# Unification of Two-Variable Family of Apostol-Type Polynomials with Applications

Authors: Beih S. El-Desouky; Rabab S. Gomaa; Alia M. Magar
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, the two-variable unified family of generalized Apostol-type polynomials is introduced, and some implicit forms and general symmetry identities are derived. Also, we obtain new degenerate Apostol-type numbers and polynomials constructed from the new 2-variable unified family. We derive explicit formulae of polynomials and identities that include some special numbers and polynomials. In addition, a probabilistic representation of the new family and some statistical properties are obtained.


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## Body

## 1. Introduction

Khan and Raza [1] defined the 2-variable general polynomials (2VGP) pnx,y by means of the following generating function:(1)extφy,t=∑n=0∞pnx,ytnn!,p0x,y=1,where φy,t has series expansion(2)φy,t=∑n=0∞φnytnn!,φ0y≠0.The 2-variable general polynomialspnx,y contain a number of important special polynomials of two variables.Generating functions for certain members that belong to the 2VGP are given as follows:The higher-order Hermite polynomials, sometimes called the Gould–Hopper polynomialsHnmx,y are defined by [2].(3)ext+ytm=∑n=0∞Hnmx,ytnn!.The 2-variable Hermite polynomialsHnx,y are defined by [3].(4)ext+yt2=∑n=0∞Hnx,ytnn!.The 2-variable generalized Laguerre polynomialsmLny,x are defined by the following generating function [4]:(5)extC0−ytm=∑n=0∞mLny,xtnn!,where C0y is the 0-th order Tricomi function [5].(6)C0y=∑r=0∞−1rxrr!2.The 2-variable Laguerre polynomialsLny,x are defined by [6](7)extC0yt=∑n=0∞Lny,xtnn!.The 2-variable truncated exponential polynomials of orderrenrx,y are defined [7](8)ext1−ytr=∑n=0∞enrx,ytnn!.In particular, we note that(9)en2x,y=n!2enx,y,2enx,1=2enx,where 2enx,y denotes the 2-variable truncated exponential polynomials [8].The 2-variable truncated exponential polynomials2enrx,y are defined by [8].(10)ext1−yt2=∑n=0∞2enx,ytnn!.Khan et al. [9] introduced the 2-variable Apostol type polynomials of order α, by(11)∑n=0∞pFnαx,y;λ;μ,νtnn!=2μtνλet+1αextφy,t,t<log−λ.Araci et al. [10] introduced and investigate a new unified class of generalized Apostol type polynomials by the following generating function:(12)∑n=0∞HFnαx,y;a,b,c;μ,ν;λtnn!=2μtνλbt+atαcxt+ytj,c>1;j>2;t<log−λlogc/b.In [11], Cartilz introduced a new study of degenerate versions of Bernoulli and Euler numbers and polynomials. Lately, many researchers have begun to study the degenerate Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate Hermite polynomials and numbers, and so on (see [12–14]).Furthermore, mathematicians have done recently a few interesting works on Apostol-type polynomials in the field of approximation theory (for more details, see [15–18]).This manuscript is a modified and extended version of [19].The paper is organized as follows: in Section2, we construct a new version of the 2-variable Apostol-type polynomials and numbers and related special polynomials and numbers. In Section 3, by using generating functions, we investigate some summation formulae, explicit expression, and some symmetric identities. In Section 4, we construct a new version of 2-variable degenerate Apostol-type polynomials and numbers and related special polynomials. In Section 5, we obtain the generating functions for new special polynomials that belong to the new version of 2-variable unified Apostol-type polynomials. In Section 6, the probabilistic representation of the new family and its statistical properties are presented.


