# Identities of Degenerate Poly-Changhee Polynomials Arising fromλ-Sheffer Sequences

Authors: Sang Jo Yun; Jin-Woo Park
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the 1970s, Gian-Carlo Rota constructed the umbral calculus for investigating the properties of special functions, and by Kim-Kim, umbral calculus is generalized calledλ-umbral calculus. In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of some special polynomials. In addition, we derive some interesting identities related to degenerate poly-Changhee polynomials and some important special functions by using λ-umbral calculus.


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## Body

## 1. Introduction

Letp be an odd prime number, and let ℤp,ℚp, and ℂp denote the ring of p-adic rational integer, the field of p-adic rational numbers, and the complication of the algebraic closure of ℚp, respectively. The p-adic norm ⋅p is normally defined by pp=1/p. As is known, the p-adic invariant integral on ℤp is defined by Kim to be(1)I−1f=∫ℤpfxdu−1x=limN⟶∞∑x=0pN−1fx−1x,where fx is a continuous functions on ℤp (see [1]).By using equation (1), the generation function of the Changhee polynomials is defined to be(2)∑n=0∞Chnxtnn!=∫ℤp1+tx+ydμ−1y=2t+21+tx,(see [2–4]). When x=0, Chn=Chn0 are called the Changhee numbers.As one of the special polynomials, the Changhee polynomials and numbers are related closely some important special polynomials in the combinatorics, applied mathematics, mathematical physics, engineering, or economics. In [3], the authors found symmetric identities for that polynomials by using the fermionic p-adic integral on ℤp, and some properties and zeros of these polynomials on the locally constant space were studied in [5]. In [4, 6], the authors generalized these polynomials to ring of p-adic integer ℤp. Kim-Kim defined different Changhee polynomials named type 2 Changhee polynomials in [7] and found some relationships between those polynomials and Changhee polynomials, the Stirling numbers of the first and second kind, Euler polynomials and type 2 Euler polynomials. Moreover, in [2, 4, 8, 9], the authors studied the properties of degenerate version of Changhee polynomials, q-Changhee polynomials, and Appell-type q-Changhee polynomials, respectively.Forn≥0, the Stirling numbers of the first kind nk and the Stirling numbers of the second kind nk, respectively, are given by(3)xn=∑k=0nnkxk,xn=∑k=0nnkxk,see10,where x0=1 and xn=xx−1,…,x−n+1,n≥1 is the falling factorial sequences.For each positive integerk, it is well known (see [10]) that(4)1k!log1+tk=∑n=k∞nktnn!,1k!et−1k=∑n=k∞nktnn!.For any nonzero real numberλ∈ℝ, the degenerate exponential function is defined to be(5)eλxt=1+λtx/λ,eλt=1+λt1/λ,(see [7, 11, 12]) and let logλt be the compositional inverse function of eλt satisfying logλeλt=t. Then, we have(6)logλ1+t=∑n=1∞λn−11n,1/λtnn!,(see [12–16] where xn,λ=xx−λx−2λ,…,x−n−1λ are λ-falling factorial of x.By using the degenerate exponential function, Carlitz defined the degenerate Bernoulli polynomials in [11, 17], and in recent years, studies of degenerate versions of some special polynomials have been increased by many researchers with their interests not only in combinatorial properties but also its applications (see [8, 9, 12–16, 18–21]).As a degenerate version of the Stirling numbers of the first and second kind in equation (3), the degenerate