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DOI: 10.32604/cmes.2022.017272

ARTICLE

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

Ghulam Muhiuddin¹,*, Waseem A. Khan² and Deena Al-Kadi³

1 Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, 71491, Saudi Arabia
2 Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, 31952, Saudi Arabia
3 Department 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. [3–5], Kim et al. [6,7], Sharma et al. [8,9], Khan et al. [10–13], and Muhiuddin et al. [14–17] 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 [18–22])

∫01(1+z)ξ+ηdη=zlog⁡(1+z)(1+z)ξ=∑j=0∞Cj(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

(zez−1)αeξz=∑j=0∞Bj(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),(j≥0), (see [23]). (3)

For λ∈R, 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(ξ−λ)(ξ−2λ)⋯(ξ−(θ−1)λ),(θ≥1).

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

zeλ(z)−1eλx(z)=∑j=0∞βj,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=0∞Cj,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=0∞Cj,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=0∞Dj,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=0∞bj,λ(ξ)zjj!.(10)

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

For i≥0, the degenerate first kind Stirling numbers are defined by (see [28])

1i!(logλ⁡(1+z))i=∑j=i∞S1,λ(j,i)zjj!. (11)

Note that limλ→0S1,λ(j,i)=S1(j,i) are the first kind Stirling numbers given by

1i!(log⁡(1+z))i=∑j=i∞S1(j,i)zjj!,  (i≥0), (see [4,26]). (12)

For i≥0, the degenerate second kind Stirling numbers are given by

1i!(eλ(z)−1)i=∑j=i∞S2,λ(j,i)zjj!, (see [27]).(13)

We note that limλ→0S2,λ(j,i)=S2(j,i) are the second kind Stirling numbers given by

1i!(ez−1)i=∑j=i∞S2(j,i)zjj!,(i≥0),(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(j−1)!jk,(∣ξ∣<1,k∈Z). (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=0∞Gj,λ(k)(ξ)zjj!,(k∈Z). (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=0∞Cj,λ(k)(ξ)zjj!,(k∈Z). (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=0j∑η=0σ(jσ)Cj−σ,λ(1)η+1,λS1,λ(σ+1,η+1)(σ+1)(η+1)ϑ−1. (19)

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

∑j=0∞Cj,λ(k)zjj!=Eik,λ(logλ⁡(1+z))log⁡(1+1λlog⁡(1+λz))=zlog⁡(1+1λlog⁡(1+λz))1z∑η=1∞(1)η,λ(logλ⁡(1+z))η(η−1)!ηk=zlog⁡(1+1λlog⁡(1+λz))1z∑η=0∞(1)η+1,λ(logλ⁡(1+z))η+1η!(η+1)k(η+1)!(η+1)!=zlog⁡(1+1λlog⁡(1+λz))1z∑η=0∞(1)η+1,λ(η+1)k−1∑i=η+1∞S1,λ(i,η+1)zii!=zlog⁡(1+1λlog⁡(1+λz))∑η=0∞(1)η+1,λ(η+1)k−1∑η=i∞S1,λ(i+1,η+1)zi(i+1)!=(∑j=0∞Cj,λzjj!)(∑i=0∞∑η=0i(1)η+1,λ(η+1)k−1S1,λ(i+1,η+1)i+1zii!)=∑j=0∞(∑σ=0j∑η=0σ(jσ)Cj−σ,λ(1)η+1,λS1,λ(σ+1,η+1)(σ+1)(η+1)ϑ−1)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=0j∑η=0σ(jσ)Cj−σ,λ(1)η+1,λS1,λ(σ+1,η+1)σ+1,(j≥0).

