# On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

Authors: Sang Jo Yun; Jin-Woo Park
Journal: Journal of Mathematics
 (2020)

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## Abstract

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we define the fully degenerate Daehee polynomials of the second kind which are different from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.


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## Body

## 1. Introduction

Letp be a fixed prime number. Throughout this paper, ℤp, ℚp, and ℂp will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp, respectively.Letfx be a uniformly differentiable function on ℤp. Then, the p-adic invariant integral onℤp is defined as(1)I0f=∫ℤpfxdμ0x=limN⟶∞1pN∑n=0pN−1fx,(see [1–3]).From (1), we have(2)I0fn−I0f=∑l=0n−1f′l,wherefnx=fx+n,f′l=dfxdxx=l.In particular, ifn=1, then(3)I0f1−I0f=f′0.TheStirling numbers of the first kind are defined by(4)xn=∑l=0nS1n,lxl,n≥0,and the Stirling numbers of the second kind are given by(5)xn=∑l=0nS2n,lxl,where x0=1 and xn=xx−1⋯x−n+1n≥1 (see [4, 5]).From (4) and (5), we can derive the following equations:(6)et−1n=n!∑l=n∞S2l,ntll!,(7)logx+1n=n!∑l=n∞S1l,nxll!,n≥0,(see [4–6]).In addition,(8)log1+t=∑n=1∞−1n+1n−1!tnn!,[4, 5].TheBernoulli polynomials of orderr are defined by the following generating function:(9)tet−1rext=∑n=0∞Bnrxtnn!,(see [7–9]).Carlitz’s degenerate Bernoulli polynomials of orderr is defined by the generating function to be(10)∑n=0∞βnrxλtnn!=t1+λt1/λ−1r1+λtx/λ,where λ∈ℝ (see [10]). By (10), we know that(11)limλ⟶0∑n=0∞βnrxλtnn!=limλ⟶0t1+λt1/λ−1r1+λtx/λ=tet−1rext=∑n=0∞Bnrxtnn!,and thus, we obtain(12)limλ⟶0βnrxλ=Bnrx.In [11], the degenerate Bernoulli polynomials are defined as(13)log1+λt1/λ1+λt1/λ−1r1+λtx/λ=∑n=0∞bn,λrxtnn!,which are different from Carlitz’s degenerate numbers and polynomials.By (13), we know that(14)limλ⟶0bn,λx=Bnrx.Note that by (3),(15)∫ℤp⋯∫ℤp1+λtx+x1+⋯+xr/λdμ0x1…dμ0xr=1/λlog1+λt1+λt1/λ−1r1+λtx/λ=∑n=0∞bn,λrxtnn!,(16)∫ℤp⋯∫ℤp1+λtx+x1+⋯+xr/λdμ0⋯dμ0xr=∫ℤp⋯∫ℤp∑n=0∞x+x1+⋯+xrλnλntndμ0x1⋯dμ0xr=∑n=0∞∫ℤp⋯∫ℤpx+x1+⋯+xrn,λdμ0x1⋯dμ0xrtnn!.By (15) and (16), we know that(17)bn,λrx=∫ℤp⋯∫ℤpx+x1+⋯+xrn,λdμ0x1⋯dμ0xr.Thehigher-order Daehee polynomials are defined by the generating function to be(18)log1+ttr1+tx=∑n=0∞Dnrxtnn!,(see [12–15]).In particular, ifr=1, then Dn1x=Dnx is called the Daehee polynomials.By replacingt as log1+t in (18), we have(19)∑n=0∞Dnrxtnn!=∑n=0∞Bnrx1n!log1+tn=∑n=0∞Bnrx1n!n!∑l=0∞S1l,ntll!=∑n=0∞∑m=0nBnrxS1n,mtnn!,and so, we obtain(20)Dnrx=∑m=0nBmrxS1n,m.Note that by (3), we have(21)∑n=0∞Dnrxtnn!=log1+ttr1+tx=∫ℤp⋯∫ℤp1+tx+x1+⋯+xrdμ0x1⋯dμ0xr=∫ℤp⋯∫ℤpx+x1+⋯+xrntndμ0x1⋯dμ0xr=∑n=0∞∫ℤp⋯∫ℤpx+x1+⋯+xrndμ0x1⋯dμ0xrtnn!,and thus, we know that(22)Dnrx=∫ℤp⋯∫ℤpx+x1+⋯+xrndμ0x1⋯dμ0xr.Carlitz introduced the degenerate Bernoulli polynomials in [10] and the degeneration of special functions have been studied (see [6, 16–29]).In particular, the degenerate Stirling numbers of the second kind with a generating function are defined as(23)1m!1+λt1/λ−1m=∑n=m∞S2,λn,mtnn!,where m is a given nonnegative integer in [6, 10, 22, 30].After introducing Daehee numbers and polynomials [31], it plays an important role of developing various generalized polynomials, and interesting properties are obtained (see [8, 15, 21, 22, 28, 30–35]).In this paper, we define the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind and investigate identities and properties of new polynomials.