## 2. Unification of Two-Variable Apostol-Type Polynomials

The two-variable unified family of generalized Apostol-type polynomials of orderr, denoted by pUnrx,y;a,b,c,ν,μ;α¯r is defined as the Apostol type convolution of the 2-variable general polynomials pnx,y.Definition 1.
Leta,b and c∈ℝ+ and a≠b. A new generalization of the Apostol Hermite–Genocchi polynomials pUnrx,y;a,b,c,ν,μ;α¯r for nonnegative integer n is defined by the generating function(13)∑n=0∞pUnrx,y;a,b,c,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atcxtφy,t,t<|logαilogb/aν,μ∈ℕ0;αi≠1;for alli=0,1,…,r−1,where r∈ℂ;α¯r=α0,α1,…,αr−1 is a sequence of complex numbers.
Settingc=e and φy,t=1 in (13), we get the following definition.Definition 2.
A unified familyUnrx;a,b,ν,μ;α¯r of generalized Apostol-type polynomials is given by(14)∑n=0∞Unrx;a,b;μ,ν;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atext.Remark 1.
Settingx=0 in (14), then we obtain the new unified family of generalized Apostol-type numbers, which is defined as(15)∑n=0∞Unra,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−at.
Also, settingc=e in (13), we get the following definition.Definition 3.
The two-variable unified family of generalized Apostol-type polynomials of orderr
pUnrx,y;a,b;ν,μ;α¯r is defined by the following generating function(16)∑n=0∞pUnrx,y;a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atextφy,t.
We obtain the series definition ofpUnrx,y;a,b,ν,μ;α¯r by the following theorem.Theorem 1.
The two-variable unified family of generalized Apostol-type of orderrpUnrx,y;a,b,ν,μ;α¯r is defined by the following series:(17)pUnrx,y;a,b,ν,μ;α¯r=∑k=0nnkUn−krν,μ,a,b,α¯rpkx,y.Proof.
Using equation (13) and the Cauchy-product rule, we can easily yield (17).
Now, by taking certain values of parameters in equations (13) and (16), we can find the generating functions and other results for the mixed special polynomials related to pUnrx,,y;a,b,ν,μ;α¯r. We present the generating function and series definitions for these polynomials in Table 1.Table 1
Special cases of thepUnrx,,y;a,b,ν,μ;α¯r.
No.Values of the parametersRelation betweenpUnrx,,y;k,a,b;α¯r and its special casesName of the results special polynomials1αi=−λ and φy,t=1 in (13)pUnrx;a,b,ν,μ;−λ=Fnrx;a,b,c,μ,ν;λThe generalized Apostol-type Gould-Hopper polynomials, [10]2αi=−λ and φy,t=cytj, j>2 in (13)pUnrx,y;a,b,ν,μ;−λ=HFnrx,y;a,b,c,μ,ν;λThe generalized Apostol type polynomials of orderr, [10]3αi=−λ,b=e and a=1 in (16)pUnrx,y;1,e,ν,μ;−λ=pFnrx,y;λ,μ,νThe 2-variable Apostol type polynomials of orderr, [9]4αi=λ,b=e,a=1μ=0 and ν=1 in (16)pUnrx,y;1,e,1,0;λ=−1rpBnrx,y;λThe 2-variable Apostol-Bernoulli polynomials of orderr, [9]5αi=−λ,b=e,a=1μ=1 and ν=0 in (16)pUnrx,y;1,e,0,1;λ=pEnrx,y;λThe 2-variable Apostol-Euler polynomials of orderr, [9]6αi=−λ,b=e,a=1 and μ=ν=1 in (16)pUnrx,y;1,e,1,1;−λ=pGnrx,y;λThe 2-variable Apostol-Genocchi polynomials of orderr, [9]7αi=−λ,b=e,a=1 and φy,t=1 in (16)pUnrx,y;1,e,ν,μ;−λ=Fnrx,y;λ,μ,νThe generalized Apostol type polynomials of orderr, [20]8a=1,b=e,μ=1−k,ν=k in (14)pUnrx,y;1,e,ν,μ;α¯r=−1rMnrx,k;α¯rThe unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, [21]9μ=1−k,ν=k,φy,t=cyt2 in (13)pUnrx,y;a,b,c,k,α¯r=HMnrx,y;a,b,c,k;α¯rThe generalization of Apostol-Hermite Genocchi polynomials, [22]Theorem 2.
Fora,b∈R+ and ν=0, the explicit formula of pUnrx,y;a,b,c,μ;α¯r can be expressed as(18)pUnrx,y;a,b,c,μ;α¯r=2rμ∑m1,m2,…,mr≥0∑m=0nnm∏i=1rαi−1milogba∑i=1rmicxarn−mφmy.Proof.
The left-hand side of (13) is equal to(19)−1r2rμ∏i=0r−1αibt−atcxtφy,t=2rμetlogcx/ar∏i=0r−11−αietlogb/aφy,t=∑m1,m2,…,mr≥0∏i=0r−1αi−1mietlogb/a∑i=1rmi+logcx/arφy,t.
By (13), we get(20)∑n=0∞pUnrx,y;a,b,c,μ;α¯rtnn!=∑n=0∞2rμ∑m1,m2,…,mr≥0∑m=0nnm∏i=0r−1αi−1milogba∑i=1rmicxarn−mφmytnn!.
By comparing the coefficients on both sides in the last equation, we obtain (18).


