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 Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.017272

ARTICLE

Some Identities of the Degenerate Poly-Cauchy and Unipoly Cauchy Polynomials of the Second Kind

1Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, 71491, Saudi Arabia
2Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, 31952, Saudi Arabia
3Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
*Corresponding Author: Ghulam Muhiuddin. Email: chishtygm@gmail.com
Received: 27 April 2021; Accepted: 11 February 2022

Abstract: In this paper, we introduce modified degenerate polyexponential Cauchy (or poly-Cauchy) polynomials and numbers of the second kind and investigate some identities of these polynomials. We derive recurrence relations and the relationship between special polynomials and numbers. Also, we introduce modified degenerate unipoly-Cauchy polynomials of the second kind and derive some fruitful properties of these polynomials. In addition, positive associated beautiful zeros and graphical representations are displayed with the help of Mathematica.

Keywords: Modified degenerate polyexponential functions; modified degenerate polyexponential Cauchy (or poly-Cauchy) polynomials of the second kind; degenerate unipoly-Cauchy polynomials of the second kind

1  Introduction

Recently, many mathematicians, specifically Carlitz [1,2], Kim et al. [35], Kim et al. [6,7], Sharma et al. [8,9], Khan et al. [1013], and Muhiuddin et al. [1417] have studied and added diverse degenerate versions of many special polynomials and numbers (like as degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Fubini polynomials, degenerate Stirling numbers of the first and second kind, and so on). In this paper, we focus on modified degenerate polyexponential Cauchy (or poly-Cauchy) polynomials and the numbers of the second type. The purpose of this paper is to introduce a degenerate model of the poly-Cauchy polynomials and numbers of the second type, the so-called degenerate poly-Cauchy polynomials, and numbers of the second type, constructed from the degenerate polyexponential feature. We derive some express expressions and identities for the one’s numbers and polynomials.

Let Cj(ξ) be the Cauchy polynomials which are given by the following generating function (see [1822])

01(1+z)ξ+ηdη=zlog(1+z)(1+z)ξ=j=0Cj(ξ)zjj!. (1)

In the case when ξ=0,Cj=Cj(0) are the Cauchy numbers.

The Bernoulli polynomials of order α are given by

(zez1)αeξz=j=0Bj(α)(ξ)zjj!, (see [21,23]). (2)

For ξ=0, Bj(α)=Bj(α)(0) are the Bernoulli numbers of order α.

We note that

Cj(ξ)=Bj(j)(ξ+1),(j0), (see [23]). (3)

For λR, the degenerate exponential functions are defined as

eλξ(z)=(1+λz)ξλ,eλ(z):=eλ1(z)=(1+λz)1λ,(see [1–17]). (4)

By (4) and binomial theorem, we have

eλξ(z)=θ=0(ξ)θ,λzθθ!, (5)

where (ξ)0,λ=1,(ξ)θ,λ=ξ(ξλ)(ξ2λ)(ξ(θ1)λ),(θ1).

The degenerate Bernoulli polynomials are defined by (see [1,2])

zeλ(z)1eλx(z)=j=0βj,λ(x)zjj!. (6)

On putting x = 0, βj,λ=βj,λ(0) are called the degenerate Bernoulli numbers.

The degenerate Cauchy polynomials Cj,λ(ξ) are defined by Kim [25] as follows:

01(1+log(1+λz)1λ)ξ+ηdη=1λ(log(1+λz))log(1+1λlog(1+λz))(1+log(1+λz)1λ)ξ=j=0Cj,λ(ξ)zjj!. (7)

Letting ξ=0,Cj,λ=Cj,λ(0) are the degenerate Cauchy numbers.

In the year 2017, Kim [24] introduced and studied the new class of degenerate Cauchy polynomials Cj,λ(ξ) of the second kind are given by

zlog(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=j=0Cj,λ(ξ)zjj!,(see [16,25,27]). (8)

At the point when ξ=0,Cj,λ=Cj,λ(0) are the degenerate Cauchy numbers of the second kind.

The degenerate Daehee polynomials Dj,λ(x) are defined by (see [6])

logλ(1+z)z(1+z)ξ=j=0Dj,λ(ξ)zjj!. (9)

On setting ξ=0,Dj,λ=Dj,λ(0) are the degenerate Daehee numbers.

