# Identities on Changhee Polynomials Arising fromλ-Sheffer Sequences

Authors: Byung Moon Kim; Taekyun Kim; Jin-Woo Park; Taha Ali Radwan
Journal: Complexity
 (2022)

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## Abstract

In this paper, authors found a new and interesting identity between Changhee polynomials and some degenerate polynomials such as degenerate Bernoulli polynomials of the first and second kind, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Bell polynomials, degenerate Lah–Bell polynomials, and degenerate Frobenius–Euler polynomials and Mittag–Leffer polynomials by usingλ-Sheffer sequences and λ-differential operators to find the coefficient polynomial when expressing the n-th Changhee polynomials as a linear combination of those degenerate polynomials. In addition, authors derive the inversion formulas of these identities.


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## Body

## 1. Introduction

Umbral calculus from 1850 to 1970 consisted primarily of symbolic techniques for sequence manipulation, and its mathematical rigor left little room for demands. In the 1970s, Gian-Carlo Rota began building a completely rigid foundation for theories based on relatively modern ideas of linear functions, linear operators, and adjacency functions (see [1–4]). Umbral calculus contributed to the generalization of Lagrange inversion formula and has been applied in many fields such as combinatorial counting with linear recurrences and lattice path counting, graph theory using chromatic polynomials, probability theory, link invariant theory, statistics, topology, and physics (see [3]). It is being actively applied in various fields by researchers (see [1–16]).In the past few years, many distinct umbral calculus types have begun to be studied (see [2, 4, 6, 10]). In particular, Kim–Kim defined the degenerate Sheffer sequences, λ-Sheffer sequence, a family of λ-linear functionals, and λ-differential operators as follows (see [2]).Letℂ be the field of complex numbers:(1)ℱ=ft=∑n=0∞aktnn!|ak∈ℂ,and let(2)ℙ=ℂx=∑k=0∞alxk|ak∈ℂwithak=0for all but finite number ofk.Letℙ∗ be the vector space of all linear functionals on ℙ.Then, each real numberλ gives rise to the linear functional ft|⋅λ on ℙ, called λ-linear functional given by ft, which is defined by (see [2])(3)ft|xn,λλ=an,n≥0,and by linear extension where x0,λ=1, xn,λ=xx−λ⋯x−n−1λ, n≥1. From (3), we have(4)tk|xn,λλ=n!δn,k,n,k≥0,where δn,k is Kronecker’s symbol (see [2]).For each real numberλ and each positive integer k, Kim and Kim defined the differential operator on ℙ in [2] as follows:(5)tkλxn,λ=nkxn−k,λ,ifk≤n,0,ifk>n,and for any ft=∑k=0∞aktk/k!∈ℱ,(6)ftλxn,λ=∑k=0nnkakxn−k,λ.In addition, they showed that forft,gt∈ℱ and px∈ℙ,(7)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 [1, 2, 12, 16]).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 (see [2])(8)gtftk|Sn,λxλ=n!δn,k,n,k≥0.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,ft if and only if(9)1gf¯teλyf¯t=∑n=0∞Sn,λytnn!,for all y∈ℂ, where f¯t is the compositional inverse of ft such that ff¯t=f¯ft=t (see [1, 2, 12, 16]).LetSn,λx∼gt,ftλ and let hx=∑l=0nalSl,λx∈ℙ. Then, by (8), we have(10)gtftk|hxλ=∑l=0nalgtftk|Sl,λxλ=k!ak,and thus we know that(11)ak=1k!gtftk|hxλ.The following theorem is proved by Kim and Kim [2] and is a very useful tool for researching degenerate versions of special polynomials and numbers.Theorem 1.