## 3. Implicit Summation Formulas for the Two-Variable Unified Family of Generalized Apostol- Type Polynomials

Theorem 3.
Leta,b>0 and a≠b. Then for x,y,z∈R and n≥0. The following implicit summation formula for pUnrx,y;a,b;μ,ν;α¯r holds true as follows:(21)pUnrx+z,y;a,b,ν,μ;α¯r=∑m=0npUnrx,y;a,b;μ,ν;α¯rzn−m.Proof.
Replacement ofx by x+z in (16) gives(22)∑n=0∞pUnrx+z,y;a,b,ν,μ;α¯rtnn!=−1rtrω2rμ∏i=0r−1αibt−atex+ztφy,t=∑n=0∞∑m=0∞pUnrx,y;a,b;μ,ν;α¯rzntn+mn!m!.
Replacingn with n−m in the right-hand side, hence equating the coefficients of t in both sides of the last equation yields (21).Theorem 4.
The next implicit summation formula forpUnrx,y;a,b;μ,ν;α¯r in terms of generalized Apostol type polynomial Unrx;a,b,ν,μ;α¯r is obtained as follows:(23)pUnrx+z,y;a,b,ν,μ;α¯r=∑k=0nnkUnrx;a,b,ν,μ;α¯rpn−kz,y.Proof.
Replacingx by x+z in (16) gives(24)∑n=0∞pUnrx+z,y;a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atex+ztφy,t=∑n=0∞Unrx;a,b,ν,μ;α¯rtnn!∑k=0∞pkz,ytkk!.
We get(25)∑n=0∞pUnrx+z,y;a,b,ν,μ;α¯rtnn!=∑n=0∞∑k=0nnkUnrx;a,b,ν,μ;α¯rpn−kz,ytnn!.
Equating the coefficients oftn on both sides, yields (23).Theorem 5.
Leta,b>0 and a≠b. Then for x,y,z∈R and n≥0, the following implicit summation formula for pUnrx,y;a,b;μ,ν;α¯r holds true as follows:(26)pUn+mrz,y;a,b,ν,μ;α¯r=∑p,q=0n,mnpmqz−xp+qpUn+m−p−qrx,y;a,b,μ,ν;α¯r.Proof.
Replacingt by t+u in the generating function (16) and using the following rule [23]:(27)∑N=0∞fNx+yNN!=∑n,m=0∞fm+nxnn!ymm!.
We obtain(28)e−xt+u∑n,m=0∞pUn+mrx,y;a,b,μ,ν;α¯rtnn!umm!=−1rt+urν2rμ∏i=0r−1αibt+u−at+uφy,t+u.
Replacingx by z in the previous equation and then equal both sides, we get(29)ez−xt+u∑n,m=0∞pUn+mrx,y;a,b,μ,ν;α¯rtnn!umm!=∑n,m=0∞pUn+mrz,y;a,b,ν,μ;α¯rtnn!umm!.
On expanding exponential function (29) gives(30)∑N=0∞z−xt+uNN!∑n,m=0∞pUn+mrx,y;a,b,μ,ν;α¯rtnn!umm!=∑n,m=0∞pUn+mrz,y,a,b,ν,μ;α¯rtnn!umm!.
Using equation (27) in the left-hand side of equation (30), we find(31)∑p,q=0∞z−xp+qtpuqp!q!∑n,m=0∞pUn+mrx,y;a,b,μ,ν;α¯rtnn!umm!=∑n,m=0∞pUn+mrz,y,a,b,ν,μ;α¯rtnn!umm!.
Now replacingn by n−p, m by m−q, and using the Cauchy-product rule in the left-hand side of (31), we get(32)∑n,m=0∞∑p,q=0n,mz−xp+qp!q!pUn+m−p−qrx,y,a,b,μ,ν;α¯rtnn−p!umm−q!=∑n,m=0∞pUn+mrz,y;a,b,ν,μ;α¯rtnn!umm!.
Finally, on equating the coefficients of the like powers oft and u in the above equation, yields (26).Theorem 6.
The following implicit summation formula forpUnrx,y;a,b;μ,ν;α¯r holds true as follows:(i)
(33)pUnrx+1,y,a,b,ν,μ;α¯r=∑m=0nnmpUn−mrx,y,a,b,μ,ν;α¯r.(ii)
(34)pUnrx+z,y;a,b,ν,μ;α¯r=∑m=0nnmUn−mrz,a,b,ν,μ;α¯rpmx,y.Proof.
Using equation (16), we can easily obtain (33) and (34).Theorem 7.
Letc,d>0 and n≥0. For x,y∈ℝ, then the following identity holds true(35)∑m=0nnmdmcn−mpUn−mrdx,y,a,b,ν,μ;αr¯pUmrcx,y,a,b,ν,μ;αr¯=∑m=0nnmcmdn−mpUn−mrcx,y;a,b,ν,μ;αr¯pUmrdx,y,a,b,ν,μ;αr¯.Proof.
Let(36)Gt=−1r2rμtrν2∏i=0r−1αibdt−adt∏i=0r−1αibct−acte2cdxtφy,dtφy,ct.
Then the expression forGt is symmetric in c and d and we can expand Gt into series in two ways.
Firstly(37)Gt=1cdrν−1r2rμdtrν∏i=0r−1αibdt−adtecxdt−1r2rμctrν∏i=0r−1αibct−actedxct=1abrν∑n=0∞pUnrcx,y;a,b,ν,μ;αr¯dtnn!∑m=0∞pUnrdx,y;a,b,ν,μ;αr¯ctmm!Gt=1abrν∑n=0∞∑m=0nnmdn−mcmpUn−mrcx,y;a,b,ν,μ;αr¯pUmrdx,y,a,b,ν,μ;αr¯tnn!.
Secondly(38)Gt=1cdrν∑n=0∞∑m=0nnmcn−mdmpUn−mrdx,y;a,b,ν,μ;αr¯pUnrcx,y;a,b,ν,μ;αr¯tnn!.
Form equations (37) and (38), by comparing the coefficients of tn on both sides, yields (35).