The degenerate Bernoulli polynomials of the second kind are defined by (see [6])

zlogλ(1+z)(1+z)ξ=j=0bj,λ(ξ)zjj!.(10)

Letting ξ=0, bj,λ=bj,λ(0) are the degenerate Bernoulli numbers of the second kind.

For i0, the degenerate first kind Stirling numbers are defined by (see [28])

1i!(logλ(1+z))i=j=iS1,λ(j,i)zjj!. (11)

Note that limλ0S1,λ(j,i)=S1(j,i) are the first kind Stirling numbers given by

1i!(log(1+z))i=j=iS1(j,i)zjj!,  (i0), (see [4,26]). (12)

For i0, the degenerate second kind Stirling numbers are given by

1i!(eλ(z)1)i=j=iS2,λ(j,i)zjj!, (see [27]).(13)

We note that limλ0S2,λ(j,i)=S2(j,i) are the second kind Stirling numbers given by

1i!(ez1)i=j=iS2(j,i)zjj!,(i0),(see [1–29]). (14)

In this paper, Section 3 incorporates the definition of degenerate poly-Cauchy polynomials of the second kind and a preliminary study of these polynomials. Section 4 is a consequence of the definition of the degenerate unipoly-Cauchy polynomials and unipoly polynomials combined with their properties and special cases. Finally, some computational values of degenerate poly-Cauchy polynomials of the second kind are given in Section 5.

2  Degenerate Poly-Cauchy Polynomials and Numbers of the Second Kind

In this segment, we introduce degenerate poly-Cauchy polynomials of the second kind, derived with the aid of modified degenerate polyexponential functions and some identities of these polynomials.

Recently, Kim et al. [4] delivered the modified degenerate polyexponential function defined by

Eik,λ(ξ)=j=1(1)j,λxj(j1)!jk,(ξ∣<1,kZ). (15)

Thus, by

Ei1,λ(ξ)=j=1(1)j,λξjj!=eλ(ξ)1. (16)

The modified degenerate polyexponential Genocchi (or poly-Genocchi) polynomials are defined by Kim et al. to be (see [7])

Eik,λ(logλ(1+z))eλ(z)+1eλξ(z)=j=0Gj,λ(k)(ξ)zjj!,(kZ). (17)

At the point when ξ=0, Gj,λ(k)=Gj,λ(k)(0) are the degenerate poly-Genocchi numbers.

By the above definitions, we introduce modified degenerate polyexponential Cauchy (or poly-Cauchy) polynomials of the second kind as

Eik,λ(logλ(1+z))log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=j=0Cj,λ(k)(ξ)zjj!,(kZ). (18)

When ξ=0, Cj,λ(k)=Cj,λ(k)(0) are the modified degenerate polyexponential Cauchy (or poly-Cauchy) numbers of the second kind.

Theorem 2.1. Let j be non negative number. Then

Cj,λ(k)=σ=0jη=0σ(jσ)Cjσ,λ(1)η+1,λS1,λ(σ+1,η+1)(σ+1)(η+1)ϑ1. (19)

Proof. Using (11) and (18), we have

j=0Cj,λ(k)zjj!=Eik,λ(logλ(1+z))log(1+1λlog(1+λz))=zlog(1+1λlog(1+λz))1zη=1(1)η,λ(logλ(1+z))η(η1)!ηk=zlog(1+1λlog(1+λz))1zη=0(1)η+1,λ(logλ(1+z))η+1η!(η+1)k(η+1)!(η+1)!=zlog(1+1λlog(1+λz))1zη=0(1)η+1,λ(η+1)k1i=η+1S1,λ(i,η+1)zii!=zlog(1+1λlog(1+λz))η=0(1)η+1,λ(η+1)k1η=iS1,λ(i+1,η+1)zi(i+1)!=(j=0Cj,λzjj!)(i=0η=0i(1)η+1,λ(η+1)k1S1,λ(i+1,η+1)i+1zii!)=j=0(σ=0jη=0σ(jσ)Cjσ,λ(1)η+1,λS1,λ(σ+1,η+1)(σ+1)(η+1)ϑ1)zjj!. (20)

Therefore, by (15) and (20), we obtain the result (19).

Corollary 2.1. Let j be non negative number. Then

Cj,λ(1)=σ=0jη=0σ(jσ)Cjσ,λ(1)η+1,λS1,λ(σ+1,η+1)σ+1,(j0).