Letsn,λ∼gt,ftλ, rn,λ=ht,ltλ. Then, we have(12)sn,λ=∑k=0ncn,krk,λ,where(13)cn,k=1k!hf¯tgf¯tlf¯tk|xn,λλ.Forn≥0, the Stirling numbers of the first kind nk and Stirling numbers of the second kind nk, respectively, are given by the following (see [11, 12, 17–20]):(14)xn=∑k=0nnkxkandxn=∑k=0nnkxk.For each positive integerk, it is well known that (see [11, 12, 17–20])(15)1k!log1+tk=∑n=k∞nktnn!,1k!et−1k=∑n=k∞nktnn!.For any nonzero real numberλ, the degenerate exponential function is defined by (see [1, 21–27])(16)eλxt=1+λtx/λ,eλt=1+λt1/λ,Note that(17)eλt+12=1+12∑n=1∞1n,λtnn!.A study of degenerate versions of some special numbers and polynomials was initiated by Carlitz who found interesting relationships connected with important numbers in combinatorics, Bernoulli polynomials, and Eulerian polynomials (see [28]). In the past decades, the study of degenerate versions of various special polynomials or numbers has been studied by many researchers (see [1, 2, 21–27, 29–32]).By using (16), the higher-order degenerate Bernoulli polynomials are defined as follows (see [1, 10, 12, 30, 33, 34]):(18)∑n=0∞Bn,λrxtnn!=teλt−1reλxt.Whenx=0, Bn,λr0=Bn,λr are called the higher-order degenerate Bernoulli numbers. In addition, when r=1, we denote Bn,λ1x=Bn,λx.On the other hand, Kim and Kim definedlogλt called the degenerate logarithm function as the compositional inverse function of eλt satisfying logλeλt=t. Then, we have (see [1, 10, 22, 24, 32])(19)logλ1+t=∑n=1∞λn−11n,1/λtnn!.By using (19), the degenerate Bernoulli polynomials of the second kind are defined by the generating function to be (see [16])(20)∑n=0∞βn,λxtnn!=tlogλ1+teλxlogλ1+t.In the special casex=0, βn,λ=βn,λ0 are called the Bernoulli numbers of the second kind.As degenerate version of the Stirling numbers of the first and second kind in (14), 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 [1, 2, 21, 22, 24, 26–30, 35, 36]) as follows:(21)1k!logλ1+tk=∑n=k∞nkλtnn!and1k!eλt−1k=∑n=k∞nkλtnn!.Letxn=∑k=0ncn,kxk,λ. Since(22)xk,λn∼1,eλt−1λ,xn,λ∼1,tλ,by Theorem 1, we obtain(23)cn,k=1k!logλ1+tk|xn,λλ=∑l=k∞S1,λl,k1l!tl|xn,λλ=nkλ,and thus, we know that(24)xn=∑k=0nnkλxk,λ,where x0=1, xn=xx−1…x−n+1,n≥1 is the falling factorial sequences. In the similar way, we also know that(25)xn,λ=∑k=0nnkλxk.The aim of this paper is to find some new and interesting identities related to the Changhee polynomials and some special polynomials by usingλ-Sheffer sequences and λ-differential operators. In more detail, we find the coefficients which are also polynomials or numbers when the n-th Changhee polynomial is expressed as a linear combination of some degenerate special polynomials by using the λ-Sheffer sequences and λ-differential operators (see Theorems 2–10), and by using the λ-Sheffer sequences and the linear combinations of those polynomials (see Theorems 5–8 and 10), and derive the inversion formulas of these identities.


## 2. Changhee Polynomials Arising fromλ-Sheffer Sequences

In this section, we find some relationships between the Changhee polynomials and some special polynomials arising fromλ-Sheffer sequences.The Changhee polynomials are given by(26)∑n=0∞Chnxtnn!=22+t1+tx.By (24) and (26), we obtain(27)∑n=0∞Chnxtnn!=∑n=0∞Chntnn!∑n=0∞xntnn!=∑n=0∞∑m=0nnmChn−mxmtnn!=∑n=0∞∑m=0n∑k=0mnmmkλChn−mxk,λtnn!,and, by (27), we have(28)Chnx=∑m=0n∑k=0mnmmkλChn−mxk,λ,n≥0.By (28), we compute the first few Changhee polynomials as follows:(29)Ch0x=1,Ch1x=x−12,Ch2x=x2−2x+12,Ch3x=−x3+92x2−5x+34,Ch4x=x4−8x3+20x2−16x+32,Ch5x=−x5+252x4−55x3+100x2−64x+154.In addition, graphs for some Changhee polynomials are shown in Figure1.Figure 1
The shapes of Changhee polynomialsChnx.Note that(30)1t+2k=t+2−k=∑l=0∞−kltl2−k−l=∑l=0∞kl−1l2l+ktll!,where x0=1, xn=xx+1x+2…x+n−1, n≥1.Theorem 2.
For each nonnegative integern, we have(31)Chnx=∑k=0n∑l=knnllkλChn−lxk,λ.
As the inversion formula of (31), we have(32)xn,λ=∑k=0nnkλ+12∑m=1n−k1m,λnmn−mkλChkx.Proof.
LetChnx=∑k=0ncn,kxk,λ. Since(33)Chnx∼eλt+12,eλt−1λ,xn,λ∼1,tλ,by Theorem 1 and (30), we have(34)cn,k=1k!1t+2/2logλ1+tk|xn,λλ=2t+2|1k!logλ1+tkλxn,λλ=∑l=knnllkλ2t+2|xn−l,λλ=∑l=knnllkλChn−l.