## 4. Applications

The 2VGP familypnx,y contains a number of important special polynomials of two variables. Some members belonging to the 2VGP family are considered in Section 1. We notice that for every member belonging to the 2VGP, there is a new special polynomial that belongs to the pUnrx,y,a,b,ν,μ;α¯r family. Thus, by selecting a suitable choice for the function φy,t in equation (16), the generating function for the corresponding member belongs to pUnrx,y,a,b,ν,μ;α¯r is a family that can be obtained.Example 1.
Settingφy,t=eytm in the left-hand side of generating function (16), gives Gould–Hopper Apostol type polynomials (GHATP), denoted by HmUnrx,y,a,b,ν,μ;α¯r are defined by(39)∑n=0∞HmUnrx,y,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atext+ytm.
Setting suitable values of the parameters in the results of the GHATPHmUnrx,y,a,b,ν,μ;α¯r, we obtain the following results:(1)
Ifαi=λ,b=e,a=1, ν=1,μ=0, then HmUnrx,y;1,e,1,0;λ=−1rHmBnrx,y;λ (Gould–Hopper-Apostol-Bernoulli polynomials of order r) (see [9]).(2)
Ifαi=−λ,b=e,a=1, ν=0,μ=1, then HmUnrx,y;1,e,0,1;−λ=HmEnrx,y;λ (Gould–Hopper-Apostol-Euler polynomials of order r), see [9].(3)
Ifαi=−λ,b=e,a=1,ν=1,μ=1, then HmUnrx,y;1,e,1,1;−λ=2−rHmGnrx,y;λ (Gould–Hopper-Apostol-Genocchi polynomials of order r), [9].
The series definitions and some results for the GHATPHmUnrx,y,a,b,ν,μ;α¯r are given in Table 2.Table 2
Results forHmUnrx,y,a,b,ν,μ;α¯r.
S.No.ResultsExpressions1Series definitionHmUnrx,y,a,b,ν,μ;α¯r=∑z=0nnzUn−zra,b,ν,μ,;α¯rHzmx,y2Summation formulaeHmUnrx+w,y,a,b;,ν,μ;α¯r=∑z=0nnzHmUzrx,y,a,b,ν,μ;α¯rwn−zHmUnrx+w,y,a,b,ν,μ;α¯r=∑z=0nnzUzrx,y,a,b,ν,μ;α¯rHn−zmw,yHmUn+prw,y,a,b,ν,μ;α¯r=∑l,z=0n,pnlpzw−xl+zHmUn+p−z−lrx,y,a,b,ν,μ;α¯rNote. We note that forλ=1 in the results derived above, we obtain the corresponding results for the Gould–Hopper–Bernoulli polynomials (GHBP) of order r, Gould–Hopper–Euler polynomials (GHEP) of order r, and Gould–Hopper–Genocchi polynomials (GHGP) of order r, respectively.Remark 2.
Form=2, the Hnmx,y reduce to Hnx,y. Therefore, setting m=2 in equation (39), we obtain the following generating function for the 2-variable Hermite Apostol type polynomials, denoted by HUnrx,y,a,b,ν,μ;α¯r as follows:(40)∑n=0∞HUnrx,y,a,b,ν,μ;α¯rtnn!=−1rtrν2μ∏i=0r−1αibt−atext+yt2.
The series definitions and some results for the 2-variable Hermite Apostol type polynomialsHUnrx,y,a,b,ν,μ;α¯r can be deduced by setting m = 2 in the results given in Table 2.Example 2.
Settingφy,t=C0−ytm (for which the pnx,y reduce to the mLny,x) in the left-hand side of generating function (16), we find that the resultant 2-variable generalized Laguerre Apostol type polynomials (2VGLATP), denoted by mLUnry,x,a,b,ν,μ;α¯r are defined by the following generating function:(41)∑n=0∞mLUnry,x,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atextC0−ytm.
The series definitions and other results for the 2VGLATPmLUnry,x,a,b,ν,μ;α¯r are given in Table 3.Table 3
Results formLUnry,x,a,b,ν,μ;α¯r.