Stirling numbers of the first kind nkλ and the degenerate Stirling numbers of the second kind nkλ are, respectively, introduced by Kim-Kim (see [8, 14–16, 19]) as follows:(7)1k!logλ1+tk=∑n=k∞nkλtnn!,1k!eλt−1k=∑n=k∞nkλtnn!.The degenerate polyexponential functions are defined by Kim-Kim as(8)Eik,λx=∑n=1∞1n,λxnn−1!nk,k∈ℤ,x<1,(see [12, 18]). In particular, if k=1, then, Ei1,λx=eλx−1.As a generalization of the degenerate exponential function, the degenerate polyexponential function is used by many researchers. In [15], Kim-Kim defined degenerate poly-Bell polynomials and provided explicit representations and combinatorial identities for these polynomials. Kurt defined the degenerate poly-Euler numbers and polynomials arising from the degenerate polyexponential functions and derived explicit relations for these numbers and polynomials (see [16]). In [14], Kim-Jang defined the degenerate poly-Genocchi polynomials and found some interesting identities of those polynomials.Letℂ be the field of complex numbers,(9)ℱ=ft=∑n=0∞aktnn!|ak∈ℂ,and let(10)ℙ=ℂx=∑k=0∞akxk|ak∈ℂwithak=0,for all but finite number ofk.Letℙ∗ be the vector space of all linear functionals on ℙ.Then, eachλ∈ℝ gives rise to the linear functional ft|⋅λ on ℙ, called λ-linear functional given by ft, which is defined by(11)ft|xn,λλ=an,n≥0,and by linear extension (see [19]). From (11), we have(12)tk|xn,λλ=n!δn,k,n,k≥0,where δn,k is Kronecker’s symbol.For eachλ∈ℝ and each k∈ℕ, Kim-Kim defined the differential operator on ℙ in [19] by(13)tkλxn,λ=nkxn−k,λ,ifk≤n,0,ifk≥n,,and for any ft=∑k=0∞aktk/k!∈ℱ,(14)ftλxn,λ=∑k=0nnkakxn−k,λ.In addition, they showed that forft,gt∈ℱ, and px∈ℙ,(15)ftgt|pxλ=gt|ftλpxλ=ft|gtλpxλ.The orderoft of ft∈ℱ−0 is the smallest integer k for which the coefficient of tk does not vanish. If oft=0, then ft is called invertible and such series has a multiplicative inverse 1/ft) of ft. If oft=1, then, ft is called delta series and it has a compositional inverse f¯t of ft with f¯ft=ff¯t=t (see [22]).Letft be a delta series and let gt be an invertible series. Then, there exists a unique sequence Sn,λxdegSn,λx=n of polynomials satisfying the orthogonality conditions:(16)gtftk|Sn,λxλ=n!δn,k,n,k≥0,(see [19]). Here, Sn,λx is called the λ-Sheffer sequence for gt,ft, which is denoted by Sn,λx∼gt,ftλ. The sequence Sn,λx is the λ-Sheffer sequence for gt,ftλ if and only if(17)1gf¯teλyf¯t=∑n=0∞Sn,λytnn!,see19,(see [19]) for all y∈ℂ, where f¯t is the compositional inverse of ft such that ff¯t=f¯ft=t.LetSn,λx∼gt,ftλ and let hx=∑l=0nalSl,λx∈ℙ. Then, by (16), we have(18)gtftk|hxλ=∑l=0nalgtftk|Sl,λxλ=k!ak,and thus, we know that(19)ak=1k!gtftk|hxλ.The following theorem is very useful tools in theλ-umbral calculus which is found by Kim and Kim (see [19]).Theorem 1.
LetSn,λx∼gt,ftλ and let rn,λ∼ht,ltλ. If(20)sn,λ=∑k=0ncn,krk,λ,then,(21)cn,k=1k!hf¯tgf¯tlf¯tk|xn,λλ.Letxn=∑k=0ncn,kxk,λ. Since(22)xn∼1,eλt−1λ,xn,λ∼1,tλ,By equation (22), we get(23)cn,k=1k!logλ1+tk|xn,λλ=∑l=k∞lkλ1l!tl|xn,λλ=nkλ,and thus, we know that(24)xn=∑k=0nnkλxk,λ.In the similar way, we also know that(25)xn,λ=∑k=0nnkλxk.In this paper, we find some interesting identities about degenerate poly-Changhee polynomials, higher-order degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Bell polynomials, degenerate Lah-Bell polynomials, degenerate Frobenius–Euler polynomials, and Mittag-Leffer polynomials by usingλ-umbral calculus.