Theorem 2.2. Let j be non negative number and k∈Z. Then

Cj,λ(k)(ξ)=∑σ=0j∑ρ=0j−σ(jσ)Cσ,λ(k)(ξ)ρλj−σ−ρS1(j−σ,ρ). (21)

Proof. Recall from (18), we have

∑j=0∞Cj,λ(k)(ξ)zjj!=Eik,λ(logλ⁡(1+z))log⁡(1+1λlog⁡(1+λz))(1+1λlog⁡(1+λz))ξ=(∑σ=0∞Cσ,λ(k)zσσ!)(∑ρ=0∞(ξρ)(1λlog⁡(1+λz))ρ)=(∑σ=0∞Cσ,λ(k)zσσ!)(∑ρ=0∞(ξ)ρλ−ρ∑s=ρ∞S1(s,ρ)λszss!)=(∑σ=0∞Cσ,λ(k)zσσ!)(∑s=0∞∑ρ=0s(ξ)ρλ−ρS1(s,ρ)λszss!)=∑j=0∞(∑σ=0j∑ρ=0j−σ(jσ)Cσ,λ(k)(ξ)ρλj−σ−ρS1(j−σ,ρ))zjj!. (22)

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

Theorem 2.3. Let j≥0. Then

Cj,λ(k)=∑r=0j(jr)∑r1+⋯+rk−1=r(rr1+⋯+rk−1)×br1,λ(λ−1)r1+1br2,λ(λ−1)r1+r2+1⋯brk−1,λ(λ−1)r1+⋯+rk−1+1Cj−r,λ.

Proof. Consider (15), we have

ddξEik,λ(logλ⁡(1+ξ))=ddξ∑j=1∞(1)j,λ(logλ⁡(1+ξ))j(j−1)!jk=(1+ξ)λ−1logλ⁡(1+ξ)∑j=1∞(1)j,λ(logλ⁡(1+ξ))j(j−1)!jk−1=(1+x)λ−1logλ⁡(1+ξ)Eik−1,λ(logλ⁡(1+ξ)).(23)

∑j=0∞Cj,λ(k)ξjj!=1log⁡(1+1λlog⁡(1+λξ))Eik,λ(logλ⁡(1+ξ))=1log⁡(1+1λlog⁡(1+λξ))∫0ξ(1+z)λ−1logλ⁡(1+z)∫0z⋯(1+z)λ−1logλ⁡(1+z)∫0z⏟(k−2)−times(1+z)λ−1logλ⁡(1+z)zdz⋯dz=ξlog⁡(1+1λlog⁡(1+λξ))∑m=0∞∑m1+⋯+mk−1=m(mm1+⋯+mk−1)×bm1,λ(λ−1)m1+1bm2,λ(λ−1)m1+m2+1⋯bmk−1,λ(λ−1)m1+⋯+mk−1+1ζmm!

∑j=0∞Cj,λ(k)ξjj!=∑j=0∞∑r=0j(jr)∑r1+⋯+rk−1=r(rr1+⋯+rk−1)×br1,λ(λ−1)r1+1br2,λ(λ−1)r1+r2+1⋯brk−1,λ(λ−1)r1+⋯+rk−1+1Cj−r,λξjj!,(24)

which complete the proof.

Corollary 2.2. Let j≥2. Then

Cj,λ(2)=∑r=0j(jr)br,λ(λ−1)r+1Cj−r,λ,(k≥2).

Theorem 2.4. The following result holds true

τk,λ(−q)=(−1)qCq,λ(k),(25)

k≥1 and q∈N⋃{0}, w∈C.

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

τk,λ(w)=1Γ(w)∫0∞zw−1log⁡(1+1λlog⁡(1+λz))Eik,λ(logλ⁡(1+z))dz=1Γ(w)∫01zw−1log⁡(1+1λlog⁡(1+λz))Eik,λ(logλ⁡(1+z))dz+1Γ(w)∫1∞zw−1log⁡(1+1λlog⁡(1+λz))Eik,λ(logλ⁡(1+z))dz. (26)

For any w∈C and absolutely converges, (26) to

limw→−q|1Γ(w)∫1∞zw−1log⁡(1+1λlog⁡(1+λz))Eik,λ(logλ⁡(1+z))dz|≤1Γ(−q)M=0. (27)

Eq. (27) can be written as

1Γ(w)∑l=0∞Cl,λ(k)l!1w+l,(ℜ(w)>0)

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

τk,λ(−q)=limw→−q1Γ(q)∫01zw−1log⁡(1+1λlog⁡(1+λz))Eik,λ(logλ⁡(1+z))dz=limw→−q1Γ(w)∫01zw−1∑l=0∞Cl,λ(k)zll!dz=limw→−q1Γ(w)∑l=0∞Cl,λ(k)w+l1l!=⋯+0+⋯+0+limw→−q1Γ(w)1w+qCq,λ(k)q!+0+0+⋯=limw→−q(Γ(1−w)sin⁡πwπ)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)Cj−σ(ξ)λ−σ−1(1)σ+1,λ(σ+1)k=∑σ=0jλj−σCσ,λ(k)(ξ)S2(j,σ).

