# Construction of Type 2 Poly-Changhee Polynomials and Its Applications

Authors: Ghulam Muhiuddin; Waseem Ahmad Khan; Jihad Younis
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce type 2 poly-Changhee polynomials by using the polyexponential function. We derive some explicit expressions and identities for these polynomials, and we also prove some relationships between poly-Changhee polynomials and Stirling numbers of the first and second kind. Also, we introduce the unipoly-Changhee polynomials by employing unipoly function and give multifarious properties. Furthermore, we provide a correlation between the unipoly-Changhee polynomials and the classical Changhee polynomials.


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## Body

## 1. Introduction

Special polynomials and their generating functions have vital roles in several branches of arithmetic, probability, statistics, mathematical physics, and additionally engineering. Since polynomials are appropriate for applying well-known operations like by-product and integral, polynomials are helpful to check real-world issues within the said areas. As an example, generating functions for special polynomials with their congruousness properties, repetition relations, process formulae, and regular add involving these polynomials are studied in recent years (see [1–4]).Forj≥0, the Stirling numbers of the first kind are defined by the following (see [1, 2, 5–14]):(1)ξj=∑l=0jS1j,lξl,where ξ0=1 and ξj=ξξ−1,…,ξ−j+1,j≥1. From (1), it is easy to see that (see [4, 15–22])(2)1r!log1+zr=∑j=r∞S1j,rzjj!,r≥0.Forj≥0, the Stirling numbers of the second kind are defined by the following (see [3, 11–16]):(3)ξj=∑l=0jS2j,lξl.From (3), we see that(4)1r!ez−1r=∑j=r∞S2j,rzjj!.The BernoulliBjξ, Euler Ejξ, and Genocchi Gjξ polynomials are defined by the following (see [1, 6, 7]):(5)zez−1eξz=∑j=0∞Bjξzjj!,z<2π,2ez+1eξz=∑j=0∞Ejξzjj!,z<π,(6)2zez+1eξz=∑j=0∞Gjξzjj!,z<π,respectively.Let the Changhee polynomialsChjξ be given by the following (see [2, 13, 14, 16]):(7)∫ℤp1+zξ+ηdμ−1η=22+z1+zξ=∑j=0∞Chjξzjj!.In the case whenξ=0,Chj=Chj0 are called the Changhee numbers.Moreover, we have the following (see [13]):(8)Chjξ=∑l=0jElξS1j,l,Ejξ=∑l=0jChlξS2j,l,j≥0.The polyexponential function as an inverse to the polylogarithm function is defined by Kim and Kim [12] to be(9)Eikξ=∑j=1∞ξjj−1!jk,k∈ℤ.We note that(10)Ei1ξ=∑j=1∞ξjj!=eξ−1.In 2019, Kim and Kim [12] introduced the poly-Bernoulli polynomials which are defined by(11)Eiklog1+zez−1eξz=∑j=0∞Bjkξzjj!.Lettingξ=0, Bjk=Bjk0 are called the poly-Bernoulli numbers.Fork∈ℤ, the polylogarithm function is defined by the following (see [5, 21]):(12)Likξ=∑j=1∞ξjjk,ξ<1.Note that(13)Li1ξ=∑j=1∞ξjj=−log1−ξ.Lee et al. [22] introduced the type 2 poly-Euler polynomials which are given by(14)Eik1+2zzez+1eξz=∑j=0∞Ejkξzjj!.In the case whenξ=0, Ejk=Ejk0 are called the type 2 poly-Euler numbers.The Daehee polynomials are defined by the following (see [7, 16]):(15)log1+zz1+zξ=∑j=0∞Djξzjj!.Whenξ=0, Dj=Dj0 are called the Daehee numbers.The following paper is as follows. In Section2, we introduce type 2 poly-Changhee polynomials and numbers and derive some identities of these polynomials. We derive some recurrence relations and relationships between Bernoulli number, Euler numbers, and Daehee numbers. In Section 3, we introduce unipoly-Changhee polynomials and investigate some identities of these polynomials.


