# Representation by Degenerate Genocchi Polynomials

Authors: Taekyun Kim; Dae San Kim; Jongkyum Kwon; Seong-Ho Park
Journal: Journal of Mathematics
 (2022)

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## Abstract

The aim of this study is to represent any polynomial in terms of the degenerate Genocchi polynomials and more generally of the higher-order degenerate Genocchi polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.


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## Body

## 1. Introduction and Preliminaries

In this study, we derive formulas expressing any polynomial in terms of the degenerate Genocchi polynomials ((14)) with the help of umbral calculus (Theorem 1) and illustrate our results with some examples (Section 5). This can be generalized to the higher-order degenerate Genocchi polynomials ((15)). Indeed, we deduce formulas for representing any polynomial in terms of the higher-order degenerate Genocchi polynomials again by using umbral calculus (Theorem 2). The contribution of this study is the derivation of such formulas which have potential applications to finding some interesting polynomial identities, as illustrated in Section 5.Many interesting identities have been derived by using similar formulas for representations by Bernoulli, Euler, and Frobenius-Euler polynomials (see [1–9]). The list in the references is far from being exhaustive. However, the interested reader can easily find more related studies in the literature. Also, we should mention here that there are other ways of obtaining the same result as the one in (2). One of them is to use Fourier series expansion of the function obtained by extending by periodicity of period 1 of the polynomial function restricted to the interval 0,1 [10].Letpx∈ℂx,withdegpx=n. Write px=∑k=0nakBkx, where Bnx are the Bernoulli polynomials defined by t/et−1ext=∑n=0∞Bnxtn/n!. Then, it is known that(1)a0=∫01ptdt,ak=1k!pk−11−pk−10,fork=1,2,…,n.Applying the formulas in (1) to the polynomial px=∑k=1n−11/kn−kBkxBn−kx, we can get an identity which yields, after slight modification, the following identity:(2)∑k=1n−112k2n−2kB2kxB2n−2kx+22n−1B1xB2n−1x=1n∑k=1n12k2n2kB2kB2n−2kx+1nH2n−1B2nx+22n−1B1xB2n−1,where n≥2, and Hn=1+1/2+⋯+1/n.Lettingx=0 in (2), we obtain a slight variant of the well-known Miki’s identity:(3)∑k=1n−112k2n−2kB2kB2n−2k=1n∑k=1n12k2n2kB2kB2n−2k+1nH2n−1B2n.Lettingx=1/2 in (2), we get the Faber–Pandharipande–Zagier (FPZ) identity:(4)∑k=1n−112k2n−2kB¯2kB¯2n−2k=1n∑k=1n12k2n2kB2kB¯2n−2k+1nH2n−1B¯2n,where B¯n=21−n−1Bn=Bn1/2, for all n=0,1,2,….Here, it should be stressed that the other proofs of Miki’s ([11–13]) and FPZ identities ([14,15]) are quite involved, while our proofs of Miki’s and Faber–Pandharipande–Zagier identities follow from the simple formulas in (1) involving only derivatives and integrals of the given polynomials.The outline of this study is as follows. In Section1, we recall some necessary facts that are needed throughout this study. In Section 2, we go over umbral calculus briefly. In