Theorem 2.2. Let j be non negative number and kZ. Then

Cj,λ(k)(ξ)=σ=0jρ=0jσ(jσ)Cσ,λ(k)(ξ)ρλjσρS1(jσ,ρ). (21)

Proof. Recall from (18), we have

j=0Cj,λ(k)(ξ)zjj!=Eik,λ(logλ(1+z))log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=(σ=0Cσ,λ(k)zσσ!)(ρ=0(ξρ)(1λlog(1+λz))ρ)=(σ=0Cσ,λ(k)zσσ!)(ρ=0(ξ)ρλρs=ρS1(s,ρ)λszss!)=(σ=0Cσ,λ(k)zσσ!)(s=0ρ=0s(ξ)ρλρS1(s,ρ)λszss!)=j=0(σ=0jρ=0jσ(jσ)Cσ,λ(k)(ξ)ρλjσρS1(jσ,ρ))zjj!. (22)

Thus by (18) and (22), the proof is completed.

Theorem 2.3. Let j0. Then

Cj,λ(k)=r=0j(jr)r1++rk1=r(rr1++rk1)×br1,λ(λ1)r1+1br2,λ(λ1)r1+r2+1brk1,λ(λ1)r1++rk1+1Cjr,λ.

Proof. Consider (15), we have

ddξEik,λ(logλ(1+ξ))=ddξj=1(1)j,λ(logλ(1+ξ))j(j1)!jk=(1+ξ)λ1logλ(1+ξ)j=1(1)j,λ(logλ(1+ξ))j(j1)!jk1=(1+x)λ1logλ(1+ξ)Eik1,λ(logλ(1+ξ)).(23)

j=0Cj,λ(k)ξjj!=1log(1+1λlog(1+λξ))Eik,λ(logλ(1+ξ))=1log(1+1λlog(1+λξ))0ξ(1+z)λ1logλ(1+z)0z(1+z)λ1logλ(1+z)0z(k2)times(1+z)λ1logλ(1+z)zdzdz=ξlog(1+1λlog(1+λξ))m=0m1++mk1=m(mm1++mk1)×bm1,λ(λ1)m1+1bm2,λ(λ1)m1+m2+1bmk1,λ(λ1)m1++mk1+1ζmm!

j=0Cj,λ(k)ξjj!=j=0r=0j(jr)r1++rk1=r(rr1++rk1)×br1,λ(λ1)r1+1br2,λ(λ1)r1+r2+1brk1,λ(λ1)r1++rk1+1Cjr,λξjj!,(24)

which complete the proof.

Corollary 2.2. Let j2. Then

Cj,λ(2)=r=0j(jr)br,λ(λ1)r+1Cjr,λ,(k2).

Theorem 2.4. The following result holds true

τk,λ(q)=(1)qCq,λ(k),(25)

k1 and qN{0}, wC.

Proof. Let us define the function τk,λ(w) as

τk,λ(w)=1Γ(w)0zw1log(1+1λlog(1+λz))Eik,λ(logλ(1+z))dz=1Γ(w)01zw1log(1+1λlog(1+λz))Eik,λ(logλ(1+z))dz+1Γ(w)1zw1log(1+1λlog(1+λz))Eik,λ(logλ(1+z))dz. (26)

For any wC and absolutely converges, (26) to

limwq|1Γ(w)1zw1log(1+1λlog(1+λz))Eik,λ(logλ(1+z))dz|1Γ(q)M=0. (27)

Eq. (27) can be written as

1Γ(w)l=0Cl,λ(k)l!1w+l,((w)>0)

In view of (26) and (27), we have

τk,λ(q)=limwq1Γ(q)01zw1log(1+1λlog(1+λz))Eik,λ(logλ(1+z))dz=limwq1Γ(w)01zw1l=0Cl,λ(k)zll!dz=limwq1Γ(w)l=0Cl,λ(k)w+l1l!=+0++0+limwq1Γ(w)1w+qCq,λ(k)q!+0+0+=limwq(Γ(1w)sinπwπ)w+qCq,λ(k)q!=Γ(1+q)cos(πq)Cq,λ(k)q!=(1)qCq,λ(k).(28)

By (28), we obtain the result.

Theorem 2.5. Let j be non-negative number. Then

σ=0j(jσ)Cjσ(ξ)λσ1(1)σ+1,λ(σ+1)k=σ=0jλjσCσ,λ(k)(ξ)S2(j,σ).