## 2. Type 2 Poly-Changhee Numbers and Polynomials

In this section, we define type 2 poly-Changhee polynomials by using the polyexponential functions and represent the usual Changhee polynomials (more precisely, the values of Changhee polynomials at 1) whenk=1. At the same time, we give explicit expressions and identities involving polynomials.Fork∈ℤ, we define type 2 poly-Changhee polynomials by means of the following exponential generating function (in a suitable neighborhood of z=0) including the polyexponential function given as follows:(16)Eiklog1+2zz2+z1+zξ=∑j=0∞Chjkξzjj!.At the pointξ=0, Chjk=Chjk0 are called type 2 poly-Changhee numbers.Fork=1, by using (9) and (16), we see that(17)Ei1log1+2zz2+z1+zξ=22+z1+zξ=∑j=0∞Chjξzjj!,j≥0,where Chjξ are called the Changhee polynomials (see equation (7)).Theorem 1.
Letj be the nonnegative number and k∈ℤ. Then,(18)Chjkξ=∑l=0jjl∑i=0l1i+1k−1S1l+1,i+12ll+1Chj−lξ.Proof.
By (16), we have(19)Eiklog1+2zz2+z1+zξ=1z2+z1+zξ∑i=0∞1i+1k−1∑l=i+1∞S1l,i+12lzll!=22+z1+zξ∑i=0∞1i+1k−1∑l=i∞S1l+1,i+1l+12lzll!=∑j=0∞Chjξzjj!∑i=0∞1i+1k−1∑l=i∞S1l+1,i+12ll+1zll!=∑j=0∞∑l=0jjl∑i=0l1i+1k−1S1l+1,i+12ll+1Chj−lξzjj!.
In view of (16) and (19), we get (17).Corollary 1.
Letj be the nonnegative number. Then,(20)Chjξ=∑l=0jjl∑i=0lS1l+1,i+12ll+1Chj−lξ.
The higher-order Bernoulli polynomials are defined by the following (see [11]):(21)zlog1+zr1+zξ−1=∑j=0∞Bjj−r+1ξzjj!,r∈ℂ,where Bjrξ are the Bernoulli polynomials of order α by(22)zez−1reξz=∑j=0∞Bjrξzjj!.Theorem 2.
Letj be the nonnegative number. Then,(23)Chjk=∑l=0jjl∑l=0∞2l×∑l1+⋯+lk−1=lll1,…,lk−1Bl1l1l1+1Bl2l2l1+l2+1⋯Blk−1lk−1l1+⋯+lk−1+1Chj−l.Proof.
From (4), we have(24)ddξEiklog1+2ξ=ddξ∑j=1∞log1+2ξjj−1!jk=21+2ξ∑j=1∞jlog1+2ξj−1j−1!jk=21+2ξlog1+2ξ∑j=1∞log1+2ξjj−1!jk−1=21+2ξlog1+2ξEik−1log1+2ξ.
Fork≥1, using equations (21) and (24), we find(25)Eiklog1+2ξ=∫0ξ11+2zlog1+2z×∫0z11+2zlog1+2z⋯∫0z︸k−2−timesz1+2zlog1+2zdzdz,…,dz.=2ξ∑l=0∞2l∑l1+⋯+lk−1=lll1,…,lk−1Bl1l1l1+1Bl2l2l1+l2+1⋯Blk−1lk−1l1+⋯+lk−1+1ξll!.
From (25), we observe that(26)∑j=0∞Chjkξjj!=1ξξ+2Eiklog1+2ξ=2ξξξ+2×∑l=0∞2l∑l1+⋯+lk−1=lll1,…,lk−1Bl1l1l1+1Bl2l2l1+l2+1⋯Blk−1lk−1l1+⋯+lk−1+1ξll!=∑j=0∞∑l=0jjl∑l=0∞2l×∑l1+⋯+lk−1=lll1,…,lk−1Bl1l1l1+1Bl2l2l1+l2+1⋯Blk−1lk−1l1+⋯+lk−1+1Chj−lξjj!.
In view of (25) and (26), we complete the theorem.Corollary 2.
