# Generalized Fubini Apostol-Type Polynomials and Probabilistic Applications

Authors: Rabab S. Gomaa; Alia M. Magar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

The paper aims to introduce and investigate a new class of generalized Fubini-type polynomials. The generating functions, special cases, and properties are introduced. Using the generating functions, various interesting identities, and relations are derived. Also, special polynomials are obtained from the general class of polynomials. Finally, probabilistic applications and some probabilistic properties are obtained.


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## Body

## 1. Introduction

In recent years, Fubini-type polynomials have gained attention not only in combinatorial mathematics but also in applications to differential equations and probability theory; so many authors have studied its properties and obtained important results [1–4]. For example, the classical Fubini polynomials Fny are defined by (see [4]).(1)Fny=∑k=0nS2n,kk!yk,where S2n,k is the Stirling numbers of the second kind [5], These polynomials satisfy the following generating function:(2)11−yet−1=∑n=0∞Fnytnn!.Puttingy=1 in (1), we obtain the nth Fubini number or the ordered Bell number Fn, defined by(3)Fn1=Fn=∑k=0nS2n,kk!.The bivariate Fubini polynomials of orderα are generalization of the classical Fubini polynomials and they are defined by the following generating function:(4)ext1−yet−1α=∑n=0∞Fnαx,ytnn!.Puttingα=1, Fnαx,y:=Fnx,y, the two-variable Fubini polynomials are obtained by(5)ext1−yet−1=∑n=0∞Fnx,ytnn!.Also, puttingx=0 in (4), we obtain(6)Fnα0,y:=Fnαy,Fnα1:=Fnα,where Fnαy and Fnα are called the higher order Fubini polynomials and the higher order Fubini numbers, respectively, (see [6, 7]). Also, Kilar and Simsek [2] obtained serval 2-variable special polynomials from the family of 2-variable general polynomials pnx,y, for more details see ([8–11]).Additionally, for anyλ∈R, the degenerate exponential functions are defined by(7)eλxt=1+λtx/λ,eλ1t=eλt=1+λt1/λ,From (7), we get(8)eλxt=∑n=0∞xn,λn!tn,where(9)x0,λ=1,xn,λ=xx−λ…x−n−1λ,n≥1.The degenerate Stirling numbers of the first kind are defined by(10)xn=∑k=0nS1,λn,kxk,λ,1k!logλ1+tk=∑n=k∞S1,λn,ktnn!,and the degenerate Stirling numbers of the second kind are given by(11)xn=∑k=0nS2,λn,kxk,1k!eλt−1k=∑n=k∞S2,λn,ktnn!.The Bell polynomials are defined by the following equation:(12)exet−1=∑n=0∞Belnxtnn!.Puttingx=1, Beln=Beln1 are called the Bell numbers. From (11), we get(13)Belnx=∑k=0nS2n,kxk.Also, the degenerate Bell polynomials are defined by (see [13]).(14)exeλt−1=∑n=0∞Beln,λxtnn!.In recent years, a lot of research on multivariable and generalized polynomials has been introduced. So, we recall the following 2-variable form of special polynomials.The 2-variable general polynomials (2VGP)pnx,y [14] are defined by(15)extφy,t=∑n=0∞pnx,ytnn!,p0x,y=1,where(16)φy,t=∑n=0∞φnytnn!,φ0y≠0.The 2VGP familypnx,y contains a number of significant 2-variable special polynomials. Based on suitable selection for the function φy,t, different members belonging to the family of 2-variable general polynomials pnx,y can be obtained, see Table 1.Table 1
Certain members of 2VGP familypnx,y.
Noφy,tGenerating functionPolynomials1.eytrext+ytr=∑n=0∞Hnrx,ytn/n!The Gould–Hopper polynomials [11]2.C0ytextC0yt=∑n=0∞Lnx,ytn/n!The 2-variable Laguerre polynomials [8]3.1/1−yts1/1−ytsext=∑n=0∞ensx,ytn/n!The 2-variable truncated exponential of orders [9]4.Ateyt2Atext+yt2=∑n=0∞AnsHx,ytn/n!The Hermite–Appell polynomials [10]5.eytC0−xt2eytC0−xt2=∑n=0∞Snx,ytn/n!The 2-variable Legendre polynomials [15]The aim of this paper is to study a new family of generalized Fubini-type polynomials and to obtain an application in probability theory. For this, we derive an explicit expression, new identities and relations with important polynomials and numbers. Furthermore, we introduce a new three-parameter discrete distribution which may be useful for modeling count data. Submodels such as Bell distribution and Bell–Touchard distribution are obtained, which have an important role in count data (for more details, see([16–18])).The article is organized as follows: In Section2, the generalized Fubini-type polynomials are introduced. The generating functions and some special cases for new families of polynomials are introduced. In Section 3, new definitions and certain important relations for new families of polynomials are derived. In Section 4, special polynomials related to the new families of polynomials are considered. In Section 5, the probabilistic application and some probabilistic properties are obtained.