Conversely, we assume thatxn,λ=∑k=0ndn,kChkx. By (6) and (17), we obtain(35)dn,k=1k!eλt+12eλt−1k|xn,λλ=1k!eλt−1k|eλt+12λxn,λλ=1k!eλt−1k|1+12∑m=1∞1m,λtmm!λxn,λλ=∑l=k∞lkλtll!|xn,λλ+12∑m=1nnm1m,λ∑l=k∞lkλtll!|xn−m,λλ=nkλ+12∑m=1n−knm1m,λn−mkλ.Since(36)∑n=0∞Bn,λrxtnn!=teλt−1reλxt=∑n=0∞Bn,λrtnn!∑n=0∞xn,λtnn!=∑n=0∞∑m=0nnmBn−m,λrxm,λtnn,and thus we know that(37)Bn,λrx=∑m=0nnmBn−m,λrxm,λ.Theorem 3.
For eachn≥0, we have(38)Chnx=∑k=0n∑l=kn∑m=0n−lnln−lmlkλBm,λChn−l−mBk,λx.
As the inversion formula of (38), we have(39)Bn,λx=∑k=0n∑m=0nnmBn−m,λmkλ+12∑l=1mmlm−lkλ1l,λChkx=∑k=0n∑m=knnmmkλBn−m,λ+12∑l=1n∑m=ln−lnln−lm1l,λmkλBn−l−m,λChkx.Proof.
LetChnx=∑k=0ncn,kBk,λx. Since(40)Chnx∼eλt+12,eλt−1λ,Bn,λx∼eλt−1t,tλ,by Theorem 1, we obtain(41)cn,k=1k!t/logλ1+tt+2/2logλ1+tk|xn,λλ=2t+2tlogλ1+t|1k!logλ1+tkλxn,λλ=∑l=knnllkλ2t+2|tlogλ1+tλxn−l,λλ=∑l=kn∑m=0n−lnln−lmlkλβm,λ2t+2|xn−l−m,λλ=∑l=kn∑m=0n−lnln−lmlkλβm,λChn−l−m.
Conversely, letBn,λx=∑k=0ndn,kChkx. By (8), (12), (15) and (37), we obtain(42)dn,k=1k!eλt+12eλt−1k|Bn,λxλ=∑m=0nnmBn−m,λ1k!eλt+12eλt−1k|xm,λλ=∑m=0nnmBn−m,λ1k!eλt−1k|eλt+12λxm,λλ=∑m=0nnmBn−m,λ1k!eλt−1k|1+12∑l=1∞1l,λtll!λxm,λλ=∑m=0nnmBn−m,λ1k!eλt−1k|xm,λ+12∑l=1mml1l,λxm−l,λλ=∑m=0nnmBn−m,λmkλ+12∑m=0n∑l=1mnmmlBn−m,λm−lkλ1l,λ.
On the other hand, by Theorem1, we have(43)dn,k=1k!eλt+1/2eλt−1/teλt−1k|xn,λλ=1k!teλt−1eλt−1k|eλt+12λxn,λλ=1k!teλt−1eλt−1k|xn,λ+12∑l=1nnl1l,λ1k!teλt−1eλt−1k|xn−l,λλ=teλt−1|1k!eλt−1kλxn,λλ+12∑l=1nnl1l,λteλt−1|1k!eλt−1kλxn−l,λλ=∑m=knnmmkλteλt−1|xn−m,λλ+12∑l=1n∑m=kn−lnln−lm1l,λmkλteλt−1|xn−l−mλ=∑m=knnmmkBn−m,λ+12∑l=1n∑m=kn−lnln−lm1l,λmkBn−l−m,λ,and hence our proofs are completed.
The degenerate Euler polynomials are defined by the generating function to be (see [28])(44)2eλt+1eλxt=∑n=0∞En,λxtnn!.
Whenx=0, En,λ=En,λ0 are called the degenerate Euler numbers.
By (44), we know that(45)∑n=0∞En,λxtnn!=2eλt+1eλxt=∑n=0∞En,λtnn!∑n=0∞xn,λtnn!=∑n=0∞∑m=0nnmEn−m,λxm,λtnn!,and so(46)En,λx=∑m=0nnmEn−m,λxm,λ.Theorem 4.
For eachn≥0, we have(47)Chnx=∑k=0nnkλEk,λx.
As the inversion formula of (47), we have(48)En,λx=∑k=0nnkλChkx=∑k=0n∑m=knnmmkλEn−m,λ+12∑m=k+1n∑l=km−1nmmllkλEn−m,λ1m−l,λChkx.Proof.