Section 3, we derive formulas expressing any polynomial in terms of the degenerate Genocchi polynomials. In Section 4, we derive formulas representing any polynomial in terms of the higher-order degenerate Genocchi polynomials. In Section 5, we illustrate our results with some examples. Finally, we conclude our study in Section 6.The Euler polynomialsEnx are defined by(5)2et+1ext=∑n=0∞Enxtnn!.Whenx=0, En=En0 are called the Euler numbers. We observe that Enx=∑j=0nnjEn−jxj,d/dxEnx=nEn−1x,Enx+1+Enx=2xn. The first few terms of En are given by(6)E0=1,E1=−12,E3=14,E5=−12,E7=178,E9=−312,…,E2k=0,k≥1.More generally, for any nonnegative integerr, the Euler polynomials Enrx of order r are given by(7)2et+1rext=∑n=0∞Enrxtnn!.The Genocchi polynomialsGnx are defined by(8)2tet+1ext=∑n=0∞Gnxtnn!.Whenx=0, Gn=Gn0 are called the Genocchi numbers. We observe that Gnx=∑j=0nnjGn−jxj, d/dxGnx=nGn−1x,  Gnx+1+Gnx=2nxn−1, and degGnx=n−1, for n≥1. Note that G0x=0. The first few terms of Gn are given by(9)G0=0,G1=1,G2=−1,G4=1,G6=−3,G8=17,G10=−155,G12=2073,…G2k+1=0,k≥1.More generally, for any nonnegative integerr, the Genocchi polynomials Gnrx of order r are given by(10)2tet+1rext=∑n=0∞Gnrxtnn!.Observe here thatG0rx=G1rx=⋯=Gr−1rx=0 and that Gnrx=n!/n−r!En−rrx has degree n−r, for n≥r.For any nonzero real numberλ, the degenerate exponentials are given by ([3, 5, 6])(11)eλxt=1+λtx/λ=∑n=0∞xn,λtnn!,eλt=eλ1t=1+λt1/λ=∑n=0∞1n,λtnn!.Carlitz [16] introduced a degenerate version of the Euler polynomials Enx, called the degenerate Euler polynomials and denoted by En,λx, which are given by(12)2eλt+1eλxt=∑n=0∞En,λxtnn!.Forx=0, En,λ=En,λ0 are called the degenerate Euler numbers.More generally, for any nonnegative integerr, the degenerate Euler polynomials En,λrx of order r are given by(13)2eλt+1reλxt=∑n=0∞En,λrxtnn!.A degenerate version of the Genocchi polynomialsGnx, called the degenerate Genocchi polynomials and denoted by Gn,λx, are given by ([17])(14)2teλt+1eλxt=∑n=0∞Gn,λxtnn!.Forx=0, Gn,λ=Gn,λ0 are called the degenerate Genocchi numbers. Note that G0,λx=0.More generally, for any nonnegative integerr, the degenerate Genocchi polynomials Gn,λrx of order r are given by ([17])(15)2teλt+1reλxt=∑n=0∞Gn,λrxtnn!.Forx=0, Gn,λr=Gn,λr0 are called the degenerate Genocchi numbers of order r. Observe that G0,λrx=G1,λrx=⋯=Gr−1,λrx=0 and that Gn,λrx=n!/n−r!En−r,λrx has degree n−r, for n≥r.We remark thatGn,λx⟶Gnx and Gn,λrx⟶Gnrx, as λ tends to 0.We recall some notations and facts about forward differences. Letf be any complex-valued function of the real variable x. Then, for any real number a, the forward difference Δa is given by(16)Δafx=fx+a−fx.Ifa=1, then we let(17)Δfx=Δ1fx=fx+1−fx.We also need(18)Δ˜fx=fx+1+fx.It is necessary to note that(19)ΔaΔ˜=Δ˜Δa.In general, thenth order forward differences are given by(20)Δanfx=∑i=0nni−1n−ifx+ia.Fora=1, we have(21)Δnfx=∑i=0nni−1n−ifx+i.It is easy to see that(22)Δ˜nfx=∑i=0nnifx+i.Finally, we recall that the Stirling numbers of the second kindS2n,k are given by(23)1k!et−1k=∑n=k∞S2n,ktnn!,k≥0.


