FAPFAκ,qu,v,w hold true:
(47)Dq,uFAκ,qu,v,w=κqFAκ−1,qu,v,w,Dq,uξFAκ,qu,v,w=κq!κ−ξq!FAκ−ξ,qu,v,w,Dq,vFAκ,qu,v,w=κqFAκ−1,qu,qv,w,Dq,vξFAκ,qu,v,w=κq!κ−ξq!qξ2FAκ−ξ,qu,qξv,w.Theorem 21.
The following relation for the 3Vq-FAPFAκ,qu,v,w holds true:
(48)∑l=0κκlqFAκ−l,qu,v,w=1ww+1FAκ,qu,v,w−Aκ,qu,v.Proof.
Consider the identity(49)wequτEqvτ1−weqτ−1eqτ=1+wequτEqvτ1−weqτ−1−equτEqvτ.
Now, multiplying both sides of the above identity byAκ,qτ and using equations (6), (38), and (39), we get
(50)w∑κ=0∞∑l=0κκlqFAκ−l,qu,v,wτκκq!=∑κ=0∞1+wFAκ,qu,v,w−Aκ,qu,vτκκq!,which on equating the coefficients of the like powers of τ yields the assertion in equation (48).Now, let us recall the generating function of the 2-variableq-generalized tangent polynomials (2Vq-GTP) Cκ,α,qu,v [19] given as
(51)2eqατ+1equτEqvτ=∑κ=0∞Cκ,α,qu,vτκκq!,ατ<π,α∈ℝ+,and Cκ,α,q≔Cκ,α,q0,0 denotes the q-generalized tangent numbers (q-GTN).Theorem 22.
The following relationships between the 3Vq-FAPFAκ,qu,v,w and 2Vq-GTP Cκ,α,qu,v holds true:
(52)FAκ,qu,v,w=12∑l=0κκlq∑σ=0llσqασCκ−l,α,qu,vFl−σ,qw+FAκ−l,qu,v,wCl,α,q.Proof.
Utilizing equations (23), (38), and (51), we have
(53)∑κ=0∞FAκ,qu,v,wτκκq!=Aqτ1−weqτ−1equτEqvτ=Aqτ1−weqτ−12eqατ+1eqατ+12equτEqvτ=12∑κ=0∞∑σ=0κκσqασFκ−σ,qwτκκq!∑κ=0∞Cκ,α,qu,vτκκq!+∑e^=0∞FAκ,qu,v,wτκκq!∑κ=0∞Cκ,α,qτκκq!=12∑κ=0∞∑l=0κκlq∑σ=0lκσqασFl−σ,qwCκ−l,α,qu,v+∑l=0κκlqFAκ−1,qu,v,wCl,α,qτκκq!,which on comparing the coefficients of τκ/κq! yield assertion in equation (52).Since forα=1, the 2-variable q-generalized tangent polynomials (2Vq-GTP) Cκ,α,qu,v reduce to 2-variable q-Euler polynomials Eκ,qu,v [20]. Therefore, setting α=1 in equation (52) gives the following result.Corollary 23.
The following relationships between the 3Vq-FAPFAκ,qu,v,w and 2Vq-EP Eκ,qu,v holds true:
(54)FAκ,qu,v,w=12∑l=0κκlq∑σ=0llσqEκ−l,α,qu,vFl−σ,qw+FAκ−l,qu,v,wEl,α,q.Let us recall the generating function of the 2-variableq-Euler-Bernoulli polynomials (2Vq-EBP) EBκ,qu,v [16] given by
(55)2τeqτ+1eqτ−1equτEqvτ=∑κ=0∞EBκ,qu,vtκκq!.Theorem 24.
The following relationships between the 3Vq-FAPFAκ,qu,v,w and 2Vq-EBP EBκ,qu,v holds true:
(56)FAκ−1,qu,v,w=12kq∑l=0κκlq∑σ=0κ∑h=0lκ−lσqlhqFAκ−σ−l,qwEBl−h,qu,v−FAκ−l,qu,v,wEBl,q.Proof.
Utilizing equations (6), (38), and (55), we have
(57)∑κ=0∞FAκ,qu,v,wτκκq!=Aqτ1−weqτ−1equτEqvτ=Aqτ1−weqτ−12teqτ+1eqτ−1eqτ+1eqτ−12tequτEqvτ=12tAqτ1−weqτ−1eqτ2tequτEqvτeqτ+1eqτ−1eqτ−AqτequτEqvτ1−weqτ−12teqτ+1eqτ−1=12t∑κ=0∞∑σ=0κκσqFAκ−σ,qwτκκq!∑l=0∞∑h=0llhqεBl−h,qu,vτllq!−∑κ=0∞∑l=0κκlqFAκ−l,qu,v,wEBl,qτκκq!=12t∑κ=0∞∑l=0κ∑σ=0κ∑h=0lκlqκ−lσqlhqFAκ−l,qwEBl−h,qu,v−∑l=0κκlqFAκ−l,qu,v,wEBl,qτκκq!,which on comparing the coefficients of τκ/κq! yield assertion in equation (56).Theorem 25.
The following relationships between the 3Vq-FAPFAκ,qu,v,w and 2Vq-AP Aκ,qu,v holds true:
(58)∑l=0κκlqFAκ−l,qu,v,w1−w=1wFAκ,qu,v,w1−w−1−wAκ,qu,v.Proof.
Replacingw by w/1−w in generating relation (38), we have
(59)∑κ=0∞FAκ,qu,v,w1−wτκκq!=Aqτ1−w/1−weqτ−1equτEqvτ.
Rewriting the above equation then using equations (38) and (39), we obtain
(60)∑κ=0∞FAκ,qu,v,w1−wτκκq!−w∑κ=0∞∑l=0κκlqFAκ−l,qu,v,w1−wτκκq!=1−w∑κ=0∞Aκ,qu,vτκκq!.which on comparing the coefficients of τκ/κq! yield assertion in equation (58).