Letk≥2, then(27)Chj2=∑l=0jjl2lBlll+1Chj−l.Theorem 3.
Forj≥0, we have(28)Chj2ξ+η=∑i=0jjiChj−ikξηi.Proof.
Considering (16), we have(29)∑j=0∞Chjkξ+ηzjj!=Eiklog1+2zz2+z1+zξ+η=∑j=0∞Chjkξzjj!∑i=0∞ηizii!=∑j=0∞∑i=0jjiChj−ikξηizjj!.
So, the proof is completed.Theorem 4.
Letj≥1, then(30)2Chjk+jChj−1k=2j+1∑i=1j+11ik−1S1j+1,ij+1.Proof.
Equation (16) can be written as(31)Eiklog1+2zz=2+z∑j=0∞Chjkzjj!=2∑j=0∞Chjkzjj!+∑j=0∞Chjkzj+1j!=2∑j=0∞Chjkzjj!+j∑j=1∞Chj−1kzjj!=∑j=1∞2Chjk+jChj−1kzjj!.
On the other hand, we have(32)Eiklog1+2zz=1z∑i=1∞log1+2zii−1!ik=1z∑i=1∞log1+2zii−1!ik=1z∑i=1∞1ik−1log1+2zii!=1z∑i=1∞1ik−1∑j=i∞S1j,i2jzjj!=1z∑j=1∞∑i=1j1ik−1S1j,i2jzjj!=∑j=1∞∑i=1j+11ik−1S1j+1,ij+12j+1zjj!.
Therefore, by (31) and (32), we obtain the result.Theorem 5.
Forj≥0, we have(33)Chjk1+Chjk=∑i=1j+11ik−1S1j+1,ij+12j+1.Proof.
By (16), we see that(34)∑j=1∞Chjk1+Chjkzjj!=Eiklog1+2zz=∑j=1∞∑i=1j+11ik−1S1j+1,ij+12j+1zjj!,which completes the proof of the theorem.Theorem 6.
Letj be the nonnegative number. Then,(35)Chjkξ=∑i=0j∑r=0ijiξrS1i,rChj−ik.Proof.
From (16), we note that(36)∑j=0∞Chjkξzjj!=Eiklog1+2zz2+z1+logezξ=∑j=0∞Chjkzjj!∑r=0∞ξrlogezr=∑j=0∞Chjkzjj!∑i=0∞∑r=0iξrS1i,rzii!=∑j=0∞∑i=0j∑r=0ijiξrS1i,rChj−ikzjj!.
By (16) and (36), we obtain the result.Theorem 7.
Letj be the nonnegative number. Then,(37)∑r=0jChjkS2j,r=∑p=0j∑q=0ppqjpBp−qEq1j−p+1k.Proof.
Replacingz by ez−1 in (16), we get(38)∑r=0∞Chrkez−1rr!=2Eikzez−1ez+1∑r=0∞Chrk∑j=r∞S2j,rzjj!=2ez+1zez−1Eikzz∑j=0∞∑r=0jChrkS2j,rzjj!=∑p=0∞Bpzpp!∑q=0∞Eqzqq!∑j=0∞zjj!j+1k.(39)=∑p=0∞∑q=0ppqBp−qEqzpp!∑j=0∞zjj!j+1k=∑j=0∞∑p=0j∑q=0ppqnpBp−qEq1j−p+1kzjj!.
Therefore, by (38) and (39), we get the result.Theorem 8.
Letj be nonnegative number. Then,(40)Chjkξ=∑s=0j−ijiEsξS1j−i,s∑r=0i1r+1k−12iS1i+1,r+1i+1.Proof.
From (16), we have(41)∑j=0∞Chjkξzjj!=Eiklog1+2zz2+z1+zξ,(42)=1zz+21+zξEiklog1+2z=1zz+21+zξ∑r=1∞log1+2zrr−1!rk=1zz+21+zξ∑r=0∞log1+2zr+1r!r+1k=1zz+21+zξ∑r=0∞1r+1k−11r+1!log1+2zr+1=2z+21+zξ∑i=0∞∑r=0i1r+1k−12iS1i+1,r+1i+1zii!=2elog1+z+1eξlog1+z∑l=0∞∑r=0i1r+1k−12iS1i+1,r+1i+1zii!=∑s=0∞Esξ1s!log1+zs∑i=0∞∑r=0i1r+1k−12iS1i+1,r+1i+1zii!=∑j=0∞∑s=0jEsξS1j,szjj!∑i=0∞∑r=0i1r+1k−12iS1i+1,r+1i+1zii!=∑j=0∞∑s=0j−ijiEsξS1j−i,s∑r=0i1r+1k−12iS1i+1,r+1i+1zjj!.
Thus, by (41) and (42), we complete the proof.