## 2. Review of Umbral Calculus

Here, we will briefly go over very basic facts about umbral calculus. For more details on this, we recommend the reader to refer to [18–20]. Let ℂ be the field of complex numbers. Then, ℱ denotes the algebra of formal power series in t over ℂ given by(24)ℱ=ft=∑k=0∞aktkk!|,ak∈ℂ,and ℙ=ℂx indicates the algebra of polynomials in x with coefficients in ℂ.The set of all linear functionals onℙ is a vector space as usual and denoted by ℙ∗. Let L|px denote the action of the linear functional L on the polynomial px.Forft∈ℱ with ft=∑k=0∞aktk/k!, we define the linear functional on ℙ by(25)ft|xk=ak.From (25), we note that(26)tk|xn=n!δn,k,n,k≥0,where δn,k is Kronecker’s symbol.Some remarkable linear functionals are as follows:(27)eyt|px=py,eyt−1|px=py−p0,eyt−1t|px=∫0ypudu.Let(28)fLt=∑k=0∞L|xktkk!.Then, by (25) and (28), we get(29)fLt|xn=L|xn.That is,fLt=L, as linear functionals on ℙ∗, shows the map L↦fLt from ℙ∗ onto ℱ is one-to-one. Additionally, the map L↦fLt is linear and also onto in view of (25) and (28). Thus, it is a vector space isomorphism from ℙ∗ onto ℱ.Henceforth,ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on ℙ. ℱ is called the umbral algebra, and the umbral calculus is the study of umbral algebra. For each nonnegative integer k, the differential operator tk on ℙ is defined by(30)tkxn=nkxn−k,ifk≤n,0,ifk>n.Extending (30) linearly, any power series(31)ft=∑k=0∞akk!tk∈ℱ,gives the differential operator on ℙ defined by(32)ftxn=∑k=0nnkakxn−k,n≥0.It should be observed that, for any formal power seriesft and any polynomial px, we have(33)ft|px=1|ftpx=ftpx|x=0.Here, we note that an elementft of ℱ is a formal power series, a linear functional, and a differential operator. Some notable differential operators are as follows:(34)eytpx=px+y,eyt−1px=px+y−px,eyt−1tpx=∫xx+ypudu.The orderoft of the power series ft≠0 is the smallest integer for which ak does not vanish. If oft=0, then ft is called an invertible series. If oft=1, then ft is called a delta series.Forft,gt∈ℱ with oft=1 and ogt=0, there exists a unique sequence snx (deg snx=n) of polynomials, such that(35)gtftk|snx=n!δn,k,n,k≥0.The sequencesnx is said to be the Sheffer sequence for gt,ft, which is denoted by snx∼gt,ft. We observe from (35) that(36)snx=1gtpnx,where pnx=gtsnx∼1,ft.In particular, ifsnx∼gt,t, then pnx=xn, and hence,(37)snx=1gtxn.It is well known thatsnx∼gt,ft if and only if(38)1gf¯texf¯t=∑k=0∞skxk!tk,for all x∈ℂ, where f¯t is the compositional inverse of ft, such that f¯ft=ff¯t=t.The following equations (39)–(41) are equivalent to the fact that snx is Sheffer for gt,ft, for some invertible gt:(39)ftsnx=nsn−1x,n≥0,(40)snx+y=∑j=0nnjsjxpn−jy,with pnx=gtsnx,(41)snx=∑j=0n1j!gf¯t−1f¯tj|xnxj.Forsnx∼gt,ft and rnx∼ht,lt, we have(42)snx=∑k=0nCn,krkx,n≥0,where(43)Cn,k=1k!hf¯tgf¯tlf¯tk|xn.