## 2. Generalized Fubini Apostol-Type Polynomials

Definition 1.
The multivariable generalized polynomials are defined by(17)cxtϕz1,z2,…,zm,t=∑n=0∞pnϕx,z1,z2,…,zmtnn!,where(18)ϕz1,z2,…,zm,t=∑n=0∞ϕnz1,z2,…,zmtnn!.Definition 2.
Leta∈R+n∈N0 , m∈N,r∈ℂ;α¯=α0,α1,…,αr−1 and β¯=β0,β1,…,βr−1 be sequences of complex numbers, then the generalized Fubini Apostol-type polynomials are defined as the following:(19)∑n=0∞Fnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt;αiβicxtϕz1,z2,…,zm,t,where(20)ψt;α=∑n=0∞Pnαtnn!,P0α≠0.Remark 1.
Puttingx=0 and zj=0,j=1,2,…,m in (19), we obtain the new Fubini Apostol-type numbers, as shown in the following equation:(21)Fnr0,0,γ,μ;α¯r,β¯r=Fnrγ,μ;α¯r,β¯r.
Settingc=e and m=1 in (19), gives the following definition.Definition 3.
(22)∑n=0∞Fnrx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt;αiβiextϕz1,t.Remark 2.
If we puty=1 in equation (22), then, we obtain the new Apostol-type polynomials, as shown in the following equation:(23)∑n=0∞Fnrx,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt;αiβiextϕz1,t.
By setting appropriate values for parameters in (19) and (22), we obtain the generating functions and some results for the mixed special polynomials related to Fnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯ . The generating function and some definitions for these polynomials are introduced in Table 2.
The series definition ofFnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯ are obtained by the next theorem:Table 2
Special cases of theFnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯.
No.ParametersRelation between theFnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯ and its special casesSpecial polynomials1αi=λ,γ=μ=0,c=e, βi=a=zj=1,j=1,…,m and ψt,αi=αiet−1 in (19)Fnrx,y;1,0,0;λ,1=Fnrx,y;λHigher order bivariate Apostol-Fubini Polynomials [19]2αi=λ,γ=μ=0,c=e, βi=a=1, zj=1,j=1,…,m,x=0, and ψt,αi=αiet−1 in (19)Fnr0,y;1,0,0;λ,1=Fnry;λHigher order univariate Apostol-Fubini Polynomial [19]3αi=1,γ=μ=0,c=e,βi=azj=1,j=1,…,m and ψt,αi=αiet−1 in (19)Fnrx,y;1,0,0;1,1=Fnrx,yA generalization of the classical Fubini Polynomials [3, 20]4αi=λ,γ=k,μ=0,βi=1zj=1,j=1,…,m and ψt,αi=αibt−at in (19)Fnrx,y;1,k;0,1=Fnx,y;a,b;λUnified generalized polynomials of higher order [21]5y=−1andψt,αi=αibt−2at in (19)Fnrx,−1;z1,z2,…,zm,γ,μ;α¯,1=Mnrpx,z1,z2,…,zm,k,a,b;α¯rMultivariable unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomial [22]6γ=a=0,αi=μ=y=1, ϕz1,z2;t=ϕz1;tcosz2t and ψt,αi=et−1 in (19)Fnrx,1,z1,z2,0,1;1,2=Fnc,rGx,z1,z2Generalized fubini-Type polynomials, [23]7γ=k,αi=β,μ=0, a=1,βi=1,c=e, ϕz1,z2,…,zm;t=1 and ψt,αi=αibet−ab in (19)Fnrx,y;1;k,0,β,1=Fnrx,y;α,b;β,kUnified degenerate aApostol-type Bernoulli, Euler, Genocchi, and Fubini polynomials [24]8y=−1,j=1 and ψt,αi=αibt−2at in (19)Fnrx,−1;z1,γ,μ;α¯,1=Unrpx,z1,γ,μ,a,b;α¯rTwo-variable unified family of generalized Apostol-type polynomials of orderr see [25]9αi=λ,γ=μ=x=0βi=a=r=1,t=txϕz,t=1 and ψtx,λ=1/xeλtx−1 in (22)Fnr0,y;1,0,0;λ,1=Fn,λx,yThe degenerate generalized Fubini polynomials [26]10γ=1μ=x=0a=1ϕz,t=ezet−1 and ψt,αi=2−αiet in (22)Fnrx,z;1,0,α,1=BnrBx,z,αThe bell based apostol-bernoulli polynomials [27]Theorem 1.
The generalized Fubini Apostol-type polynomialsFnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯ is defined by the series:(24)Fnrx,y,z1,z2,…,zm,γ,μ;α¯,β¯=∑k=0nnkFnrγ,μ;α¯,β¯pkx,z1,z2,…,zm.Proof.
Using equation (19) and by using Cauchy-product rule, yields (2.8).