## 2. Degenerate Poly-Changhee Polynomials

The degenerate poly-Changhee polynomials are defined by the generating function:(26)∑n=0∞Chn,λkxtnn!=2Eik,λlogλ1+ttt+2eλxlogλ1+t.In the special casex=0, Chn,λk=Chn,λk0 are called the degenerate poly-Changhee numbers.By equation (17), we know that the λ-Sheffer sequences of degenerate poly-Changhee polynomials is(27)Chn,λkx∼eλt−1eλt+12Eik,λt,eλt−1λ.Theorem 2.
For each non-negative integern, we have(28)Chn,λkx=∑n=0∞∑m=lnnlmlλChn−m,λkxn,λ.Proof.
LetChn,λkx=∑l=0ncn,lxl,λ. Since(29)xn,λ∼1,tλ,
By Theorem1, we get(30)cn,l=1l!1eλlogλ1+t−1eλlogλ1+t+1/2Eik,λlogλ1+tlogλ1+tl|xn,λλ=2Eik,λlogλ1+teλlogλ1+t−1eλlogλ1+t+1|1l!logλ1+tlλ|xn,λλ=2Eik,λlogλ1+ttt+2|∑m=l∞mlλtmm!λ|xn,λλ=∑m=lnnmmlλ∑r=0∞Chr,λktrr!|xn−m,λλ=∑m=lnnmmlλChn−m,λk,and thus, our theorem is proved.
Letsn,λx∼gt,ftλ. By (15) and (16),(31)n!δn,k=gtftk|sn,λxλ=ftk|gtλsn,λxλ,and so(32)gtλsn,λx∼1,ftλ.
By equations (24) and (27), we know that(33)∑n=0∞eλ2t−12Eik,λtλChn,λkxtnn!=1+tx=∑n=0∞∑l=0nnlλxl,λtnn!,and so(34)Chn,λkx=2Eik,λteλ2t−1λ∑l=0nnlλxl,λ,n≥0.
Note that(35)2Eik,λteλ2t−1=2∑n=1∞1n,λtnn−1!nk1eλ2t−1=2∑n=1∞1n,λtn−1nk−1n!1eλtteλteλ2t−1=2∑n=1∞1n,λtn−1nk−1n!∑n=0∞−1n1n,λtnn!∑n=0∞bn,λtnn!=2∑n=0∞∑r=0n∑m=0rnrrm1m+1,λ1r−m,λm+1k−1r−mbn−r,λtnn!,where x0,λ=1, xn,λ=xx+λx+2λ,…,x−n−1λ,n≥1, and bn,λx are type 2 Bernoulli numbers which are defined by the generating function to be(36)∑n=0∞bn,λtnn!=teλt−eλ−1t,see7.(see [7]) By equations (13) and (35), we know that(37)Chn,λkx=2Eik,λteλ2t−1λ∑l=0nnlλxl,λ=2∑l=0n∑a=0l∑r=0a∑m=0rlaarrm1m+1,λ1r−m,λm+1knlλ−1r−mba−r,λxl−a,λ.
By equation (36), we obtain the following theorem.Theorem 3.
For eachn∈ℕ∪0, we have(38)Chn,λkx=∑l=0n∑a=0l∑r=0a∑m=0rlaarrm1m+1,λ1r−m,λm+1knlλ−1r−mba−r,λxl−a,λ.Note that by Theorem1 and equation (27), we get(39)Chn,λky=2Eik,λlogλ1+ttt+21+ty|xn,λλ=Eik,λlogλ1+tt|2t+21+tyλ|xn,λλ=∑l=0nnlChlyEik,λlogλ1+tt|xn−l,λλ=∑l=0nnlChlyn−l+1Eik,λlogλ1+t|xn−l+1,λλ=∑l=0nnlChlyn−l+1∑m=1∞1m,λmk−1m!logλ1+tm|xn,−l+1,λλ=∑l=0nnlChlyn−l+1∑a=1∞∑m=1a1m,λmk−1amλtaa!|xn−l+1,λλ=∑l=0n∑m=1n−l+1nl1m,λn−l+1mk−1n−l+1mλChly.Thus, by (39), we obtain the following theorem.Theorem 4.
For each non-negative integern, we have(40)Chn,λky=∑l=0n∑m=1n−l+1nl1m,λn−l+1mk−1n−l+1mλChly.
By using equation (17), the higher-order degenerate Bernoulli polynomials are defined by the generating function:(41)∑n=0∞βn,λrxtnn!=teλt−1reλxt,(see [24, 25]). In the special case x=0, βn,λr=βn,λr0 are called the higher-order degenerate Bernoulli numbers.
By the definition of higher-order degenerate Bernoulli polynomials (41), we know that(42)βn,λrx∼eλt−1tr,tλ.