## 3. Representation by Degenerate Genocchi Polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the degenerate Genocchi polynomials.From (11), we first observe that(44)xn,λ∼1,ft=1λeλt−1.From (39), we note that(45)ftxn,λ=nxn−1,λ.It is immediate to see from (14) that(46)Δ˜Gn,λx=Gn,λx+1+Gn,λx=2nxn−1,λ.Now, we assume thatpx∈ℂx has degree n, and write px=∑k=1n+1akGn,λx. Let ax=px+1+px=Δ˜px. Then, from (45) and (46), we have(47)ax=∑k=1n+1akGk,λx+1+Gk,λx=2∑k=1n+1akkxk−1,λ=2ft∑k=1n+1akxk,λ.Fork≥1, from (45) and (47), we obtain(48)ftk−1ax=2ftk∑l=1n+1alxl,λ=2∑l=kn+1ll−1⋯l−k+1alxl−k,λ.Lettingx=0 in (48), we finally get(49)ak=12k!ftk−1ax|x=0=12k!ftk−1|ax,1≤k≤n+1.An alternative expression of (49) is given by(50)ak=12k!λk−1Δλk−1ax|x=0=12k!λk−1Δλk−1p1+Δλk−1p0=12k!λk−1Δλk−1Δ˜p0,since ftax=1/λeλt−1ax=1/λΔλax.From (50), we have another alternative expression of (49) which is given by(51)ak=12k!λk−1Δλk−1ax|x=0=12k!λk−1∑l=0k−1k−1l−1k−1−lax+lλ|x=0=12k!λk−1∑l=0k−1k−1l−1k−1−lp1+lλ+plλ.From (49), we get yet another expression of ak as follows:(52)ak=12kλk−11k−1!eλt−1k−1ax|x=0=12k∑l=k−1nS2l,k−1λl−k+1l!pl1+pl0,where plx=d/dxlpx.Finally, from (49)–(52), we get the following theorem.where ft=1/λeλt−1.Theorem 1.
Letpx∈ℂx,degpx=n. Then, we have px=∑k=1n+1akGk,λx, where ak’s are given by the following various expressions:(53)ak=12k!ftk−1px+1+px|x=0=12k!λk−1eλt−1k−1|px+1+px=12k!λk−1Δλk−1Δ˜p0=12k!λk−1∑l=0k−1k−1l−1k−1−lp1+lλ+plλ=12k∑l=k−1nS2l,k−1λl−k+1l!pl1+pl0,fork=1,2,…,n+1,Remark 1.
Letpx∈ℂx,withdegpx=n. Write px=∑k=1n+1akGkx. As λ tends to 0, ft⟶t. Thus, we get the following result, namely, we have(54)ak=12k!pk−11+pk−10,fork=1,2,…,n+1.Remark 2.
The formulas in (1) and (54) and analogous formula for representation by Euler polynomials have been applied to many polynomials in order to obtain interesting identities for certain special polynomials and numbers. Some of the polynomials that have been considered are as follows:(a)
(55)∑Bi1x⋯BirxEj1x⋯EjsxGk1+1x⋯Gkt+1xxl,Where the sum is over all nonnegative integersi1,…,ir,j1,…,js,k1,…,kt,l, such that i1+⋯+ir+j1+⋯+js+k1+⋯+kt+l=n, and r,s,t,l are the nonnegative integers with r+s+t≥1.(b)