## 3. Identities and Relations

Theorem 2.
Leta,c∈ℝ and x,ℓ∈R. Then,(25)Fnrx+ℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncpn−kFkrℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯.Proof.
From (19)(26)∑n=0∞Fnrx+ℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!=trγ2rμ∏i=0r−1at−yψit;αiβicx+ℓtϕz1,z2,…,zm,t=∑k=0∞Fkrℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tkk!∑n=0∞txlncnn!,
by using Cauchy-product rule, yields (25).Corollary 1.
Settingℓ=0 in (25), we obtain the recurrence relation(27)Fnrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkry,z1,z2,…,zm,γ,μ;αr¯,βr¯,(28)=∑k=0nnn−kxklnckFn−kry,z1,z2,…,zm,γ,μ;αr¯,βr¯.Theorem 3.
Leta>0 , then, for x,z∈R and n≥0. The following identity holds true:(29)Fnrx+ℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯.Proof.
Changingx by x+ℓ in (19)(30)∑n=0∞Fnrx+ℓ,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβicx+ℓtϕz1,z2,…,zm,t=∑k=0∞Fkrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tkk!∑n=0∞tllncnn!,
by using Cauchy-product rule, yields (29).Theorem 4.
Forn,j∈ℕ0, we have the relationship(31)Fnrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑j=0nxj¯∑ℓ=jnnn−ℓlncℓSℓ,j,Fnry,z1,z2,…,zm,γ,μ;αr¯,βr¯,
between the generalized Fubini Apostol-type polynomials and Stirling number of second kind.Proof.
Using (28) and from definition of Stirling number of second kind, see [28, 29], we have(32)Fkrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑ℓ=0nnn−ℓlncℓ∑j=0lxℓ¯Sℓ,j,Fn−ℓry,z1,z2,…,zm,γ,μ;αr¯,βr¯.
By using Cauchy-product rule, yields (31).Theorem 5.
Forx,k∈R and n≥0. The following identity for Fnrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯ holds true:(33)Fn+ℓrk,y,z1,z2,…,zm,γ,μ;αr¯,βr¯=∑i,j=0n,ℓniℓjlnci+jk−xi+jFn+ℓ−i−jrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯.Proof.
changingt by t+v in equation (19) and using the following rule [30](34)∑M=0∞fMQ+RMM!=∑n,m=0∞fm+nQnn!Rmm!,
in the left hand side becomes(35)∑n,ℓ=0∞Fn+ℓrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!uℓℓ!=t+vrγ2rμ∏i=0r−1at+v−yψt+v;αiβicxt+vϕz1,z2,…,zm,t+v.
Rewriting (35) as(36)c−xt+v∑n,ℓ=0∞Fn+ℓrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!=t+vrγ2rμ∏i=0r−1at+v−yψt+v;αiβiϕz1,z2,…,zm,t+v.
Replacingx by k and equating the resulting equation to the previous equation, we obtain(37)ck−xt+v∑n,ℓ=0∞Fn+ℓrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!=∑n,ℓ=0∞Fn+ℓrk,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!.
On expanding exponential function in (37) we obtain(38)∑N=0∞k−xt+vlncNN!∑n,ℓ=0∞Fn+ℓrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!uℓℓ!=Fn+ℓrk,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!.
Using (34) in the left hand side of (38), we get(39)∑i,j=0∞lnci+jk−xi+jtivji!j!∑n,ℓ=0∞Fn+ℓrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!=∑n,ℓ=0∞Fn+ℓrk,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!.