S.No.ResultsExpressions1Series definitionmLUnry,x,a,b,ν,μ;α¯r=∑z=0nnzUn−zra,b,ν,μ;α¯rmLzy,x2Summation formulaemLUnry+w,x,a,b,ν,μ;α¯r=∑z=0nnzmLUzry,x,a,b,ν,μ;α¯rwn−zmLUnrx+w,a,b,ν,μ;α¯r=∑z=0nnzUzrx,y,a,b,ν,μ;α¯rmLn−zw,ymLUn+prw,x,a,b,ν,μ;α¯r=∑l,z=0n,pnlpzw−yl+zmLUn+p−z−lry,x,a,b,ν,μ;α¯rRemark 3.
Since form=1 and y⟶−y, then mLnx,y reduce to the Lnx,y. Therefore, setting m=1 and y⟶−y in equation (41), we obtain the generating function for the 2-variable Laguerre Apostol type polynomials, denoted by LUnry,x,a,b,ν,μ;α¯r as(42)∑n=0∞LUnry,x,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atextC0yt.
The series definitions and other results for theLUnry,x,a,b,ν,μ;α¯r can be obtained by setting m=1 and y⟶−y in the results given in Table 3.Remark 4.
Since forx=1, the Lnx,y reduce to the classical Laguerre polynomials Lny. Therefore, setting x=1 in equation (42), we obtain the following generating function for the Laguerre Apostol type polynomials, denoted by LUnry,a,b,ν,μ;α¯r:(43)∑n=0∞LUnry,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atetC0yt.
The series definitions and other results for theLUnry,a,b,ν,μ;α¯r can be obtained by setting m=1, y⟶−y, and x=1 in the results given in Table 3.
Setting suitable values of the parameters in the results of themLUnry,x,a,b,ν,μ;α¯r, we obtain results for 2-variable generalized Laguerre-Apostol polynomials related to mLUnry,x,a,b,ν,μ;α¯r (for more details see [9]).Example 3.
Settingφy,t=1/1−ytβ (for which the pnx,y reduce to the enβx,y) in the left-hand side of generating function (16), we obtain the 2-variable truncated exponential Apostol type polynomials of order β (2VTEATP), denoted by eβUnrx,y,a,b,ν,μ;α¯r are defined by(44)∑n=0∞eβUnrx,y,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atext1−ytβ.
The series definitions and other results for the 2VTEATPeβUnrx,y,a,b,ν,μ;α¯r are given in Table 4.
Setting suitable values of the parameters in the results of themLUnry,x,a,b,ν,μ;α¯r, we obtain results for 2-variable truncated exponential Apostol polynomials related to mLUnry,x,a,b,ν,μ;α¯r (for more details see [9]).Table 4
Results foreβUnrx,y,a,b,ν,μ;α¯r.
S.No.ResultsExpressions1Series definitioneβUnrx,y,a,b,ν,μ;α¯r=∑z=0nnzUn−zra,b,ν,μ;α¯rezβx,y2Summation formulaeeβUnrx+w,y,a,b,ν,μ;α¯r=∑z=0nnzeβUzrx,y,a,b,ν,μ;α¯rwn−zeβUnrx+w,a,b,ν,μ;α¯r=∑z=0nnzUzrx,y,a,b,ν,μ;α¯ren−zβw,yeβUn+prw,y,a,b,ν,μ;α¯r=∑l,z=0n,pnlpzw−xl+zeβUn+p−z−lrx,y,a,b,ν,μ;α¯rRemark 5.
Since forβ=2, the enβx,y of order β reduce to the 2enx,y. Therefore, taking β=2 in equation (44), we obtain the following generating function for the 2-variable truncated exponential Apostol type polynomials, denoted by 2eUnrx,y,a,b,ν,μ;α¯r as(45)∑n=0∞2eUnrx,y,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atext1−yt2.
The series definitions and other results for the 2VTEATP2eUnrx,y,a,b,ν,μ;α¯r can be obtained by taking β=2 in the results given in Table 4.Remark 6.
Since fory=1, the 2enx,y of order β reduce to the truncated exponential polynomials 2enx. Therefore, taking y=1 in equation (45), we get the following generating function for the truncated exponential polynomials, denoted by 2eUnrx,a,b,ν,μ;α¯r:(46)∑n=0∞2eUnrx,a,b,ν,μ;α¯rtnn!=−1rtrν2rμ∏i=0r−1αibt−atext1−t2.