LetChnx=∑k=0ncn,kEk,λx. Since(49)Chnx∼eλt+12,eλt−1λ,En,λx∼eλt+12,tλ,by Theorem 1, we obtain(50)cn,k=1k!t+2/2t+2/2logλ1+tk|xn,λλ=1k!logλ1+tk|xn,λλ=∑l=k∞lkλtll!|xn,λλ=nkλ.
Conversely, we assume thatEn,λx=∑k=0ndn,kChkx. Then,(51)dn,k=1k!eλt+1/2eλt+1/2eλt−1k|xn,λλ=1k!eλt−1k|xn,λλ=∑l=k∞lkλtll!|xn,λλ=nkλ.
On the other hand, by (11) and (46), we obtain(52)dn,k=1k!eλt+12eλt−1k|En,λxλ=∑m=0nnmEn−m,λeλt+12|1k!eλt−1kλxm,λλ=∑m=0n∑l=kmnmmllkλEn−m,λeλt+12|xm−l,λλ=∑m=0n∑l=kmnmmllkλEn−m,λ1+12∑a=1∞1a,λtaa!|xm−l,λλ=∑m=0n∑l=kmnmmllkλEn−m,λ×1|xm−l,λλ+12∑a=1∞1a,λa!ta|xm−l,λλ=∑m=knnmmkλEn−m,λ+12∑m=k+1n∑l=km−1nmmllkλEn−m,λ1m−l,λ,and so our proofs are completed.
By (46), we compute the first few degenerate Euler polynomials as follows:(53)E0,λx=1,E1,λx=x−12,E2,λx=x2−λ+1x+12λ,E3,λx=−x3+3λ+32x2−2λ2+3λx+λ2−14,E4,λx=−x4+2+6λx3−11λ2+9λx2+6λ3+11λ2−1x−3λ2+32λ.
AlthoughCh2x=∑l=02alxEl,λx, it is difficult to find aix, i=0,1,2 through Figure 2. But by Theorem 4, we see that aix=2i0.5, i=0,1,2.
The degenerate Daehee polynomials are defined by the generating function to be(54)logλ1+tteλxlogλ1+t=∑n=0∞Dn,λxtnn!.
In the special case ofx=0, Dn,λ=Dn,λ0 are called the degenerate Daehee numbers (see [32, 37]).
Note that(55)∑n=0∞Dn,λxtnn!=logλ1+tt1+tx=∑n=0∞Dn,λtnn!∑n=0∞xntnn!=∑n=0∞∑l=0nnlDn−l,λxltnn!,and by (24), we have(56)Dn,λx=∑l=0n∑k=0lnllkλDn−l,λxk,λ.Figure 2
The shapes ofCh2x, E0,λx, E1,λx, and E2,λx when λ=0.5.Theorem 5.
For each nonnegative integern, we have(57)Chnx=∑k=0nnk∑m=0n−kn−kmChmβn−k−m,λDk,λx=∑k=0n∑l=0n∑m=0l∑a=kmnlma×1m−a+1,λChn−lm−a+1lmλakλDk,λx.
As the inversion formula of (57), we have(58)Dn,λx=∑k=0nnkDn−k,λ+n−k2Dn−k−1,λChkx=∑k=0n∑l=0n∑r=0lnllrλDn−l,λrkλ+12∑a=1rra1a,λr−akλChkx.Proof.
LetChnx=∑k=0ncn,kDk,λx. Since(59)Chnx∼eλt+12,eλt−1λ,Dn,λx∼eλt−1t,eλt−1λ,by Theorem 1, we have(60)cn,k=1k!t/logλ1+tt+2/2tk|xn,λλ=1k!tlogλ1+t2t+2|tkλxn,λλ=nktlogλ1+t2t+2|xn−k,λλ=nktlogλ1+t|2t+2λxn−k,λλ=nk∑m=0n−kn−kmChmtlogλ1+t|xn−k−m,λλ=nk∑m=0n−kn−kmChmβn−k−m,λ.
On the other hand, by (11) and (28), we obtain(61)cn,k=1k!eλt−1teλt−1k|Chnx=∑l=0n∑m=0lnllmλChn−leλt−1t|1k!eλt−1kλxm,λλ=∑l=0n∑m=0l∑a=kmnlmalmλakλChn−leλt−1t|xm−a,λλ=∑l=0n∑m=0l∑a=kmnlmalmλakλChn−l×∑b=0∞1b+1,λb+1tbb!|xm−a,λλ=∑l=0n∑m=0l∑a=kmnlma1m−a+1,λChn−lm−a+1lmλakλ.
Conversely, we assume thatDn,λx=∑k=0ndn,kChkx. Then, by Theorem 1 and (6), we obtain(62)dn,k=1k!t+2/2t/logλ1+ttk|xn,λλ=1k!t+22logλ1+tt|tkλxn,λλ=nklogλ1+tt|t+22λxn−k,λλ=nklogλ1+tt|xn−k,λ+n−k2xn−k−1,λλ=nklogλ1+tt|xn−k,λλ+n−k2logλ1+tt|xn−k−1,λλ=nkDn−k,λ+n−k2Dn−k−1,λ.