Puttingn by n−i, ℓ by ℓ−j and by using Cauchy-product rule in the left hand side of (39), we get(40)∑n,ℓ=0∞∑i,j=0n,ℓlnci+jk−xi+ji!j!Fn+ℓ−i−jrx,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn−i!vℓℓ−j!=∑n,ℓ=0∞Fn+ℓrk,y,z1,z2,…,zm,γ,μ;αr¯,βr¯tnn!vℓℓ!.
Finally, comparing the similar coefficients oftn and vn in the previous equation yields (33).Theorem 6.
Forx,z∈R and n,m≥0, we have the relationship(41)Fnrx,1,z,0,0;αr¯,1=∑m=0∞Sα¯m,rHn,ℓx+m,z,
between the generalized Fubini Apostol-type polynomials, generalized Stirling number of first kind and generalized Hermite polynomials.Proof.
Settingγ=μ=0,βi=y=a=1,ψt,αi=αiet,andϕz,t=eztℓ in equation (22), from the generating function of generalized Stirling numbers and from the generating function of generalized Hermite polynomials, we have(42)∑n=0∞Fnrx,1,z,0,0;α¯,1tnn!=1∏i=0r−11−αietexteztm=∑m=0∞Sα¯m,rem+xt+ztℓ=∑∞n=0∑m=0∞Sα¯m,rHn,ℓx+m,ztnn!.
By comparing the similar coefficients oftn in the previous equation, yields (41).Theorem 7.
Forx,z∈R and n,m≥0, we have the relationship(43)Fnrx,y,ξ,0,0;α,1=∑m=0n∑r=0n−mn−mmS2,αn−m,rr!yrHm,ℓξ1,ξ2,between the generalized Fubini Apostol-type polynomia, degenerate Stirling number of second kind and generalized Hermite polynomials.Proof.
Puttingγ=μ=0,βi=a=1,ψt,αi=1/xeαtx−1,ϕξ,t=etξℓ in equation (22), from the generating function of degenerate generalized Fubini polynomials, see ([26]) and from the generating function of Hermite polynomials, we obtain(44)∑n=0∞Fnrx,1,z,0,0;α,1tnn!=∑n=0∞∑m=0nn!nmFn−m,αy,xHm,ℓξ1,ξ2tnn!.
By comparing the coefficient oftn in the previous (44), we obtain(45)Fnrx,1,z,0,0;α,1=∑m=0nn!nmFn−m,αy,xHn,ℓξ1,ξ2.
Since(46)Fn−m,αy,x=∑k=0n−mS2,αn−m,kk!ykxn−k.
Therefore, we get (43).Theorem 8.
Forx,z∈R and n,m≥0, we have the relationship(47)Fnrx,1,z,0,0;α¯,β¯=∑m=0n∑n1+n2+⋯+nr=mnn1,n2,…,nr∏i=1rBelni,αiβi−1Hn−m,ℓx,z,
between the generalized Fubini Apostol-type polynomials, degenerate Bell polynomials and generalized Hermite polynomials.Proof.
Puttingγ=μ=0,βi=a=y=1,ψt,αi=1−eλt,andϕz,t=eztℓ in equation (22) and using equation (14), we obtain (47).Corollary 2.
Settingαi⟶0 in (47), we obtain relation between generalized Apostol-type polynomials, Bell polynomials and generalized Hermite polynomials(48)Fnrx,z,β¯=∑m=0n∑n1+n2+⋯+nr=mnn1,n2,…,nr∏i=1rBelniβi−1Hn−m,ℓx,z.Theorem 9.
Forn≥0, we have the relationship(49)Fnrα¯,β¯=∑n1+n2+⋯+nr=nnn1,n2,…,nr∏i=1rBelni,αiβi,
between the generalized Apostol-type numbers and Bell polynomials.Proof.
Puttingγ=μ=x=0,βi=a=y=1,ψt,αi=1−eλt,andϕz,t=1 in equation (23) and using equation (14), we obtain (49).Corollary 3.
Settingαi⟶0 in (49), we obtain relation between generalized Apostol numbers and Bell polynomials.(50)Fnrβ¯=∑n1+n2+⋯+nr=nnn1,n2,…,nr∏i=1rBelniβi−1.