## 5. Two-Variable Degenerate Apostol-type Polynomials

In this section, replacinget⟶1+λt1/λ, and b=1,a=c=e in (13), a new class of two-variable degenerate Apostol type polynomials pDUnrx,y;μ;α¯r,λ is given. Some identities and properties are obtained.Definition 4.
For non-negativen, pDUnrx,y;ν,μ;λ,α¯r is given by means of the generating function(47)∑n=0∞pDUnrx,y;ν,μ;λ,α¯rtnn!=trν2rμ∏i=0r−11+λt1/λ−αi1+λtx/λφy,t.


### 5.1. Special Cases

(1)
Ifαi=1,r=1,ν=1,μ=0 and φy,t=1 in (47), we obtain     (48)pDUn1x,y;1,0;λ,1=Bn,λx.(Degenerate Bernoulli polynomials, see [13]).(2)
Ifαi=−1,r=1,ν=0,μ=1 and φy,t=1 in (47), we obtain(49)pDUn1x;−1,1;λ,0=En,λx.(Degenerate Euler polynomials, see [13]).(3)
Ifαi=1,i=0,1,…,r−1,ν=μ=1 and φy,t=1 in (47), we obtain(50)pDUnrx;1,1;λ,1=Gn,λrx.(Degenerate Genocchi polynomials of orderr, see [13]).(4)
Ifαi=0,r=1,ν=μ=0 and φy,t=eyt2 in (47), we obtain(51)pDUn1x,y;0,0;λ,0=Hnx−1,y,λ.(Two-variable partially degenerate Hermite polynomials, see [12]).Theorem 8.
The explicit formula of degenerate generalized Apostol Hermit polynomials is given by(52)pDUnrα,β,λ;α¯r=∑x1,…,xn=0∞∏i=1rαi−1xi∑k=0n/2βkn!k!n−2k!∏i=0n−2k−1α−∑i=0rxi−r−iλ.Proof.
Puttingx=α,y=β, and φβ,t=eβt2 in (47), we have(53)∑n=0∞pDUnrα,β,λ;α¯rtnn!=∑x1,…,xn=0∞∏i=1rαi−1xi1+λtα−∑i=1rxi−r/λeβt2.
The generating function of degenerate generalized Hermite polynomials is given by [12].(54)1+λtα/λeβt2=∑n=0∞Hnα,β,λtnn!,
Then,(55)∑n=0∞pDUnrα,β,λ;α¯rtnn!=∑n=0∞∑x1,…,xn=0∞∏i=1rαi−1xiHnα−∑i=1rxi−r,β,λtnn!.
Equating the coefficients oftn on both sides in the last equation, yields (52).