On the other hand, by (11) and (56), we obtain(63)dn,k=1k!eλt+12eλt−1k|Dn,λxλ=1k!∑l=0n∑r=0lnllrλDn−l,λeλt+12eλt−1k|xr,λλ=∑l=0n∑r=0lnllrλDn−l,λ1k!eλt−1k|eλt+12λxr,λλ=∑l=0n∑r=0lnllrλDn−l,λ×1k!eλt−1k|1+∑a=1∞1a,λ2taa!λxr,λλ=∑l=0n∑r=0lnllrλDn−l,λ×∑b=k∞bkλtbb!|xr,λ+12∑a=1rra1a,λxr−a,λλ=∑l=0n∑r=0lnllrλDn−l,λrkλ+12∑a=1rra1a,λr−akλ.
The degenerate Bell polynomials are defined by the generating function to be (see [1, 23])(64)eλxeλt−1=∑n=0∞Beln,λxtnn!.
Note that(65)∑n=0∞Beln,λxtnn!=eλxeλt−1=∑m=0∞xm,λ1m!eλt−1m=∑m=0∞xm,λ∑l=m∞lmλtll!=∑n=0∞∑m=0nnmλxm,λtnn!,and thus(66)Beln,λx=∑m=0nn\bracemλxm,λ.
In addition, we know that(67)1k!logλ1+logλ1+tk=∑l=k∞lkλ1l!logλ1+tl=∑l=k∞∑m=l∞lkλmlλtmm!=∑l=k∞∑m=klmkλlmλtll!.(68)eλeλt−1+12=121+∑a=0∞1a,λ1a!eλt−1a=1+12∑a=1∞∑b=a∞1a,λbaλtbb!=1+12∑a=1∞∑b=1a1b,λabλtaa!.Theorem 6.
For each nonnegative integern, we have(69)Chnx=∑k=0n∑l=kn∑m=klnllkλlmλChn−lBelk,λx.
As the inversion formula of (69), we have(70)Beln,λx=∑k=0n∑m=0n−km+kkλnm+kλ+12∑l=kn−1∑m=0l−k∑b=1n−l1b,λnlm+kkλlm+kλn−lbλChkx=∑k=0n∑m=knnmλmkλ+12∑m=k+1n∑l=km−1mlnmλlkλ1m−l,λChkx.Proof.
LetChnx=∑k=0ncn,kBelk,λx. Since(71)Chnx∼eλt+12,eλt−1λ,Beln,λx∼1,logλ1+tλ,by Theorem 1and (67), we obtain(72)cn,k=1k!1t+2/2logλ1+logλ1+tk|xn,λλ=2t+2|1k!logλ1+logλ1+tkλxn,λλ=2t+2|∑l=k∞∑m=klmkλlmλtll!λxn,λλ=∑l=kn∑m=klnlmkλlmλ2t+2|xn−l,λλ=∑l=kn∑m=klnlmkλlmλChn−l.
Conversely, we assume thatBeln,λx=∑k=0ndn,kChkx. By (68), we obtain(73)dn,k=1k!eλeλt−1+12eλeλt−1−1k|xn,λλ=eλeλt−1+12|1k!eλeλt−1−1kλxn,λλ=eλeλt−1+12|∑l=k∞lkλ1l!eλt−1lλxn,λλ=eλeλt−1+12|∑l=k∞∑m=l∞lkλmlλtmm!λxn,λλ=∑l=kn∑m=0l−knlm+kkλlm+kλeλeλt−1+12|xn−l,λλ=∑l=kn∑m=0l−knlm+kkλlm+kλ×1|xn−l,λ+12∑a=1∞∑b=1a1b,λabλtaa!|xn−l,λλ=∑m=0n−km+kkλnm+kλ+12∑l=kn−1∑m=0l−k∑b=1n−l1b,λnlm+kkλlm+kλn−lbλ.
On the other hand, by (11) and (66), we have(74)dn,k=1k!eλt+12eλt−1k|Beln,λxλ=1k!∑m=0nnmλeλt+12eλt−1k|xm,λλ=∑m=0nnmλeλt+12|1k!eλt−1kλxm,λλ=∑m=0n∑l=kmmlnmλlkλeλt+12|xm−l,λλ=∑m=0n∑l=kmmlnmλlkλ1+12∑a=1∞1a,λtaa!|xm−l,λλ=∑m=0n∑l=kmmlnmλlkλ1|xm−l,λλ+12∑a=1∞1a,λa!ta|xm−l,λλ=∑m=knnmλmkλ+12∑m=k+1n∑l=km−1mlnmλlkλ1m−l,λ,and thus our proofs are completed.