Proof. By changing z with 1λeλ(z)1 in (18) that

i=0Ci,λ(k)(ξ)λi(eλ(z)1)ii!=Eik,λ(zλ)log(1+z)(1+z)ξ=(zlog(1+z)(1+z)ξ)(Eik,λ(zλ)z)=(j=0Cj(ξ)zjj!)(1zi=1λi(1)i,λzi(i1)!ik)=(j=0Cj(ξ)zjj!)(i=0λi1(1)i+1,λzii!(i+1)k)=j=0(σ=0j(jσ)Cjσ(ξ)λσ1(1)σ+1,λ(σ+1)k)zjj!.(29)

On the other hand, we see that

i=0Ci,λ(k)(x)λi(eλ(z)1)ii!=i=0Ci,λ(k)(ξ)λij=iS2(j,i)λjzjj!=j=0(σ=0jλjσCσ,λ(k)(ξ)S2(j,σ))zjj!. (30)

In view of (29) and (30), we obtain the result.

Theorem 2.6. Let η be non-negative number. Then

σ=1η(1)σ,λS1,λ(η,σ)σk1

=σ=1ηρ=1σ(ησ)Cησ,λ(k)(ρ1)!(1)ρ1λσρS1(σ,ρ). (31)

Proof. Consider the Eq. (18), we have

Eik,λ(logλ(1+z))=η=1(σ=1η(1)σ,λS1,λ(η,σ)σk1)zηη!. (32)

=(η=0Cη,λ(k)zηη!)(log(1+1λlog(1+λz)))=(η=0Cη,λ(k)zηη!)(log(1+1λlog(1+λz)))=(η=0Cη,λ(k)zηη!)(ρ=1(1)ρ1ρλρ(log(1+λz))ρ)=(η=0Cη,λ(k)zηη!)(ρ=1(ρ1)!(1)ρ1λρσ=ρS1(σ,ρ)λσzσσ!)=(η=0Cη,λ(k)zηη!)(σ=1(ρ=1σ(ρ1)!(1)ρ1λσρS1(σ,ρ))zσσ!)=η=1(σ=1ηρ=1σ(ησ)Cησ,λ(k)(ρ1)!(1)ρ1λσρS1(σ,ρ))zηη!. (33)

The complete of the Proof.

Corollary 2.3. Let η be non-negative number, we have

σ=1η(1)σ,λS1,λ(η,σ)=σ=1ηρ=1σ(ησ)Cησ,λ(1)(ρ1)!(1)ρ1λσρS1(σ,ρ).

Theorem 2.7. Let j be non-negative number. Then

Cj,λ(k)=s=0j(js)r=0s(sr)q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k1Csr,λλjsBjs(js).

Proof. We observe that

Eik,λ(logλ(1+z))log(1+1λlog(1+λz))=(λzlog(1+λz))(Eik,λ(logλ(1+z))z)(1λlog(1+λz)log(1+1λlog(1+λz)))=(j=0λlBj(j)zjj!)(1zq=1(1)q,λ(logλ(1+z))q(q1)!qk)(l=0Cl,λzll!)=(j=0λlBj(j)zjj!)(1zq=0(1)q+1,λ(logλ(1+z))q+1q!(q+1)k(q+1)!(q+1)!)(l=0Cl,λzll!)=(j=0λlBj(j)zjj!)(1zq=0(1)q+1,λ(q+1)k1r=q+1S1,λ(r,q+1)zrr!)(l=0Cl,λzll!)=(j=0λjBj(j)zjj!)(r=0q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k1zrr!)(s=0Cs,λzss!)=(j=0λjBj(j)zjj!)(l=0r=0l(lr)q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k1Csr,λ)zss!=j=0(s=0j(jl)r=0s(sr)q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k1Csr,λλjlBjq(jq))zjj!. (34)

Therefore, by (15) and (34), we acquire the desired result.

Theorem 2.8. Let j be non-negative number. Then

Cj+1,λ(k)(1)Cj+1,λ(k)j+1=r=0j(jr)η=0r(1)η+1,λS1,λ(r+1,η+1)(r+1)(η+1)k1Cjr,λ.