## 4. Examples

Selecting appropriate choice forφy,t in (22), the new special families of polynomials are obtained from Fnrx,y,z1,γ,μ;α¯,β¯.


### 4.1. Gould-Hopper Fubini Apostol Type Polynomials

Settingφz1,t=ez1tm in (22), we obtain Gould–Hopper Fubini Apostol-type polynomials (GHFATP), denoted by FHmnrx,y,z1,γ,μ;α¯,β¯ which are defined by(51)∑n=0∞HmFnrx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+z1tm.We can obtain some definitions and some results for the GHFATP, these results are listed in Table3.Table 3
Identity and relation forFnrHmx,y,z1,γ,μ;α¯,β¯.
S.No.ResultsExpressions1Series definitionFnrHmx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrHmγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrHmx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrHmℓ,y,z1,γ,μ;αr¯,βr¯.FnrHmx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0k=0ℓn−klncn−kFkrHmx,y,z1,γ,μ;αr¯,βr¯.


### 4.2. Laguerre Fubini Apostol-Type Polynomials

Settingφz1,t=C0z1 in (22), gives Laguerre Fubini Apostol-type polynomials (LFATP), denoted by FnrLx,y,z1,γ,μ;α¯,β¯ which are defined by(52)∑n=0∞FnrLx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiextC0z1.We can obtain some definitions and some results for the LFATP, these results are listed in Table4.Table 4
Identity and relation forFnrLx,y,z1,γ,μ;α¯,β¯.
S.No.ResultsExpressions1Series definitionFnrLx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrLγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrLℓ,y,z1,γ,μ;αr¯,βr¯.FnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkrLx,y,z1,γ,μ;αr¯,βr¯.


### 4.3. Truncated Exponential Fubini Apostol-Type Polynomials

Settingφz1,t=1/1−z1ts in (22), gives Truncated exponential Fubini Apostol-type polynomials (TEFATP), denoted by Fnresx,y,z1,γ,μ;α¯,β¯ which are defined by(53)∑n=0∞Fnresx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext1−z1ts.We can obtain some definitions and some results for the TEFATP, these results are listed in Table5.Table 5
Identity and relation for.Fnresx,y,z1,γ,μ;α¯,β¯
S.No.ResultsExpressions1Series definitionFnresx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnresγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkresℓ,y,z1,γ,μ;αr¯,βr¯.Fnresx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkresx,y,z1,γ,μ;αr¯,βr¯FnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkresx,y,z1,γ,μ;αr¯,βr¯.


### 4.4. Generalized Hermite Fubini Apostol-Type Polynomials

Settingφz,t=eztm in (22), gives generalized Hermite Fubini Apostol-type polynomials (GHFATP), denoted by FnrHx,y,z1,γ,μ;α¯,β¯ which are defined by(54)∑n=0∞FnrHx,y,z,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+ztm.


### 4.5. Hermite-Appell Fubini Apostol Type Polynomials

Settingφz1,t=Atez1t2 in (22), gives Hermite-Appell Fubini Apostol-type polynomials (HAFATP), denoted by FnrHAx,y,z1,γ,μ;α¯,β¯ which are defined by(55)∑n=0∞FnrHAx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+z1t2At.We can obtain some definitions and some results for the TEFATP, these results are listed in Table6.Table 6
Identity and relation for.FnrHAx,y,z1,γ,μ;α¯,β¯
S.No.ResultsExpressions1Series definitionFnrHAx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrHAγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrHAx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrHAℓ,y,z1,γ,μ;αr¯,βr¯.FnrHAx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkrHAx,y,z1,γ,μ;αr¯,βr¯.