## 2. Fully Degenerate Daehee Polynomials of the Second Kind

Let us assume thatλ∈ℝ. By (3), we have(24)∫ℤp1+λlog1+tx+y/λdμ0y=1/λlog1+λlog1+t1+λlog1+t1/λ−11+λlog1+tx/λ.By (24), we define the degenerate Daehee polynomials of the second kind by the generating function to be(25)1/λlog1+λlog1+t1+λlog1+t1/λ−11+λlog1+tx/λ=∑n=0∞Dnxλtnn!.In the special case,x=0, and Dnλ=Dn0λ are called the degenerate Daehee numbers of the second kind.Note that(26)limλ⟶0∑n=0∞Dnxλtnn!=limλ⟶01/λlog1+λlog1+t1+λlog1+t1/λ−11+λlog1+tx/λ=log1+tt1+tx=∑n=0∞Dnxtnn!,and thus, we know that(27)limλ⟶0Dnxλ=Dnx,n≥0.From (7) and (24), we have(28)∫ℤp1+λlog1+tx+y/λdμ0y=∑l=0∞∫ℤpx+yλlλllog1+tldμ0y=∑n=0∞∑k=0nS1n,lλll!∫ℤpx+yλldμ0ytnn!.By (25) and (28), we have(29)Dn,λ=∑k=0nS1n,lλll!∫ℤpx+yλldμ0y,n≥0.Since(30)x+yλl=x+y/λx+y/λ−1x+y/λ−2…x+y/λ−l+1l!=x+yx+y−λx+y−2λ…x+y−l−1λλll!,by (29) and (30), we have(31)Dnλ=∑k=0nS1n,l∫ℤpx+yl,λdμ0y,n≥0,where xn,λ=xx−λx−2λ⋯x−n−1λ.Thus, by (29) and (30), we have the following theorem which is Witt’s type formula about degenerate Daehee polynomials of the second kind.Theorem 1.
For eachn≥0, we have(32)Dnλ=∑k=0nS1n,lλll!∫ℤpx+yλldμ0y=∑k=0nS1n,l∫ℤpx+yl,λdμ0y.By replacingt as et−1 in (25), we obtain the following:(33)∑n=0∞Dnxλet−1nn!=∑n=0∞Dnx|λ∑m=n∞S2m,ntmm!=∑n=0∞∑m=0nDmx|λS2m,ntnn!.On the other hand,(34)∑n=0∞Dnxλet−1nn!=1/λlog1+λt1+λt1/λ−11+λtx/λ=∑n=0∞Bnxλtnn!,where Bnxλ is the degenerate Bernoulli polynomials of the second kind of order r∈ℤ which are defined by the generating function to be(35)1/λlog1+λt1+λt1/λ−1r1+λtx/λ=∑n=0∞Bnrxλtnn!(see [20, 25]).In particular, ifr=1, Bn1xλ=Bnxλ is called the degenerate Bernoulli polynomials of the second kind.For positive integerd with d≡1mod2, if we put fx=1+λlog1+tx/λ, then, by (2), we obtain(36)∫ℤp1+λlog1+tx+d/λdμ0y−∫ℤp1+λlog1+tx/λdμ0y=∑l=0d−11λ1+λlog1+tx/λlog1+λlog1+t.By (36), we have(37)∫ℤp1+λlog1+tx/λdμ0x=1/λlog1+λlog1+t1+λlog1+td/λ−1∑l=0d−11+λlog1+tl/λ=d∑l=0d−11/λlog1+λ/ddlog1+t1+λ/ddlog1+td/λ−11+λddlog1+td/λl/d.Note that by (6),(38)1/λlog1+λ/ddlog1+t1+λ/ddlog1+td/λ−11+λ/ddlog1+td/λl/d=∑n=0∞Bnldλddlog1+tnn!