### 5.2. Some Identities ofpDUnrx,y;ν,μ;λ,α¯r

Theorem 9.
Forn≥0, we have(56)∑m=0npDUnrx,y;ν,μ;λ,α¯rSn,m=∑k=0n∑m=0knkpDUnrx;ν,μ;λ,α¯rSk,mBeln−k,λy,where Beln−k,λx are degenerate Bell polynomials, see ([14]) and Sn,m are Stirling numbers of the second kind, see ([24]).Proof.
Replacingt by et−1 in (47), we get(57)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λφy,et−1=∑m=0∞pDUmrx,y;ν,μ;λ,α¯ret−1mm!.
It is well known that the Stirling numbers of the second kind are defined by (see [24])(58)et−1mm!=∑n=m∞Sn,mtnn!.
We get(59)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λφy,et−1=∑n=0∞∑m=0npDUnrx,y;ν,μ;λ,α¯rSn,mtnn!.
Let us put, [14].(60)φy,et−1=1+λet−1y/λ=∑ℓ=0∞Belℓ,λytℓℓ!,in the left-hand side of (59), we have(61)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λ1+λet−1y/λ=∑n=0∞∑k=0nnk∑m=0kpDUnrx;ν,μ;λ,α¯rSk,mBeln−k,λytnn!.
Therefore, by comparing the coefficients oftn on both sides of (59) and (61), we obtain (56).Theorem 10.
Forn≥0, we have(62)pDUnrx,y;ν,μ;λ,α¯r=∑m=0nnmpDUn−mry;ν,μ;λ,α¯r∑ℓ=0msm,ℓxℓλm−ℓ,where sn,m are the Stirling numbers of the first kind, see [24].Proof.
Form equation (47), we have(63)trν2rμ∏i=0r−11+λt1/λ−αi1+λtx/λφy,t=∑n=0∞pDUmry;ν,μ;λ,α¯rtnn!∑m=0∞xλmλtmm!=∑n=0∞∑m=0nnmpDUn−mry;ν,μ;λ,α¯r∑ℓ=0msm,ℓxℓλm−ℓtnn!.
Therefore, by equating the right-hand side of (47) and the last equation, we obtain (62).


## 5.1. Special Cases

(1)
Ifαi=1,r=1,ν=1,μ=0 and φy,t=1 in (47), we obtain     (48)pDUn1x,y;1,0;λ,1=Bn,λx.(Degenerate Bernoulli polynomials, see [13]).(2)
Ifαi=−1,r=1,ν=0,μ=1 and φy,t=1 in (47), we obtain(49)pDUn1x;−1,1;λ,0=En,λx.(Degenerate Euler polynomials, see [13]).(3)
Ifαi=1,i=0,1,…,r−1,ν=μ=1 and φy,t=1 in (47), we obtain(50)pDUnrx;1,1;λ,1=Gn,λrx.(Degenerate Genocchi polynomials of orderr, see [13]).(4)
Ifαi=0,r=1,ν=μ=0 and φy,t=eyt2 in (47), we obtain(51)pDUn1x,y;0,0;λ,0=Hnx−1,y,λ.(Two-variable partially degenerate Hermite polynomials, see [12]).Theorem 8.
The explicit formula of degenerate generalized Apostol Hermit polynomials is given by(52)pDUnrα,β,λ;α¯r=∑x1,…,xn=0∞∏i=1rαi−1xi∑k=0n/2βkn!k!n−2k!∏i=0n−2k−1α−∑i=0rxi−r−iλ.Proof.
Puttingx=α,y=β, and φβ,t=eβt2 in (47), we have(53)∑n=0∞pDUnrα,β,λ;α¯rtnn!=∑x1,…,xn=0∞∏i=1rαi−1xi1+λtα−∑i=1rxi−r/λeβt2.
The generating function of degenerate generalized Hermite polynomials is given by [12].(54)1+λtα/λeβt2=∑n=0∞Hnα,β,λtnn!,
Then,(55)∑n=0∞pDUnrα,β,λ;α¯rtnn!=∑n=0∞∑x1,…,xn=0∞∏i=1rαi−1xiHnα−∑i=1rxi−r,β,λtnn!.
Equating the coefficients oftn on both sides in the last equation, yields (52).