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

∑i=0∞Ci,λ(k)(ξ)λ−i(eλ(z)−1)ii!=Eik,λ(zλ)log⁡(1+z)(1+z)ξ=(zlog⁡(1+z)(1+z)ξ)(Eik,λ(zλ)z)=(∑j=0∞Cj(ξ)zjj!)(1z∑i=1∞λ−i(1)i,λzi(i−1)!ik)=(∑j=0∞Cj(ξ)zjj!)(∑i=0∞λ−i−1(1)i+1,λzii!(i+1)k)=∑j=0∞(∑σ=0j(jσ)Cj−σ(ξ)λ−σ−1(1)σ+1,λ(σ+1)k)zjj!.(29)

On the other hand, we see that

∑i=0∞Ci,λ(k)(x)λ−i(eλ(z)−1)ii!=∑i=0∞Ci,λ(k)(ξ)λ−i∑j=i∞S2(j,i)λjzjj!=∑j=0∞(∑σ=0jλj−σCσ,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)σk−1

=∑σ=1η∑ρ=1σ(ησ)Cη−σ,λ(k)(ρ−1)!(−1)ρ−1λσ−ρS1(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)σk−1)zηη!. (32)

=(∑η=0∞Cη,λ(k)zηη!)(log⁡(1+1λlog⁡(1+λz)))=(∑η=0∞Cη,λ(k)zηη!)(log⁡(1+1λlog⁡(1+λz)))=(∑η=0∞Cη,λ(k)zηη!)(∑ρ=1∞(−1)ρ−1ρλ−ρ(log⁡(1+λz))ρ)=(∑η=0∞Cη,λ(k)zηη!)(∑ρ=1∞(ρ−1)!(−1)ρ−1λ−ρ∑σ=ρ∞S1(σ,ρ)λσzσσ!)=(∑η=0∞Cη,λ(k)zηη!)(∑σ=1∞(∑ρ=1σ(ρ−1)!(−1)ρ−1λσ−ρS1(σ,ρ))zσσ!)=∑η=1∞(∑σ=1η∑ρ=1σ(ησ)Cη−σ,λ(k)(ρ−1)!(−1)ρ−1λσ−ρS1(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=1η∑ρ=1σ(ησ)Cη−σ,λ(1)(ρ−1)!(−1)ρ−1λσ−ρS1(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)k−1Cs−r,λ∗λj−sBj−s(j−s).

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=0∞λlBj(j)zjj!)(1z∑q=1∞(1)q,λ(logλ(1+z))q(q−1)!qk)(∑l=0∞Cl,λ∗zll!)=(∑j=0∞λlBj(j)zjj!)(1z∑q=0∞(1)q+1,λ(logλ(1+z))q+1q!(q+1)k(q+1)!(q+1)!)(∑l=0∞Cl,λ∗zll!)=(∑j=0∞λlBj(j)zjj!)(1z∑q=0∞(1)q+1,λ(q+1)k−1∑r=q+1∞S1,λ(r,q+1)zrr!)(∑l=0∞Cl,λ∗zll!)=(∑j=0∞λjBj(j)zjj!)(∑r=0∞∑q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k−1zrr!)(∑s=0∞Cs,λ∗zss!)=(∑j=0∞λjBj(j)zjj!)(∑l=0∞∑r=0l(lr)∑q=0r(1)q+1,λS1,λ(r+1,q+1)(r+1)(q+1)k−1Cs−r,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)k−1Cs−r,λ∗λj−lBj−q(j−q))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)k−1Cj−r,λ∗.