The unsigned Lah numberLn,k counts the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets and has the explicit formula (see [1, 20, 23, 38, 39])(75)Ln,k=n−1k−1n!k!.
By (75), we can derive the generating function of Ln,k to be (see [1, 20, 23, 38, 39])(76)1k!t1−tk=∑n=k∞Ln,ktnn!,k≥0.
Recently, Kim–Kim introduced the degenerate Lah–Bell polynomials as follows (see [1, 20]):(77)eλx11−t−1=∑n=0∞Bn,λLxtnn!.
In the special case ofx=1, BnL=BnL1 are called Lah–Bell numbers. Note that n-th Lah–Bell number BnLn≥0 is the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets. By (77), we can derive the following:(78)∑n=0∞Bn,λLxtnn!=eλxt1−t=∑n=0∞xn,λ1n!t1−tn=∑n=0∞xn,λ∑m=n∞Lm,ntmm!=∑n=0∞∑m=0nLn,mxm,λtnn!,and thus, we obtain(79)Bn,λLx=∑m=0nLn,mxm,λ.Theorem 7.
For each nonnegative integern, we have(80)Chnx=∑k=0n∑l=0n∑m=0l−1m−knlmkkm−klmλChn−lBkLx.
As the inversion formula of (80), we have(81)BnLx=∑k=0n∑l=knlkλLn,l+12∑m=kn−1∑l=km∑b=1n−mnm1b,λlkλm,lLn−m,bChkx=∑k=0n∑m=knmkλLn,m+12∑m=0n∑l=km−1ml1m−l,λlkλLn,mChkx.Proof.
LetChnx=∑k=0ncn,kBkLx. Since(82)Chnx∼eλt+12,eλt−1λ,BnLx∼1,t1+tλ,by (11) and (28), we obtain(83)cn,k=1k!t1+tk|Chnxλ=1k!∑l=0n∑m=0lnllmλChn−lt1+tk|xm,λλ=1k!∑l=0n∑m=0lnllmλChn−l1+t−k|tkλxm,λλ=∑l=0n∑m=0lnlmklmλChn−l∑a=0∞−1akaa!ta|xm−k,λλ=∑l=0n∑m=0lnlmk−1m−kkm−klmλChn−l.
Conversely, we assume thatBn,λLx=∑k=0ndn,kChkx. Then,(84)dn,k=1k!eλt/1−t+12eλt/1−t−1k|xn,λλ=eλt/1−t+12|1k!eλt/1−t−1kλxn,λλ=eλt/1−t+12|∑l=k∞lkλ1l!t/1−tlλxn,λλ=eλt/1−t+12|∑l=k∞∑m=l∞lkλLm,ltmm!λxn,λλ=∑m=kn∑l=kmnmlkλLm,leλt/1−t+12|xn−m,λλ=∑m=kn∑l=kmnmlkλLm,l1+12∑a=1∞∑b=a∞1a,λLb,atbb!|xn−m,λλ=∑m=kn∑l=kmnmlkλLm,l×1|xn−m,λλ+12∑a=1∞∑b=1a1b,λLa,btaa!|xn−m,λλ=∑l=knlkλLn,l+12∑m=kn−1∑l=km∑b=1n−mnm1b,λlkλLm,lLn−m,b.
On the other hand, by (11) and (79), we obtain(85)dn,k=1k!eλt+12eλt−1k|Bn,λLx=1k!∑m=0nLn,meλt+12eλt−1k|xm,λλ=∑m=0nLn,meλt+12|1k!eλt−1kλxm,λλ=∑m=0n∑l=kmmlLn,mlkλeλt+12|xm−l,λλ=∑m=0n∑l=kmmlLn,mlkλ1+12∑a=1∞1a,λtaa!|xm−l,λλ=∑m=0n∑l=kmmlLn,mlkλ×1|xm−l,λλ+12∑a=1∞1a,λa!ta|xm−l,λλ=∑m=knmkλLn,m+12∑m=0n∑l=km−1ml1m−l,λlkλLn,m,and so our proofs are completed.
By the definition of the degenerate Bernoulli polynomials of the second kind, we note that(86)∑n=0∞βn,λxtnn!=tlogλ1+teλxlogλ1+t=∑n=0∞∑m=0n∑l=0mnmmlλβn−m,λxl,λtnn!,and thus, we obtain(87)βn,λx=∑m=0n∑l=0mnmmlλβn−m,λxl,λ.Theorem 8.