If we putChn,λrx=∑l=0ncn,lβl,λrx, then, by the Theorem 1 and (42), we have(43)cn,l=1l!t/logλ1+trtt+2/2Eik,λlogλ1+tlogλ1+tl|xn,λλ=2Eik,λlogλ1+tt+2tlogλ1+tr|1l!logλ1+tlλxn,λλ=∑a=lnnaalλ2Eik,λlogλ1+ttt+2|tlogλ1+trλxn−a,λλ=∑a=ln∑s=0n−anan−asalλBs,λr2Eik,λlogλ1+ttt+2|xn−a−s,λλ=∑a=ln∑s=0n−anan−asalλBs,λrChn−a−s,λk,where the higher-order degenerate Bernoulli numbers of the second kind are given.(44)tlogλ1+tr=∑n=0∞Bn,λrtnn!,(see [26]). By equation (43), we obtain the following theorem.Theorem 5.
For each non-negative integern, we have(45)Chn,λrx=∑l=0∞∑a=ln∑s=0n−anan−asalλBs,λrChn−a−s,λkβl,λrx.
The degenerate Euler polynomials are defined by the generating function:(46)2eλt+1eλxt=∑n=0∞En,λxtnn!,(see [24]). When x=0, En,λ=En,λ0 are called the degenerate Euler numbers.
By the definition of the degenerate Euler polynomials, we know that(47)En,λx∼eλt+12,tλ.
LetChn,λkx=∑l=0ncn,lEl,λx. By Theorem 1 and (47), we get(48)cn,l=1l!t+2/2tt+2/2Eik,λlogλ1+tlogλ1+tl|xn,λλ=Eik,λlogλ1+tt|1l!logλ1+tlλxn,λλ=∑a=lnnaalλEik,λlogλ1+tt|xn−a,λλ=∑a=lnnaalλn−a+1Eik,λlogλ1+t|xn−a+1,λλ=∑a=ln∑s=1n−a+1na1s,λsk−1n−a+1alλn−a+1sλ.Therefore, by equation (48), we obtain the following theorem.Theorem 6.
For each non-negative integern, we have(49)Chn,λkx=∑n=0n∑a=ln∑s=1n−a+1na1s,λsk−1n−a+1alλn−a+1sλEl,λx.The fully degenerate Bell polynomials are defined as(50)eλxeλt−1=∑n=0∞Beln,λxtnn!,(see [12, 20]). By the definition of fully degenerate Bell polynomials, we note that(51)Beln,λx∼1,logλ1+tλ.Note that(52)1l!logλ1+logλ1+tl=∑r=l∞S1,λr,l1r!logλ1+tr=∑r=l∞S1,λr,l∑s=r∞srλtss!=∑m=l∞∑r=lmrlλmrλtmm!.LetChn,λkx=∑l=0ncn,lBell,λx. By Theorem 1, equations (51), and (52), we get(53)cn,l=1l!2Eik,λlogλ1+ttt+2logλ1+logλ1+tl|xn,λλ=2Eik,λlogλ1+ttt+2|1l!logλ1+logλ1+tlλxn,λλ=∑m=ln∑r=lmnmrlλmrλ2Eik,λlogλ1+ttt+2|xn−m,λλ=∑m=ln∑r=lmnmrlλmrλ∑s=0∞Chs,λktss!|xn−m,λλ=∑m=ln∑r=lmnmrlλmrλChn−m,λk,and thus, by equation (53), we obtain the following theorem.Theorem 7.
For each non-negative integern, we have(54)Chn,λkx=∑l=0n∑m=ln∑r=lmnmrlλmrλChn−m,λkBell,λx.The degenerate Lah-Bell polynomials are defined by the generating function:(55)eλxt1−t=∑n=0∞Bn,λLxtnn!,(see [27]). In the special case x=0, Bn,λL=Bn,λL0 are called the Lah-Bell numbers.By equation (55), we know the λ-Sheffer sequence of degenerate Lah-Bell polynomials is(56)Bn,λLx∼1,t1+tλ.Note that(57)logλ1+t1+logλ1+tk=∑l=0∞−1lkll!logλ1+tl+k=∑l=0∞−1lkll!l+k!1l+k!logλ1+tl+k=∑r=0∞∑l=0r−1lkll+kkr+kl+kλtr+kr+k!.LetChn,λkx=∑l=0ncn,lBl,λLx. By Theorem 1, (56) and (57), we get(58)cn,l=1l!1tt+2/2Eik,λlogλ1+tlogλ1+t1+logλ1+tl|xn,λλ=1l!2Eik,λlogλ1+ttt+2|logλ1+t1+logλ1+tlλxn,λλ=∑r=0n∑a=0r−1anr+aa+lllar+ll+aλ2Eik,λlogλ1+ttt+2|xn−r−l,λλ=∑r=0n−l∑a=0r−1anr+la+lllar+ll+aλChn−r−l,λk.By equation (58), we obtain the following theorem.Theorem 8.