=∑n=0∞Bnldλddnn!n!∑m=n∞S1m,nxmm!=∑n=0∞∑m=0nBmldλddmS1n,mtnn!.By (24), (37), and (38), we have(39)∑n=0∞Dnλtnn!=∫ℤp1+λlog1+tx/λdμ0x=∑n=0∞∑m=0n∑l=0d−1Bmldλddm+1S1n,mtnn!.Hence, by (33), (34), and (39), we obtain the following theorem which shows the relationship between degenerate Daehee polynomials of the second kind and degenerate Bernoulli polynomials of the second kind.Theorem 2.
For nonnegative integern and d∈ℕ with d≡1mod2, we have(40)Bnxλx=∑m=0nDm,λxS2n,m,Dn,λ=∑m=0n∑l=0d−1Bmldλddm+1S1n,m.
By (25), we note that(41)∑n=0∞Dnxλtnn!=1/λlog1+λlog1+t1+λlog1+t1/λ−11+λlog1+tx/λ=∑n=0∞Dnλtnn!∑n=0∞xλnλnlog1+tn=∑n=0∞Dnλtnn!∑n=0∞xλnλnn!∑l=n∞S1l,nxll!=∑n=0∞Dnλtnn!∑n=0∞∑l=0nxl,λS1n,lxnn!=∑n=0∞∑k=0n∑l=0knkxl,λS1k,lDn−kλtnn!.By comparing the coefficients on both sides of (41), we obtain the following theorem.Theorem 3.
For nonnegative integern, we have(42)Dnxλ=∑k=0n∑l=0knkxl,λS1k,lDn−kλ.Note that if we putfx=1+λlog1+tx/λ, then(43)f′0=1λlog1+λlog1+t=1λ∑n=1∞−1n+1λnnlog1+tn=∑n=1∞−1n+1λn−1nn!∑l=n∞S1l,ntll!=∑n=0∞∑m=1n−1m+1λm−1m−1!S1n,mtnn!,and thus, by (3), we have(44)∫ℤpfx1dμ0x−∫ℤpfxdμ0x=1/λlog1+λlog1+t1+λlog1+t1/λ−11+λlog1+t1/λ−1/λlog1+λlog1+t1+λlog1+t1/λ−1=∑n=0∞Dn1λ−Dnλtnn!.Moreover,(45)1/λlog1+λlog1+t∑l=0n−11+λlog1+tl/λ=1/λlog1+λlog1+t1+λlog1+t1/λ1+λlog1+tn−1/λlog1+λlog1+t1+λlog1+t1/λ=∑r=0∞Drnλ−Drλtrr!,and, by (43), we obtain(46)1/λlog1+λlog1+t∑l=0n−11+λlog1+tl/λ=∑p=0∞∑m=1p−λm−1m−1!S−1p,mtpp!∑l=0n−1∑s=0∞∑q=0slq,λS1s,qtss!=∑r=0∞∑l=0n−1∑p=0r∑m=1p∑q=0r−prplq,λ−λm−1m−1!S1p,mS1r−p,qtrr!.From (43), (44), and (46), we obtain the following theorem which represents a recurrence relations between degenerate Daehee polynomials of the second kind and degenerate Daehee numbers of the second kind.Theorem 4.
For each nonnegative integerr, we have(47)Dr1λ−Drλ=∑m=1r−λm−1m−1!S1r,m.Moreover, for each positive integern≥2,(48)Drnλ−Drλ=∑l=0n−1∑p=0r∑m=1p∑q=0r−prplq,λ−λm−1m−1!S1p,mS1r−p,q.