### 4.6. Generalized Legendre-fubini Based Apostol Type Polynomials

Settingφz1,z2,t=ez1t2C0−z2t2 and c=e in (19), gives Generalized Legendre-Fubini based Apostol-type polynomials (GLFATP), denoted by FnrLx,y,z1,z2,γ,μ;α¯,β¯ which are defined by(56)∑n=0∞FnrLx,y,z1,z2,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiez1t2+xtC0z2t2.Example 1.
Ify=1,γ=1−μ,a=1 and ψt,αi=ab−αbet−1, we obtain generalized Legendre-Based Apostol-type polynomials, see [31].


## 4.1. Gould-Hopper Fubini Apostol Type Polynomials

Settingφz1,t=ez1tm in (22), we obtain Gould–Hopper Fubini Apostol-type polynomials (GHFATP), denoted by FHmnrx,y,z1,γ,μ;α¯,β¯ which are defined by(51)∑n=0∞HmFnrx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+z1tm.We can obtain some definitions and some results for the GHFATP, these results are listed in Table3.Table 3
Identity and relation forFnrHmx,y,z1,γ,μ;α¯,β¯.
S.No.ResultsExpressions1Series definitionFnrHmx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrHmγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrHmx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrHmℓ,y,z1,γ,μ;αr¯,βr¯.FnrHmx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0k=0ℓn−klncn−kFkrHmx,y,z1,γ,μ;αr¯,βr¯.


## 4.2. Laguerre Fubini Apostol-Type Polynomials

Settingφz1,t=C0z1 in (22), gives Laguerre Fubini Apostol-type polynomials (LFATP), denoted by FnrLx,y,z1,γ,μ;α¯,β¯ which are defined by(52)∑n=0∞FnrLx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiextC0z1.We can obtain some definitions and some results for the LFATP, these results are listed in Table4.Table 4
Identity and relation forFnrLx,y,z1,γ,μ;α¯,β¯.
S.No.ResultsExpressions1Series definitionFnrLx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrLγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrLℓ,y,z1,γ,μ;αr¯,βr¯.FnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkrLx,y,z1,γ,μ;αr¯,βr¯.


## 4.3. Truncated Exponential Fubini Apostol-Type Polynomials

Settingφz1,t=1/1−z1ts in (22), gives Truncated exponential Fubini Apostol-type polynomials (TEFATP), denoted by Fnresx,y,z1,γ,μ;α¯,β¯ which are defined by(53)∑n=0∞Fnresx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext1−z1ts.We can obtain some definitions and some results for the TEFATP, these results are listed in Table5.Table 5
Identity and relation for.Fnresx,y,z1,γ,μ;α¯,β¯
S.No.ResultsExpressions1Series definitionFnresx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnresγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkresℓ,y,z1,γ,μ;αr¯,βr¯.Fnresx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkresx,y,z1,γ,μ;αr¯,βr¯FnrLx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkresx,y,z1,γ,μ;αr¯,βr¯.


## 4.4. Generalized Hermite Fubini Apostol-Type Polynomials

Settingφz,t=eztm in (22), gives generalized Hermite Fubini Apostol-type polynomials (GHFATP), denoted by FnrHx,y,z1,γ,μ;α¯,β¯ which are defined by(54)∑n=0∞FnrHx,y,z,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+ztm.


## 4.5. Hermite-Appell Fubini Apostol Type Polynomials

Settingφz1,t=Atez1t2 in (22), gives Hermite-Appell Fubini Apostol-type polynomials (HAFATP), denoted by FnrHAx,y,z1,γ,μ;α¯,β¯ which are defined by(55)∑n=0∞FnrHAx,y,z1,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiext+z1t2At.We can obtain some definitions and some results for the TEFATP, these results are listed in Table6.Table 6
Identity and relation for.FnrHAx,y,z1,γ,μ;α¯,β¯
S.No.ResultsExpressions1Series definitionFnrHAx,y,z1,γ,μ;α¯,β¯=∑k=0nnkFnrHAγ,μ;α¯,β¯pkx,z1.2Identity and relationFnrHAx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nnkxn−klncn−kFkrHAℓ,y,z1,γ,μ;αr¯,βr¯.FnrHAx+ℓ,y,z1,γ,μ;αr¯,βr¯=∑k=0nℓn−klncn−kFkrHAx,y,z1,γ,μ;αr¯,βr¯.