(56)∑Bi1x⋯BirxEj1x⋯EjsxGk1+1x⋯Gkt+1xxli1!⋯ir!j1!⋯js!k1+1!⋯kt+1!l!,Where the sum is over all nonnegative integersi1,…,ir,j1,…,js,k1,…,kt,l, such that i1+⋯+ir+j1+⋯+js+k1+⋯+kt+l=n, and r,s,t,l are the nonnegative integers with r+s+t≥1.(c)
(57)∑Bi1x⋯BirxEj1x⋯EjsxGk1+1x⋯Gkt+1xxli1⋯irj1⋯jsk1+1⋯kt+1l,Where the sum is over all positive integersi1,…,ir,j1,…,js,l and nonnegative integers k1,…,kt, such that i1+⋯+ir+j1+⋯+js+k1+⋯+kt+l=n, and r,s,t,l are the nonnegative integers with r+s+t≥1.


## 4. Representation by Higher-Order Degenerate Genocchi Polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the higher-order degenerate Genocchi polynomials.As we noted in (45), we have ftxn,λ=nxn−1,λ, with ft=1/λeλt−1. Let gt=et+1.It is immediate to see from (15) that(58)Δ˜Gn,λrx=Gn,λrx+1+Gn,λrx=2nGn−1,λr−1x.Repeated application of (58) gives us(59)Δ˜rGn,λrx=2n2n−1⋯2n−r+1Gn−r,λ0x=2rnn−1⋯n−r+1xn−r,λ,n≥r.Now, we assume thatpx∈ℂx has degree n, and write px=∑k=rn+rakGk,λrx. Then, from (59), we observe that(60)gtrpx=Δ˜rpx=∑l=rn+ralΔ˜rGl,λrx=2r∑l=rn+rll−1⋯l−r+1alxl−r,λ.By using (45) and (60), for k≥0, we observe that(61)ftkgtrpx=2r∑l=r+kn+rll−1⋯l−r−k+1alxl−r−k,λ.By evaluating (61) at x=0, we obtain(62)ak+r=12rk+r!ftkgtrpx|x=0=12rk+r!ftk|gtrpx,0≤k≤n.Several alternative expressions of (62) follow from (20) and (22), which are given by(63)ak=12rk!gtrftk−rpx|x=0=12rk!λk−rΔ˜rΔλk−rp0=12rk!λk−r∑j=0r∑l=0k−rrjk−rl−1k−r−lpj+lλ=12rk!λk−rΔ˜r−1Δλk−rpx+1+px|x=0,r≤k≤n+r.Another expression forak follows from (23):(64)ak=12rλk−rk!∑j=0rrjΔλk−rpx+j|x=0=k−r!2rλk−rk!∑j=0rrj1k−r!eλt−1k−rpx+j|x=0=k−r!2rk!∑j=0r∑l=k−rnrjλl−k+rl!S2l,k−rplj,r≤k≤n+r.Summarizing the results so far, from (62) and (64), we obtain the following theorem.(65)px=∑k=rn+rakGk,λrx,where ak’s are given by the following various expressions:Theorem 2.
Letpx∈ℂx,degpx=n. Let gt=et+1,ft=1/λeλt−1. Then, we have(66)ak=12rk!gtrftk−rpx|x=0=12rk!λk−rΔ˜rΔλk−rp0=12rk!λk−r∑j=0r∑l=0k−rrjk−rl−1k−r−lpj+lλ=12rk!λk−r∑j=0r−1r−1jΔλk−rpx+j+1+px+j|x=0=k−r!2rk!∑j=0r∑l=k−rnrjλl−k+rl!S2l,k−rplj,r≤k≤n+r.Remark 3.
Letpx∈ℂx,withdegpx=n. Write px=∑k=rn+rakGkrx. As λ tends to 0, ft⟶t. Thus, from Theorem 2, we have the following result.(67)ak=12rk!gtrpk−r0=12rk!∑j=0r−1r−1jpk−rj+1+pk−rj=12rk!∑j=0rrjpk−rj,r≤k≤n+r.