## 3. Type 2 Unipoly-Changhee Numbers and Polynomials

Recently, Kim and Kim [12] introduced the unipoly function ukξ|p by(43)ukξ|p=∑j=1∞pjjkξj,k∈ℤ,where ukξ|p is attached to polynomials pξ and p is any arithmetic function which is real or complex.Puttingk=1, (43) to get (see [5])(44)ukx|1=∑j=1∞ξjjk=Likξ,and it is called the polylogarithm function.By using (7) and (43), we consider the unipoly-Changhee polynomials attached to polynomials pξ by(45)1z2+zuklog1+2z|p1+zξ=∑j=0∞Chj,pkξzjj!.Whenξ=0, Chj,pk=Chj,pk0 are called the unipoly-Changhee numbers attached to p.Theorem 9.
Letj be the nonnegative number. Then,(46)Chj,1/Γk=Chjk.Proof.
Takingpj=1/Γj in (45), we have(47)∑j=0∞Chj,1/Γkzjj!=1z2+z1+zuklog1+2z|1Γ=1z2+z∑r=1∞log1+2zrrkr−1!=1z2+zEiklog1+2z=∑j=0∞Chjkzjj!.
Thus, we complete the proof.Theorem 10.
Forj≥0 and k∈ℤ, we have(48)Chj,pkξ=∑i=0j∑r=0ijipr+1r+1!r+1kS1,λi+1,r+12ii+1Chj−iξ.Proof.
From (45), we have(49)∑j=0∞Chj,pkξzjj!=1z2+z1+zξuklog1+2z|p=1z2+z1+zξ∑r=1∞prrklog1+2zr=1z2+z1+zξ∑r=0∞pr+1r+1klog1+2zr+1=1z2+z1+zξ∑r=0∞pr+1r+1!r+1k∑i=r+1∞S1i,r+12izii!=22+z1+zξ∑r=0∞pr+1r+1!r+1k∑i=r∞S1i+1,r+12ii+1zii!=∑j=0∞Chjξzjj!∑i=0∞∑r=0ipr+1r+1!r+1kS1i+1,r+12ii+1zii!=∑j=0∞∑i=0j∑r=0ijipr+1r+1!r+1kS1i+1,r+12ii+1Chj−iξzjj!.
So, the proof is completed.Corollary 3.
Letj be the nonnegative number. Then,(50)Chj,1/Γkξ=∑i=0j∑r=0iji1r+1k−1S1i+1,r+12ii+1Chj−iξ.Theorem 11.
Letj be the nonnegative number. Then,(51)Chj,pkξ=∑r=0jjrChr,pkξj−r.Proof.
Using (45), we have(52)∑j=0∞Chj,pkξzjj!=1z2+zuklog1+2z|p1+zξ=1z2+zuklog1+2z|p∑j=0∞ξjzjj!=∑r=0∞Chr,pkzrr!∑j=0∞ξjzjj!=∑j=0∞∑r=0jjrChr,pkξj−rzjj!.
So, we complete the theorem.Theorem 12.
Letj be the nonnegative number. Then,(53)Chj,pk=∑r=0j−i∑i=0jj−irji2i+rDrChj−r−i∑s=0ips+1s!s+1kS1i,s.Proof.
By (45), we have(54)∑j=0∞Chj,pkzjj!=1z2+zuklog1+2z|p=1z2+z∑s=0∞ps+1s+1klog1+2zs+1=log1+2zz2+z∑s=0∞ps+1s!s+1k∑l=s∞S1l,m2lzll!=log1+2z2z22+z∑m=0∞pm+1m!m+1k∑l=m∞S1l,m2lzll!=∑r=0∞Dr2rzrr!∑j=0∞Chjzjj!∑i=0∞∑m=0ipm+1m!m+1kS1i,m2izii!=∑n=0∞∑r=0nnr2rDrChn−rtnn!∑l=0∞∑m=0lpm+1m!m+1kS1l,m2ltll!=∑j=0∞∑r=0j−i∑i=0jj−irji2i+rDrChj−r−i∑s=0ips+1s!s+1kS1i,szjj!.
Therefore, by (45) and (54), we obtain the result.


## 4. Conclusion

In the previous sections, we have touched on the problem of recognizing the algebraic structure underlying the poly-Changhee polynomials as given by definition (16). The analysis is aimed at accounting for the wealth of the properties exhibited by these polynomials within the context of the poly-Changhee numbers and polynomials which provide a unifying formalism where the theory of special functions can be framed inherently. Some analogies with the theory of poly-Changhee numbers and polynomials can be recognized and usefully exploited to infer further properties of these polynomials and links with other special functions. Let us stress that the scheme suggested by the following properties of Changhee numbers and polynomials Chnx studied in detail by Kim and Kim [14] can be applied to connect other special functions of relevance in mathematical physics.



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