## 4.6. Generalized Legendre-fubini Based Apostol Type Polynomials

Settingφz1,z2,t=ez1t2C0−z2t2 and c=e in (19), gives Generalized Legendre-Fubini based Apostol-type polynomials (GLFATP), denoted by FnrLx,y,z1,z2,γ,μ;α¯,β¯ which are defined by(56)∑n=0∞FnrLx,y,z1,z2,γ,μ;α¯,β¯tnn!=trγ2rμ∏i=0r−1at−yψt,αiβiez1t2+xtC0z2t2.Example 1.
Ify=1,γ=1−μ,a=1 and ψt,αi=ab−αbet−1, we obtain generalized Legendre-Based Apostol-type polynomials, see [31].


## 5. Probabilistic Application

### 5.1. Degenerate Bell Distribution (DBE)

Definition 4.
LetX be a random variable which follows DBE distribution with parameters α,β and λ. It is denoted by X∼DBEα,β,λ. Then its probability mass function (pmf) is(57)PX=x=Belx,αβλxe−βeαλ−1x!,x=0,1,….where α,β,λ≥0, Belx,αβ is degenerate Bell polynomial in (14) and eαλ is exponential functions in (7)
Submodels:1
Ifα⟶0,β⟶1, we obtain Bell distribution, see [16].2
Ifα⟶0, we obtain Bell–Touchard distribution, see [17].


#### 5.1.1. Some Structural Properties of DBE Distribution

Some statistical properties were presented through the following theorems and remarks.Theorem 10.
LetX∼DBEα,β,λ where α,β,λ≥0, then, the probability generating function is(58)Gt=EtX=e−βeαλ−1eβeαλt−1.Proof.
From(59)Gt=∑x=0∞txPX=x,
and using the expansion of degenerate Bell polynomials, we obtain (58).Corollary 4.
The mean and variance ofX∼DBEα,β,λ can be derived from Gt. We have(60)EX=G′1=βλ1+αλ1/α−1,(61)VarX=G″1+G′1−G′12=βλ21−α1+αλ1/α−2+βλ1+αλ1/α−1.Remark 3.
The over dispersion index of DBE is given by(62)OD=VarXEX=λ+11+αλ.
It follows thatOD>1 for 0<α<1,λ>0. Then, DBE may be suitable for modelling count data with over dispersion.Theorem 11.
LetX1,X2,…,Xk be independent DBE distribution and X=∑i=1rXi. Then, pmf of X is(63)PX=x=λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!,where α¯=α0,…,αr−1 and β¯=β0,…,βr−1 are positive parameters and Fxrα¯,β¯ is generalized Apostol-type numbers.Proof.
(64)=EtX=EtX1+X2+⋯+Xr=∏i=1rEtXi=∑x=0∞∑x1+x2+⋯+xr=xxx1,x2,…,xrλxx!∏irBelxi,αiβie−βieαiλ−1tx.
From (49), we obtain(65)EtX=∑x=0∞PX=xtx∑x=0∞λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!tx.
By comparing coefficients oftx on both sides, we obtain (63).Theorem 12.
IfX is sum of zero truncated poisson random variables with parameter α,λ>0, see [32] and Y is generalized Hermite random variable with two parameter a,b≥0. Then, Z=X+Y is a random variable with parameters θ=α,λ,a,b, has probability mass function is defined as(66)PZ=z=∑x=rzr!αxS2,λx,rHz−x,ma,be−a+bx!eλα−1r,z≥x.


## 5.1. Degenerate Bell Distribution (DBE)

Definition 4.
LetX be a random variable which follows DBE distribution with parameters α,β and λ. It is denoted by X∼DBEα,β,λ. Then its probability mass function (pmf) is(57)PX=x=Belx,αβλxe−βeαλ−1x!,x=0,1,….where α,β,λ≥0, Belx,αβ is degenerate Bell polynomial in (14) and eαλ is exponential functions in (7)
Submodels:1
Ifα⟶0,β⟶1, we obtain Bell distribution, see [16].2
Ifα⟶0, we obtain Bell–Touchard distribution, see [17].