Proof. Consider the following expression:

j=0Cj,λ(k)(1)zjj!=Eik,λ(logλ(1+z))log(1+1λlog(1+λz))(1+1λlog(1+λz)).=Eik,λ(logλ(1+z))log(1+1λlog(1+λz))+Eik,λ(logλ(1+z))1λlog(1+λz)log(1+1λlog(1+λz))=j=0Cj,λ(k)zjj!+zEik,λ(logλ(1+z))z1λlog(1+λz)log(1+1λlog(1+λz))j=1[ Cj,λ(k)(1)Cj,λ(k) ]zjj!=zEik,λ(logλ(1+z))z1λlog(1+λz)log(1+1λlog(1+λz))=(1zη=1(1)η,λ(logλ(1+z))η(η1)!ηk)(l=0Cl,λzll!)=(η=0(1)η+1,λ(logλ(1+z))η+1η!(η+1)k(η+1)!(η+1)!)(j=0Cj,λzjj!)=(r=0η=0r(1)η+1,λS1,λ(r+1,η+1)(r+1)(η+1)k1zrr!)(j=0Cj,λzjj!)(35)

=j=0(r=0j(jr)η=0r(1)η+1,λS1,λ(r+1,η+1)(r+1)(η+1)k1Cjr,λ)zjj!.(36)

From Eq. (35), we have

j=1[Cj,λ(k)(1)Cj,λ(k)]tj1j!

=j=0[Cj+1,λ(k)(1)Cj+1,λ(k)j+1]zjj!. (37)

Thus, by (36) and (37), we complete the proof.

3  DegenerAte Unipoly-Cauchy Polynomials of the Second Kind

In this section, we introduce degenerate unipoly-Cauchy polynomials of the second kind by using degenerate unipoly function and derive the relationships between degenerate Daehee polynomials and degenerate Cauchy polynomials of the second kind.

In [25], Dolgy and Khan introduced degenerate unipoly function given by

uk,λ(ξ|p)=j=1p(j)(1)j,λξjjk(38)

Note that, we have

uk,λ(ξ|1Γ)=Eik,λ(ξ)(39)

is the modified degenerate polyexponential function, where Γ:=Γ(j) is well-known as Gamma function.

It is clear that

limλ0uk,λ(ξ|p)=j=1limλ0p(j)(1)j,λξjjk=uk(ξ|p)=j=1p(j)ξjjk,(kZ)(40)

are called the unipoly function attached to polynomials p(x) (see [3]).

From (40), we have

uk(ξ|1)=j=1ξjjk=Lik(ξ), (see [26]), (41)

is the ordinary polylogarithm function.

By using (15) and (38), the degenerate unipoly-Cauchy polynomials of the second kind is given by the following generating function

uk,λ(logλ(1+z)|p)log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=j=0Cj,λ,p(k)(ξ)zjj!. (42)

In the case when ξ=0,Cj,λ,p(k)=Cj,λ,p(k)(0) are the degenerate unipoly-Cauchy numbers of the second kind.

Theorem 3.1. Let j0. Then

Cj,λ,1Γ(k)(ξ)=Cj,λ(k)(ξ).

Proof. On taking p(j)=1Γλ. Then we have

j=0Cj,λ,1Γ(k)(ξ)zjj!=uk,λ(logλ(1+z)|1Γp)log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=1log(1+1λlog(1+λz))m=1(1)m,λ(logλ(1+z))mmk(m1)!(1+1λlog(1+λz))ξ=Eik,λ(logλ(1+z))log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ=j=0Cj,λ(k)(ξ)zjj!. (43)

In view of (43), we obtain the result.

Theorem 3.2. Let j be non-negative number. Then

=j=0Cj,λ(k)zjj!+zEik,λ(logλ(1+z))z1λlog(1+λz)log(1+1λlog(1+λz)) (44)

Proof. Consider the Eq. (42), we have

j=0Cj,λ,p(k)zjj!=uk,λ(logλ(1+z)|p)log(1+1λlog(1+λz))=1log(1+1λlog(1+λz))η=1p(η)(1)η,ληk(logλ(1+z))η=1log(1+1λlog(1+λz))η=0p(η+1)(1)η+1,λ(η+1)!(η+1)kζ=η+1S1,λ(η+1,ζ)zζζ!=(j=0Cj,λzjj!)(η=0ζ=0ηp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1)zζζ!)=j=0(ζ=0η=0ζ(jζ)p(η+1)(1)η+1,λ(η+1)!S1,λ(ζ+1,η+1)Cjζ,λ(η+1)k(ζ+1))zjj!.(45)

By (42) and (45), we complete the proof.