## 3. Higher-Order Degenerate Daehee Polynomials of the Second Kind

In this section, we consider thehigher-order degenerate Daehee polynomials of the second kind given by the generating function as follows: for the given positive real number r,(49)1/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=∑n=0∞Dnrxλtnn!.In particular, ifx=0, Dnr0λ=Dnrλ are called the higher-order degenerate Daehee numbers of the second kind.Note that(50)limλ⟶0∑n=0∞Dnrxλtnn!=limλ⟶01/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=log1+ttr1+tx=∑n=0∞Dnrxtnn!.From (35), we note that(51)1/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=∑n=0∞Bnrxλlog1+tnn!=∑n=0∞Bnrxλn!n!∑l=n∞S1l,ntll!=∑n=0∞∑m=0nBmrxλS1n,mtnn!.In addition, by replacingt by et−1 in (49), we have(52)∑n=0∞Dnrxλet−1nn!=∑n=0∞Dnrxλn!n!∑m=n∞S2n,mtmm!=∑n=0∞∑m=0nDmrxλS2n,mtnn!,(53)∑n=0∞Dnrxλet−1nn!=1/λlog1+λt1+λt1/λ−1r1+λtx/λ=∑n=0∞Bnrxλtnn!.Hence, by (51)–(53), we obtain the following theorem.Theorem 5.
Forn≥0, we have(54)Dnrxλ=∑m=0nBmrxλS1n,m.Moreover,(55)Bnrxλ=∑m=0nDmrxλS2n,m.In particular, ifr=1, then we know that Theorem 5 is a generalization of Theorem 1 and Theorem 2.Note that for eachk,r∈ℕ,(56)1/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=1/λlog1+λlog1+t1+λlog1+t1/λ−1k1/λlog1+λlog1+t1+λlog1+t1/λ−1r−k1+λlog1+tx/λ=∑n=0∞Dnkλtnn!∑n=0∞Dnr−kxλtnn!=∑n=0∞∑m=0nnmDmkλDn−mr−kxλtnn!.It is well known that for eachk∈ℤ,(57)tlog1+tk1+tx−1=∑n=0∞Bnn−k+1xtnn!.By (23), (7), and (57), we have(58)1/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=1/λlog1+λlog1+t1+λlog1+t1/λ−1−r1+λlog1+tx/λ=∑n=0∞Bnn+r+1x+11n!1+λlog1+t1/λ−1n=∑n=0∞Bnn+r+1x+1∑m=n∞S2,λm,nlog1+tmm!=∑p=0∞∑n=0pBnn+r+1x+1S2,λp,n1p!log1+tp=∑q=0∞∑p=0q∑n=0pBnn+r+1x+1S2,λq−l,nS1q,ptqq!.By (56) and (58), we obtain the following theorem.Theorem 6.
For eachn,q≥0, we have(59)Dnrxλ=∑m=0nnmDmkλDn−mr−kxλ,Dqrxλ=∑p=0q∑n=0pBnn+r+1x+1S2,λq−l,nS1q,p.Note that by (3),(60)∫ℤp⋯∫ℤp1+λlog1+tx+x1+⋯+xr/λdμ0x1⋯dμ0xr=1/λlog1+λlog1+t1+λlog1+t1/λ−1r1+λlog1+tx/λ=∑n=0∞Dnrxλtnn!.By (17) and (60), we have(61)∫ℤp⋯∫ℤp1+λlog1+tx+x1+⋯+xr/λdμ0x1⋯dμ0xr=∫ℤp⋯∫ℤp∑n=0∞x+x1+⋯+xrλnλnlog1+tndμx1⋯dμ0xr=∑n=0∞∫ℤp⋯∫ℤpx+x1+⋯+xrn,λ1n!n!∑l=n∞S1l,nxll!=∑n=0∞∑m=0n∫ℤp⋯∫ℤp+x1+⋯+xrm,λS1n,mtnn!=∑n=0∞∑m=0nS1n,mβmrxtnn!.Thus, by (60) and (61), we obtain the following theorem which shows that higher-order degenerate Daehee polynomials of the second kind are represented by linear combination of the higher-order Carlitz’s type degenerate Bernoulli polynomials.Theorem 7.
For each nonnegative integern and each integer r,(62)Dnrxλ=∑m=0nS1n,mβmrx.By (7) and (23), we have(63)1+λlog1+tx+x1+⋯+xr/λ=1+λlog1+t1/λ−1+1x+x1+⋯+xr=∑m=0∞x+x1+⋯+xrm1+λlog1+t1/λ−1m=∑m=0∞x+x1+⋯+xrm∑n=m∞S2,λn,mlog1+tnn!=∑n=0∞∑m=0nx+x1+⋯+xrmS2,λn,mlog1+tnn!=∑n=0∞∑m=0nx+x1+⋯+xrmS2,λn,m1n!n!∑l=n∞S1l,ntll!=∑n=0∞∑k=0n∑m=0kx+x1+⋯+xrmS2,λk,mS1n,ktnn!,and so by (22), (60), and (63), we obtain(64)∫ℤp⋯∫ℤp1+λlog1+tx+x1+⋯+xr/λdμ0x1⋯dμ0xr=∑n=0∞∑k=0n∑m=0kS2,λk,mS1n,k∫ℤp⋯∫ℤpx+x1+⋯+xrmdμ0x1⋯dμ0xrtnn!=∑n=0∞∑k=0n∑m=0kS2,λk,mS1n,kDmrxtnn!.By (60) and (64), we obtain the following theorem.Theorem 8.
For each nonnegative integerm,(65)Dnrxλ=∑k=0n∑m=0kS2,λk,mS1n,kDmrx.Theorem8 shows that higher-order degenerate Daehee polynomials are related closely to Daehee polynomials of order r.


## 4. Conclusion

In the past two decades, the degenerations of special functions and their applications have been studied as a new area of mathematics. In this paper, we considered the degenerate Daehee numbers and polynomials by usingp-adic invariant integral on ℤp which are different from Kim’s degenerate Daehee polynomials. We derive some new and interesting properties of those polynomials.Next, from the definition of the higher-order degenerate Daehee numbers and Daehee polynomials of the second kind, we found the relationship between the degenerate Bernoulli polynomials, the first and second Stirling numbers, the Bernoulli polynomials, degenerate Stirling numbers of the second kind, and those numbers and polynomials.



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