For eachn≥0, we have(88)Chnx=∑l=0n−knkn−klChlDn−k−l,λβk,λx.
As the inversion formula of (88), we have(89)βn,λx=∑k=0nnkβn−k,λ+nkn−k2βn−k−1,λChkx=∑k=0n∑m=0n∑l=0mnmβn−m,λmlλlkλ+12∑m=0n∑l=k+1m∑a=kl−1nmlaβn−m,λmlλakλ1l−a,λChkx.Proof.
LetChnx=∑k=0ncn,kβk,λx. Since(90)Chnx∼eλt+12,eλt−1λ,βn,λx∼teλt−1,eλt−1λ,by Theorem 1, we obtain(91)cn,k=1k!logλ1+t/tt+2/2tk|xn,λλ=1k!2t+2logλ1+tt|tkλxn,λλ=nk2t+2logλ1+tt|xn−k,λλ=nklogλ1+tt|2t+2λxn−k,λλ=∑l=0n−knkn−klChllogλ1+tt|xn−k−l,λλ=∑l=0n−knkn−klChlDn−k−l,λ.
Conversely, we assume thatβn,λx=∑k=0ndn,kChkx. Then,(92)dn,k=1k!t+2/2logλ1+t/ttk|xn,λλ=1k!tlogλ1+tt+22|tkλxn,λλ=nktlogλ1+t|1+12txn−k,λλ=nk∑m=0∞βm,λtmm!|xn−k,λ+n−k2xn−k−1,λλ=nkβn−k,λ+nkn−k2βn−k−1,λ.
On the other hand, by (11) and (87), we obtain(93)dn,k=1k!eλt+12eλt−1k|βn,λxλ=1k!∑m=0n∑l=0mnmβn−m,λmlλeλt+12eλt−1k|xl,λλ=∑m=0n∑l=0mnmβn−m,λmlλeλt+12|1k!eλt−1kλxl,λλ=∑m=0n∑l=0m∑a=klnmlaβn−m,λmlλakλeλt+12|xl−a,λλ=∑m=0n∑l=0m∑a=klnmlaβn−m,λmlλakλ1+12∑b=1∞1b,λtbb!|xl−a,λλ=∑m=0n∑l=0mnmβn−m,λmlλlkλ+12∑m=0n∑l=k+1m∑a=kl−1nmlaβn−m,λmlλakλ1l−a,λ,and thus our proofs are completed.
The Mittag–Leffler polynomials are defined by the generating function to be see ([14, 34, 36])(94)1+t1−tx=∑n=0∞Mnxtnn!.Theorem 9.
For each nonnegative integern, we have(95)Chnx=∑k=0nnk−1n−kk+1n−k2nMkx.
As the inversion formula of (95), we have(96)Mnx=∑k=0n2k∑l=knnlLl,kn−l!Chkx.Proof.
LetChnx=∑k=0ncn,kMkx. Since(97)Chnx∼eλt+12,eλt−1λ,Mnx∼1,eλt−1eλt+1λ,by Theorem 1 and (30), we obtain(98)cn,k=1k!1t+2/2tt+2k|xn,λλ=1k!2tkt+2k+1|xn,λλ=1k!∑m=0∞−1mk+1m2k+mtmm!|tkλxn,λλ=nk∑m=0∞−1mk+1m2k+mtmm!|xn−k,λλ=nk−1n−kk+1n−k2n.
Conversely, we assume thatMnx=∑k=0ndn,kChkx. Then, by (76), we obtain(99)dn,k=1k!eλlogλ1+t/1−t+12eλlogλ1+t1−t−1k|xn,λλ=12k!21−t2t1−tk|xn,λλ=2k11−t|1k!t1−tkλxn,λλ=2k∑l=knnlLl,k11−t|xn−l,λλ=2k∑l=knnlLl,k∑m=0∞1mtmm!|xn−l,λλ=2k∑l=knnlLl,kn−l!,and so our proofs are completed.
The degenerate Frobenius–Euler polynomials of orderα are defined by the generating function to be (see [1, 2, 31, 40])(100)1−ueλt−uαeλxt=∑n=0∞hn,λαx|utnn!,u≠1,u∈ℂ,r≥0.
In the special casex=0, hn,λαu=hn,λα0|u are called the degenerate Frobenius–Euler numbers of order α. By the definition of the Frobenius–Euler polynomials of order α, we note that(101)∑n=0∞hn,λαx|utnn!=1−ueλt−uαeλxt=∑n=0∞hn,λαutnn!∑n=0∞xn,λtnn!=∑n=0∞∑l=0nnlhn−l,λαuxl,λtnn!,and thus we have(102)hn,λαx|u=∑l=0nnlhn−l,λαuxl,λ.