## 4. Determinant Representations

One of the significant representations of theq-special polynomials is the determinant representation due to its importance for the computational and applied purposes. In 2015, Keleshteri and Mahmudov [18] established the determinant representation of the q-Appell polynomials. In the section, the determinant representations of the q-FAP FAκ,qw and the 3Vq-FAP FAκ,qu,v,w are introduced.Definition 26.
The determinant representation for theq-FAP FAκ,qw of degree κ is given as
(61)FA0,qw=1B0,q,(62)FAκ,qw1F1,qF2,qw⋯Fκ−1,qwFκ,qwB0,qB1,qB2,q⋯Bκ−1,qBκ,q0B0,q21qB1,q⋯κ−11qBκ−2,qκ1qBκ−1,q00B0,q⋯κ−12qBκ−3,qκ2qBκ−2,q⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯B0,qκκ−1qB1,q,(63)Bκ,q=−1FA0,q∑ν=1κκνFAν,qBκ−ν,q,B0,q≠0,κ=1,2,3,⋯.SettingB0,q=1 and Bδ,q=1/δ+1qδ=1,2,⋯,κ in equations (61) and (62) gives the determinant representation of the q-FBP FBκ,qw as:Definition 27.
The determinant representation for theq-FBPFBκ,qw of degree κ is given as
(64)FB0,qw=1,FBκ,qw=−1κ1F1,qwF2,qw⋯Fκ−1,qwFκ,qw112q13q⋯1κq1κ+1q0121q12q⋯κ−11q1κ−1qκ1q1κq001⋯κ−12q1κ−2qκ2q1κ−1q⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯1κκ−1q12q,κ=1,2⋯.SettingB0,q=1 and Bδ,q=1/2δ=1,2,⋯,κ in equations (61) and (62) gives the determinant representation of the q-FEP FEκ,qw as:Definition 28.
The determinant representation for theq-FEP FEκ,qw of degree κ is given as
(65)Fε0,qw=1,Fεκ,qw=−1κ1F1,qwF2,qw⋯Fκ−1,qwFκ,qw11212⋯12120121q12⋯κ−11q12κ1q12001⋯κ−12q12κ2q12⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯1κκ−1q12,κ=1,2,3,⋯.Similarly, the determinant representation of the 3Vq-FAPFAκ,qu,v,w, 3Vq-FBP FBκ,qu,v,w, and 3Vq-FEP FEκ,qu,v,w are established as:Definition 29.
The determinant representation for the 3Vq-FAPFAκ,qu,v,w of degree κ is given as
(66)FA0,qu,v,w=1B0,q,FAκ,qu,v,w=−1κB0,qκ+11F1,qu,v,wF2,qu,v,w⋯Fκ−1,qu,v,wFκ,qu,v,wB0,qB1,qB2,q⋯Bκ−1,qBκ,q0B0,q21qB1,q⋯κ−11qBκ−2,qκ1qBκ−1,q00B0,q⋯κ−12qBκ−3,qκ2qBκ−2,q⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯B0,qκκ−1B1,q,Bκ,q=−1FA0,q∑ν=1κκνqFAν,qBκ−ν,q,B0,q≠0,κ=1,2,3,⋯.Definition 30.
The determinant representation for the 3Vq-FBPFBκ,qu,v,w of degree κ is given as
(67)FB0,qu,v,w=1,FB0,qu,v,w=−1κ1F1,qu,v,wF2,qu,v,w⋯Fκ−1,qu,v,wFκ,qu,v,w1⋯1κq1κ+1q0121q12q⋯κ−11q1κ−1qκ1q1κq001⋯κ−12q1κ−2qκ2q1κ−2q⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯1κκ−1q12q,κ=1,2⋯.Definition 31.
The determinant representation for the 3Vq-FEPFEκ,qu,v,w of degree κ is given as
(68)FE0,qu,v,w=1,Fεκ,qu,v,w=−1κ1F1,qu,v,wF2,qu,v,w⋯Fκ−1,qu,v,wFκ,qu,v,w11212⋯12120121q12⋯κ−11q12κ1q12001⋯κ−12q12κ2q12⋅⋅⋅⋯⋅⋅⋅⋅⋅⋯⋅⋅000⋯1κκ−1q12,κ=1,2,3⋯.


## 5. Conclusions

Recently, the Fubini polynomials and theirq-analogue have been studied and investigated by many researchers. Motivated by various recent studies related to these type of polynomials (see for example [8, 21, 22]), in this article, we introduced two important families of q-hybrid special polynomials, namely, the q-Fubini-Appell polynomials and 3-variable q-Fubini-Appell polynomials. Certain properties related to these families are derived.Further investigations along the results obtained in this article, which are associated with many recent generalizations and extensions of theq-Appell polynomial family, especially, the parametric types, may be worthy of consideration in future investigations.



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