For each non-negative integern, we have(59)Chn,λkx=∑l=0n∑r=0n−l∑a=0r−1anr+la+lllar+ll+aλChn−r−l,λkBl,λLx.The degenerate Daehee polynomials are defined by the generating function:(60)logλ1+tteλlogλ1+t=∑n=0∞Dn,λxtnn!,see28,29.By equation (60), we know that(61)Dn,λ∼eλt−1t,eλt−1λ.LetChn,λkx=∑l=0ncn,lDl,λLx. Then, by Theorem 1 and (61), we get(62)cn,l=1l!t/logλ1+ttt+2/2Eik,λlogλ1+ttl|xn,λλ=1l!t/logλ1+ttt+2/2Eik,λlogλ1+t|tlλxn,λλ=nl2Eik,λlogλ1+ttt+2|tlogλ1+tλxn−l,λλ=∑m=0n−lnln−lmBm,λ2Eik,λlogλ1+ttt+2|xn−l−m,λλ=∑m=0n−lnln−lmBm,λChn−l−m,λk.By equation (62), we obtain the following theorem.Theorem 9.
For each non-negative integern, we have(63)Chn,λkx=∑l=0n∑m=0n−lnln−lmBm,λChn−l−m,λkDl,λx.The degenerate Frobenius–Euler polynomials of orderα defined by the generating function to be(64)1−ueλt−uαeλxt=∑n=0∞hn,λαx|utnn!,u≠1,u∈ℂ,r≥0,(see [1, 19, 30]). In the special case x=0, hn,λαu=hn,λα0|u are called the degenerate Frobenius–Euler numbers of order α.By equation (64), we know the λ-Sheffer sequence of degenerate Frobenius–Euler polynomials is(65)hn,λαx|u∼eλt−u1−uα,tλ.LetChn,λkx=∑l=0ncn,lhl,λLxu|. Then, by Theorem 1, equations (27) and (65), we have(66)cn,l=1l!t+1−u/1−uαtt+2/2Eik,λlogλ1+tlogλ1+tl|xn,λλ=2Eik,λlogλ1+ttt+2t+1−u1−uα|1l!logλ1+tlλxn,λλ=∑m=lnnmmlλ2Eik,λlogλ1+ttt+2|t+1−u1−uαλxn−m,λλ=∑m=ln∑r=0n−mnmαrn−mr1−ur2Eik,λlogλ1+ttt+2|xn−m−r,λλ=∑m=ln∑r=0n−mnmαrn−mr1−urChn−m−r,λk,and so by (66), we obtain the following theorem.Theorem 10.
For each non-negative integern, we have(67)Chn,λkx=∑l=0n∑m=ln∑r=0n−mnmαrn−mr1−urChn−m−r,λkhl,λαx|u.The Mittag-Leffler polynomials are defined by the generating function to be(68)1+t1−tx=∑n=0∞Mnxtnn!,see19.By the definition of Mittag-Leffler polynomials, we see that(69)Mnx∼1,eλt−1eλt+1λ,tt+2k=∑l=0∞−1lkl2k+ltl+kl!.LetChn,λkx=∑l=0ncn,lMlx. Then, by Theorem 1, equations (27) and (69), we have(70)cn,l=1l!1tt+2/2Eik,λlogλ1+ttt+2l|xn,λλ=1l!2Eik,λlogλtt+2|tt+2lλxn,λλ=∑r=0n−l−1rlr2l+rnl+rl+rr∑m=0∞Chm,λktmm!|xn−l−r,λλ=∑r=0n−l−1rlr2l+rnl+rl+rrChn−l−r,λk,and so by (70), we obtain the following theorem.Theorem 11.
For each non-negative integern, we have(71)Chn,λkx=∑l=0n∑r=0n−l−1rlr2l+rnl+rl+rrChn−l−r,λkMlx.


## 3. Conclusion

In 1970s Gian-Carlo Rota constructed the important tools, called umbral calculus, for investigating the properties of special functions which was based on a linear functional, a linear differential operator, and adjoint (see [22]). In 2021, Kim-Kim generalized the umbral calculus to λ-umbral calculus, and λ-umbral calculus has been used as an important tool to study special functions by many researchers.In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of higher-order degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Bell polynomials, degenerate Lah-Bell polynomials, degenerate Frobenius–Euler polynomials or Mittag-Leffer polynomials.If the methods used in this paper are applied to expandable polynomials using polyexponential functions, many new and interesting identities can be found.



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