Furthermore, we see that(103)eλt−u−α=eλt−1+1−u−α=∑a=0∞−αa1−u−α−aeλt−1a=∑a=0∞−1aαa1−u−α−a1a!eλt−1a=∑m=0∞∑a=0m−1aαa1−uα+amaλtmm!.Theorem 10.
For each nonnegative integern, we have(104)Chnx=∑k=0n∑l=kn∑m=0n−lnlαmn−lm1−umlkλChn−l−mhk,λαx|u.
As the inversion formula of (104), we have(105)hn,λαx=∑k=0n∑m=kn∑b=0n−mnm−1bαb1−ubmkλn−mbλ+12∑m=kn−1∑a=0n−m−1∑b=0anmn−ma×−1bαb1−ubmkλabλ1n−m−a,λChkx=∑k=0n∑l=0nnllkλhn−l,λαu+12∑l=k+1n∑m=kl−1nllmmkλhn−l,λαu1l−m,λChkx.Proof.
LetChnx=∑k=0ncn,khk,λαx|u. Since(106)Chnx∼eλt+12,eλt−1λ,hn,λαx|u∼eλt−u1−uα,tλ,by Theorem 1, we obtain(107)cn,k=1k!1+t−u/1−uαt+2/2logλ1+tk|xn,λλ=2t+2t+1−u1−uα|1k!logλ1+tkλxn,λλ=11−uα∑l=knnllkλ2t+2t+1−uα|xn−l,λλ=11−uα∑l=knnllkλ2t+2|t+1−uαλxn−l,λλ=∑l=kn∑m=0n−lnlαmlkλn−lm1−um2t+2|xn−l−m,λλ=∑l=kn∑m=0n−lnlαmn−lm1−umlkλChn−l−m.
Conversely, we assume thathn,λαx|u=∑k=0ndn,kChkx. Then, by Theorem 1 and (103), we have(108)dn,k=1k!eλt+1/2eλt−u/1−uαeλt−1k|xn,λλ=1−ueλt−uαeλt+12|1k!eλt−1kλxn,λλ=∑m=knnmmkλ1−ueλt−uαeλt+12|xn−m,λλ=1−uα∑m=knnmmkλeλt+12|eλt−u−αλxn−m,λλ=∑m=kn∑a=0n−m∑b=0anmn−ma−1bαb1−ubmkλabλeλt+12|xn−m−a,λλ=∑m=kn∑a=0n−m∑b=0anmn−ma−1bαb1−ubmkλabλ×1+12∑l=1∞1l,λtll!|xn−m−a,λλ=∑m=kn∑b=0n−mnm−1bαb1−ubmkλn−mbλ+12∑m=kn−1∑a=0n−m−1∑b=0anmn−ma−1bαb1−ubmkλabλ1n−m−a,λ.
In addition, by (11) and (102), we obtain(109)dn,k=1k!eλt+12eλt−1k|hn,λαx|uλ=1k!∑l=0nnlhn−l,λαueλt+12eλt−1k|xl,λλ=∑l=0nnlhn−l,λαueλt+12|1k!eλt−1kλxl,λλ=∑l=0n∑m=klnllmmkλhn−l,λαueλt+12|xl−m,λλ=∑l=0n∑m=klnllmmkλhn−l,λαu1+12∑a=1∞1a,λtaa!|xl−m,λλ=∑l=0nnllkλhn−l,λαu+12∑l=k+1n∑m=kl−1nllmmkλhn−l,λαu1l−m,λ.


## 3. Conclusion

In this paper, we studied the Changhee polynomials related to the lambda falling factorial (Theorem2), the degenerate Bernoulli polynomials (Theorem 3), the degenerate Euler polynomials (Theorem 4), the degenerate Daehee polynomials (Theorem 5), the degenerate Bell polynomials (Theorem 6), the degenerate Lah–Bell polynomials (Theorem 7), the degenerate Bernoulli polynomials of the second kind (Theorem 8), the Mittag–Leffler polynomials (Theorem 9), and the degenerate Frobenius–Euler polynomials (Theorem 10) by finding the coefficients which are also polynomials or numbers when the n-th Changhee polynomial is expressed as a linear combination of those degenerate special polynomials by using the λ-Sheffer sequences and λ-differential operators. In addition, we derive the inversion formulas of these identities.Umbral calculus has been applied in many fields such as combinatorial counting with linear recurrences and lattice path counting, graph theory using chromatic polynomials, probability theory, link invariant theory, statistics, topology, and physics. It is being actively applied in various fields by researchers. As one of our future projects, we would like to continue to study degenerate versions of certain special polynomials and numbers by usingλ-Umbral calculus. [41].



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