### 5.1.1. Some Structural Properties of DBE Distribution

Some statistical properties were presented through the following theorems and remarks.Theorem 10.
LetX∼DBEα,β,λ where α,β,λ≥0, then, the probability generating function is(58)Gt=EtX=e−βeαλ−1eβeαλt−1.Proof.
From(59)Gt=∑x=0∞txPX=x,
and using the expansion of degenerate Bell polynomials, we obtain (58).Corollary 4.
The mean and variance ofX∼DBEα,β,λ can be derived from Gt. We have(60)EX=G′1=βλ1+αλ1/α−1,(61)VarX=G″1+G′1−G′12=βλ21−α1+αλ1/α−2+βλ1+αλ1/α−1.Remark 3.
The over dispersion index of DBE is given by(62)OD=VarXEX=λ+11+αλ.
It follows thatOD>1 for 0<α<1,λ>0. Then, DBE may be suitable for modelling count data with over dispersion.Theorem 11.
LetX1,X2,…,Xk be independent DBE distribution and X=∑i=1rXi. Then, pmf of X is(63)PX=x=λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!,where α¯=α0,…,αr−1 and β¯=β0,…,βr−1 are positive parameters and Fxrα¯,β¯ is generalized Apostol-type numbers.Proof.
(64)=EtX=EtX1+X2+⋯+Xr=∏i=1rEtXi=∑x=0∞∑x1+x2+⋯+xr=xxx1,x2,…,xrλxx!∏irBelxi,αiβie−βieαiλ−1tx.
From (49), we obtain(65)EtX=∑x=0∞PX=xtx∑x=0∞λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!tx.
By comparing coefficients oftx on both sides, we obtain (63).Theorem 12.
IfX is sum of zero truncated poisson random variables with parameter α,λ>0, see [32] and Y is generalized Hermite random variable with two parameter a,b≥0. Then, Z=X+Y is a random variable with parameters θ=α,λ,a,b, has probability mass function is defined as(66)PZ=z=∑x=rzr!αxS2,λx,rHz−x,ma,be−a+bx!eλα−1r,z≥x.


## 5.1.1. Some Structural Properties of DBE Distribution

Some statistical properties were presented through the following theorems and remarks.Theorem 10.
LetX∼DBEα,β,λ where α,β,λ≥0, then, the probability generating function is(58)Gt=EtX=e−βeαλ−1eβeαλt−1.Proof.
From(59)Gt=∑x=0∞txPX=x,
and using the expansion of degenerate Bell polynomials, we obtain (58).Corollary 4.
The mean and variance ofX∼DBEα,β,λ can be derived from Gt. We have(60)EX=G′1=βλ1+αλ1/α−1,(61)VarX=G″1+G′1−G′12=βλ21−α1+αλ1/α−2+βλ1+αλ1/α−1.Remark 3.
The over dispersion index of DBE is given by(62)OD=VarXEX=λ+11+αλ.
It follows thatOD>1 for 0<α<1,λ>0. Then, DBE may be suitable for modelling count data with over dispersion.Theorem 11.
LetX1,X2,…,Xk be independent DBE distribution and X=∑i=1rXi. Then, pmf of X is(63)PX=x=λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!,where α¯=α0,…,αr−1 and β¯=β0,…,βr−1 are positive parameters and Fxrα¯,β¯ is generalized Apostol-type numbers.Proof.
(64)=EtX=EtX1+X2+⋯+Xr=∏i=1rEtXi=∑x=0∞∑x1+x2+⋯+xr=xxx1,x2,…,xrλxx!∏irBelxi,αiβie−βieαiλ−1tx.
From (49), we obtain(65)EtX=∑x=0∞PX=xtx∑x=0∞λxFxrα¯,β¯∏i=1re−βiαieαiλ−1x!tx.
By comparing coefficients oftx on both sides, we obtain (63).Theorem 12.
IfX is sum of zero truncated poisson random variables with parameter α,λ>0, see [32] and Y is generalized Hermite random variable with two parameter a,b≥0. Then, Z=X+Y is a random variable with parameters θ=α,λ,a,b, has probability mass function is defined as(66)PZ=z=∑x=rzr!αxS2,λx,rHz−x,ma,be−a+bx!eλα−1r,z≥x.


## 6. Conclusions

In this paper, we introduced and investigated a new class of generalized Fubini-type polynomials. We proved some relations, identities, and summation formulae for these polynomials. Special families of polynomials are obtained for the general class of polynomials. Also, we derived explicit formulae for polynomials. Finally, probabilistic applications and some probabilistic properties are obtained.



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