Corollary 3.1. Let j0. Then

Cj,λ,1Γ(k)=Cj,λ(k)=ζ=0η=0ζ(jl)S1,λ(j+1,ζ+1)Cjζ,λ(j+1)k1(ζ+1).

Theorem 3.3. Let j0. Then

Cj,λ,p(k)(ξ)=ζ=0jη=0jζCζ,λ,p(k)(ξ)ηλsηS1(jζ,η).(46)

Proof. Recall from (42), we see that

j=0Cj,λ(k,p)(ξ)zjj!=uk,λ(logλ(1+z)|p)log(1+1λlog(1+λz))(1+1λlog(1+λz))ξ

=uk,λ(logλ(1+z)|p)log(1+1λlog(1+λz))η=0(ξη)(1λlog(1+λz))η

=(ζ=0Cζ,λ,p(k)zζζ!)(η=0(ξ)ηλsηs=ηS1(s,η)zss!)

=(ζ=0Cζ,λ,p(k)zζζ!)(s=0η=0s(ξ)ηλsηS1(s,η)zss!)

=j=0(ζ=0jη=0jζCζ,λ,p(k)(ξ)ηλsηS1(jζ,η))zjj!. (47)

Thus, by (47), we get the desired result.

Theorem 3.4. Let j0. Then

Cj,λ,p(k)=ζ=0ja=0jζη=0ζ(jζ)(ja)Djζa,λCa,λp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1). (48)

Proof. Using (42), we have

j=0Cj,λ,p(k)zjj!=uk,λ(logλ(1+z)|p)log(1+1λlog(1+λz))

=1log(1+1λlog(1+λz))η=1p(η)(1)η,ληk(logλ(1+z))η+1

=1log(1+1λlog(1+λz))η=0p(η+1)(1)η+1,λ(η+1)k(logλ(1+z))η+1

=logλ(1+z)log(1+1λlog(1+λz))η=0p(η+1)(1)η+1,λ(η+1)!(η+1)k(η+1)!(logλ(1+z))η+1

=logλ(1+z)zzlog(1+1λlog(1+λz))η=0p(η+1)(1)η+1,λ(η+1)!(η+1)kζ=ηS1,λ(ζ,η+1)zζζ!

=(s=0Ds,λzss!)(a=0Ca,λzaa!)(ζ=0η=0ζp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1)zζζ!)

=(b=0a=0b(ba)Dba,λCa,λzbb!)(ζ=0η=0ζp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1)zζζ!)

=j=0(ζ=0ja=0jζη=0ζ(jζ)(ja)Djζa,λCa,λp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1))zjj!. (49)

In view of (49), we complete the proof.

4  Computational Values and Graphical Representation of Degenerate Poly-Cauchy Polynomials of the Second Kind

In this section, sure numerical computations are carried out to calculate sure contributors of the degenerate poly-Cauchy polynomials of the second kind and display some graphical representations. The first six individuals of Cj,λ(k)(ξ) are calculated and given as

C0,λ(k)(ξ)=1,C1,λ(k)(ξ)=15932+ξC2,λ(k)(ξ)=437233888+63ξ16+ξ2C3,λ(k)(ξ)=84455318432+3661ξ32499ξ232+ξ3C4,λ(k)(ξ)=19534760142599720000000+263515ξ13824+55549ξ2648129ξ38+ξ4C5,λ(k)(ξ)=170447475823913888000000281099ξ900000046995065ξ227648+416095ξ39721125ξ432+ξ5

To show the behavior of Cj,λ(k)(ξ), we display the graph of Cj,λ(k)(ξ) for k = 7 and λ=5, this graph is presented in Fig. 1.

Figure 1: Graph of Cj,λ(k)(ξ), j=1,2,...,12

5  Conclusions

In this paper, we have presented the degenerate poly-Cauchy numbers and polynomials of the second kind and discussed, in particular, some interesting series representations. We have deduced some relevant properties by using the structure and the relations satisfied by the recently degenerate polyexponential functions. Section 3 incorporates the definition of degenerate poly-Cauchy polynomials of the second kind and a preliminary study of these polynomials. Section 4 is a consequence of the definition of the degenerate unipoly-Cauchy polynomials and unipoly polynomials combined with their properties and special cases. Finally, some computational values of degenerate poly-Cauchy polynomials of the second kind are given in Section 5.

Acknowledgement: The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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