## 5.2. Some Identities ofpDUnrx,y;ν,μ;λ,α¯r

Theorem 9.
Forn≥0, we have(56)∑m=0npDUnrx,y;ν,μ;λ,α¯rSn,m=∑k=0n∑m=0knkpDUnrx;ν,μ;λ,α¯rSk,mBeln−k,λy,where Beln−k,λx are degenerate Bell polynomials, see ([14]) and Sn,m are Stirling numbers of the second kind, see ([24]).Proof.
Replacingt by et−1 in (47), we get(57)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λφy,et−1=∑m=0∞pDUmrx,y;ν,μ;λ,α¯ret−1mm!.
It is well known that the Stirling numbers of the second kind are defined by (see [24])(58)et−1mm!=∑n=m∞Sn,mtnn!.
We get(59)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λφy,et−1=∑n=0∞∑m=0npDUnrx,y;ν,μ;λ,α¯rSn,mtnn!.
Let us put, [14].(60)φy,et−1=1+λet−1y/λ=∑ℓ=0∞Belℓ,λytℓℓ!,in the left-hand side of (59), we have(61)et−1rν2rμ∏i=0r−11+λet−11/λ−αi1+λet−1x/λ1+λet−1y/λ=∑n=0∞∑k=0nnk∑m=0kpDUnrx;ν,μ;λ,α¯rSk,mBeln−k,λytnn!.
Therefore, by comparing the coefficients oftn on both sides of (59) and (61), we obtain (56).Theorem 10.
Forn≥0, we have(62)pDUnrx,y;ν,μ;λ,α¯r=∑m=0nnmpDUn−mry;ν,μ;λ,α¯r∑ℓ=0msm,ℓxℓλm−ℓ,where sn,m are the Stirling numbers of the first kind, see [24].Proof.
Form equation (47), we have(63)trν2rμ∏i=0r−11+λt1/λ−αi1+λtx/λφy,t=∑n=0∞pDUmry;ν,μ;λ,α¯rtnn!∑m=0∞xλmλtmm!=∑n=0∞∑m=0nnmpDUn−mry;ν,μ;λ,α¯r∑ℓ=0msm,ℓxℓλm−ℓtnn!.
Therefore, by equating the right-hand side of (47) and the last equation, we obtain (62).


## 6. Probabilistic Application

Definition 11.
LetX1,X2,…,Xr be a nonnegative random variable. Then X¯=X1,X2,…,Xr is said to be generalized degenerate Hermite distribution (GDHD), if its probability mass function is(64)PX¯=∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ,β,γ≥0,λ≥0;n∈ℕ0,where(65)Dα¯,γ,β,λ=∑ℓ1,ℓ2,…,ℓr=0∞∏i=1rαi−1ℓiHnγ−∑i=1rℓi−r,β,λ,and(66)Hnγ−∑i=1rℓi−r,β,λ=∑k=0n/2βkk!n!n−2k!∏i=0n−2k−1γ−∑i=0rxi−r−iλ.
Dα¯,γ,β,λ is convergent and positive for α¯r=α0,α1,…,αr−1,0<αr<1.


### 6.1. Statistical Properties of GDHD Model

#### 6.1.1. Cumulative Distribution Function

The cumulative distribution function (cdf) of GDHD is given by(67)PX≤xi=1−αixi+1Dα¯,γ+xi+1,β,λDα¯,γ,β,λ.


#### 6.1.2. Moments and Related Measures

The moment generating function is given by(68)MX¯t=Eetx=Dα¯et,γ,β,λDα¯,γ,β,λ,where X¯=x1,x2,…,xr and t=t1,t2,…,tr.Ther−th moments μr′ of DGHD is(69)μr′=∑x1,x2,…,xr=0∞xir∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.The mean and the variance of DGHD are(70)EX¯=∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.(71)VarX¯=∑x1,x2,…,xr=0∞xi2∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ−∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ2.


## 6.1. Statistical Properties of GDHD Model

### 6.1.1. Cumulative Distribution Function

The cumulative distribution function (cdf) of GDHD is given by(67)PX≤xi=1−αixi+1Dα¯,γ+xi+1,β,λDα¯,γ,β,λ.


### 6.1.2. Moments and Related Measures

The moment generating function is given by(68)MX¯t=Eetx=Dα¯et,γ,β,λDα¯,γ,β,λ,where X¯=x1,x2,…,xr and t=t1,t2,…,tr.Ther−th moments μr′ of DGHD is(69)μr′=∑x1,x2,…,xr=0∞xir∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.The mean and the variance of DGHD are(70)EX¯=∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.(71)VarX¯=∑x1,x2,…,xr=0∞xi2∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ−∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ2.


## 6.1.1. Cumulative Distribution Function

The cumulative distribution function (cdf) of GDHD is given by(67)PX≤xi=1−αixi+1Dα¯,γ+xi+1,β,λDα¯,γ,β,λ.


## 6.1.2. Moments and Related Measures

The moment generating function is given by(68)MX¯t=Eetx=Dα¯et,γ,β,λDα¯,γ,β,λ,where X¯=x1,x2,…,xr and t=t1,t2,…,tr.Ther−th moments μr′ of DGHD is(69)μr′=∑x1,x2,…,xr=0∞xir∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.The mean and the variance of DGHD are(70)EX¯=∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ.(71)VarX¯=∑x1,x2,…,xr=0∞xi2∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ−∑x1,x2,…,xr=0∞xi∏i=1rαi−1xiHnγ−∑i=1rxi−r,β,λDα¯,γ,β,λ2.



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