Proof. Consider the following expression:

∑j=0∞Cj,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=0∞Cj,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))=(1z∑η=1∞(1)η,λ(logλ(1+z))η(η−1)!ηk)(∑l=0∞Cl,λ∗zll!)=(∑η=0∞(1)η+1,λ(logλ(1+z))η+1η!(η+1)k(η+1)!(η+1)!)(∑j=0∞Cj,λ∗zjj!)=(∑r=0∞∑η=0r(1)η+1,λS1,λ(r+1,η+1)(r+1)(η+1)k−1zrr!)(∑j=0∞Cj,λ∗zjj!)(35)

=∑j=0∞(∑r=0j(jr)∑η=0r(1)η+1,λS1,λ(r+1,η+1)(r+1)(η+1)k−1Cj−r,λ∗)zjj!.(36)

From Eq. (35), we have

∑j=1∞[Cj,λ(k)(1)−Cj,λ(k)]tj−1j!

=∑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=1∞p(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

limλ→0uk,λ(ξ|p)=∑j=1∞limλ→0p(j)(1)j,λξjjk=uk(ξ|p)=∑j=1∞p(j)ξjjk,(k∈Z)(40)

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

From (40), we have

uk(ξ|1)=∑j=1∞ξjjk=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=0∞Cj,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 j≥0. Then

Cj,λ,1Γ(k)(ξ)=Cj,λ(k)(ξ).

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

∑j=0∞Cj,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(m−1)!(1+1λlog⁡(1+λz))ξ=Eik,λ(logλ⁡(1+z))log⁡(1+1λlog⁡(1+λz))(1+1λlog⁡(1+λz))ξ=∑j=0∞Cj,λ(k)(ξ)zjj!. (43)

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

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

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

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

Corollary 3.1. Let j≥0. Then

Cj,λ,1Γ(k)=Cj,λ(k)=∑ζ=0∞∑η=0ζ(jl)S1,λ(j+1,ζ+1)Cj−ζ,λ(j+1)k−1(ζ+1).

Theorem 3.3. Let j≥0. Then

Cj,λ,p(k)(ξ)=∑ζ=0j∑η=0j−ζCζ,λ,p(k)(ξ)ηλs−ηS1(j−ζ,η).(46)

Proof. Recall from (42), we see that

∑j=0∞Cj,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=0∞Cζ,λ,p(k)zζζ!)(∑η=0∞(ξ)ηλs−η∑s=η∞S1(s,η)zss!)

=(∑ζ=0∞Cζ,λ,p(k)zζζ!)(∑s=0∞∑η=0s(ξ)ηλs−ηS1(s,η)zss!)

=∑j=0∞(∑ζ=0j∑η=0j−ζCζ,λ,p(k)(ξ)ηλs−ηS1(j−ζ,η))zjj!. (47)

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

Theorem 3.4. Let j≥0. Then

Cj,λ,p(k)=∑ζ=0j∑a=0j−ζ∑η=0ζ(jζ)(ja)Dj−ζ−a,λCa,λp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1). (48)

Proof. Using (42), we have

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

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

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

=logλ⁡(1+z)log⁡(1+1λlog⁡(1+λz))∑η=0∞p(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=0∞p(η+1)(1)η+1,λ(η+1)!(η+1)k∑ζ=η∞S1,λ(ζ,η+1)zζζ!

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

=(∑b=0∞∑a=0b(ba)Db−a,λCa,λzbb!)(∑ζ=0∞∑η=0ζp(η+1)(1)η+1,λ(η+1)!(η+1)kS1,λ(ζ,η+1)zζζ!)

=∑j=0∞(∑ζ=0j∑a=0j−ζ∑η=0ζ(jζ)(ja)Dj−ζ−a,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+3661ξ324−99ξ232+ξ3C4,λ(k)(ξ)=−19534760142599720000000+263515ξ13824+55549ξ2648−129ξ38+ξ4C5,λ(k)(ξ)=170447475823913888000000−281099ξ9000000−46995065ξ227648+416095ξ3972−1125ξ432+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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