## 5. Examples

(a)
Here, we illustrate Theorem1, with px=Gnx. Let Gnx=∑k=1n+1akGk,λx.Fork with 1≤k≤n+1, we have(68)ak=12k!λk−1eλt−1k−1|Gnx+1+Gnx=12k!λk−1eλt−1k−1|2nxn−1=nk!λk−1Δλk−1xn−1|x=0=nk!λk−1Δλk−10n−1.Thus, from (68), we have(69)Gnx=∑k=1n+1nk!λk−1Δλk−10n−1Gk,λx.(b)
Here, we illustrate Theorem1, for px=∑k=1n−11/kn−kBkxBn−kx,n≥2. For this, we first recall from [8] that(70)px=∑k=0n−22nknHn−1−Hn−k−1Bn−k+n−1k∑l=k+1n−1Bl−kBn−ll−kn−l+n−1kBn−1−kn−1−kEkx+n−12En−2x+2nHn−1Enx,WhereHn=1+1/2+⋯+1/n is the harmonic number. Let px=∑k=1n+1akGk,λx. For k, with 1≤k≤n+1, we obtain(71)2k!λk−1ak=eλt−1k−1|px+1+px=∑m=0n−22nmnHn−1−Hn−m−1Bn−m+n−1m∑l=m+1n−1Bl−mBn−ll−mn−l+n−1mBn−1−mn−1−meλt−1k−1|Emx+1+Emx+n−12eλt−1k−1|En−2x+1+En−2x+2nHn−1eλt−1k−1|Enx+1+Enx.By proceeding as we did in (a), from (71), we see that(72)k!λk−1ak=∑m=0n−22nmnHn−1−Hn−m−1Bn−m+n−1m∑l=m+1n−1Bl−mBn−ll−mn−l+n−1mBn−1−mn−1−mΔλk−10m+n−12Δλk−10n−2+2nHn−1Δλk−10n.Thus, from (72) and for n≥2, we have(73)∑k=1n−11kn−kBkxBn−kx=∑k=1n+11k!λk−1∑m=0n−22nmnHn−1−Hn−m−1Bn−m+n−1m∑l=m+1n−1Bl−mBn−ll−mn−l+n−1mBn−1−mn−1−mΔλk−10m+n−12Δλk−10n−2+2nHn−1Δλk−10nGk,λx.(c)
In [4], it is shown that the following identity holds for n≥2:(74)∑k=1n−11kn−kEkxEn−kx=∑k=0n−2∑l=k+1n−1n−1kl−kn−lEl−kEn−lEkx+2nHn−1Enx,WhereHn=1+1/2+⋯+1/n is the harmonic number.Write∑k=1n−11/kn−kEkxEn−kx=∑k=1n+1akGk,λx.By proceeding similarly to (b), we obtain the following identity:(75)∑k=1n−11kn−kEkxEn−kx=∑k=1n+11k!λk−1∑m=0n−2∑l=m+1n−1n−1ml−mn−lEl−mEn−lΔλk−10m+2nHn−1Δλk−10nGk,λx.(d)
In [4], it is proved that the following identity is valid for n≥2:(76)∑k=1n−11kn−kBkxEn−kx=∑k=0n−21nnkHn−1−Hn−k−1Bn−k−12n−1kEn−k−1n−k−1Ekx+2nHn−1Enx.Again, by proceeding analogously to (b), we get the following identity:(77)∑k=1n−11kn−kBkxEn−kx=∑k=1n+11k!λk−1∑m=0n−21nnmHn−1−Hn−m−1Bn−m−12n−1mEn−m−1n−m−1Δλk−10m+2nHn−1Δλk−10nGk,λx.(e)
Nielsen [21, 22] expressed the product of a Bernoulli polynomial and an Euler polynomial in terms of Euler polynomials, namely, for nonnegative integers m and n,(78)BmxEnx=mEm+n−1x+∑r=0mmrBrEm+n−rx−m2∑s=0nnsEsEm+n−s−1x.Again, in a similar way to (b), we can show that(79)BmxEnx=∑k=1m+n+11k!λk−1mΔλk−10m+n−1+∑r=0mmrBrΔλk−10m+n−r−m2∑s=0nnsEsΔλk−10m+n−s−1Gk,λx.We also representBmxEnx in terms of the degenerate Genocchi polynomials of order r. Indeed, from Theorem 2, we obtain three expressions:(80)BmxEnx=∑k=rm+n+r12rk!λk−r∑j=0r∑l=0k−rrjk−rl−1k−r−l×Bmj+lλEnj+lλGk,λrx=∑k=rm+n+r12r−1k!λk−r∑j=0r−1r−1jmΔλk−rjm+n−1+∑r=0mmrBrΔλk−rjm+n−r−m2∑s=0nnsEsΔλk−rjm+n−s−1Gk,λrx=∑k=rm+n+rk−r!2rk!∑j=0r∑l=k−rm+n∑rjmanbλl−k+r×S2l,k−rBm−ajEn−bjGk,λrx,r≤k≤n+r,Where the innermost sum in (80) is over all integers a and b satisfying a+b=l,0≤a≤m, and 0≤b≤n.(f)
In (4.20) of [1], it is shown that the following identity holds for n≥s:(81)∑i1+⋯+is=ni1,…,is≥1Gi1x…Gisx=1n+s∑l=1ssl−2l−1∑j0+j1+⋯+js−l=n+1−lj0,j1,…,js−l≥1n+sj0Gj1⋯GjsGj0x.Then, proceeding similarly to (b), we can show that(82)∑i1+⋯+is=ni1,…,is≥1Gi1x…Gisx=∑k=1n−s+11k!λk−11n+s∑l=1ssl−2l−1×∑j0+j1+⋯+js−l=n+1−lj0,j1,…,js−l≥1j0n+sj0Gj1⋯GjsΔλk−10j0−1Gk,λx.In general, in terms of the degenerate Genocchi polynomials of orderr, we obtain the following:(83)∑i1+⋯+is=ni1,…,is≥1Gi1x…Gisx=∑k=rn−s+r12r−1k!λk−r∑m=0r−1r−1m1n+s∑l=1ssl−2l−1×∑j0+j1+⋯+js−l=n+1−lj0,j1,…,js−l≥1j0n+sj0Gj1⋯GjsΔλk−rmj0−1Gk,λrx.


## 6. Conclusion and Future Work

In this study, we were interested in representing any polynomial in terms of the degenerate Genocchi polynomials and of the higher-order degenerate Genocchi polynomials. We were able to derive formulas for such representations with the help of umbral calculus. By lettingλ tend to zero, we obtained the formulas for representations by the Genocchi polynomials and by the higher-order Genocchi polynomials. Furthermore, we illustrated the formulas with some examples.Even though the method adopted in this study is simple, they are very useful and powerful. Indeed, as mentioned in Section1, both a variant of Miki’s identity and Faber–Pandharipande–Zagier (FPZ) identity follow from the identity in (2) that can be derived from the formulas in (1), involving only derivatives and integrals of the given polynomial, while all the other known proofs are quite involved.It is one of our future research projects to continue to find formulas for representing polynomials in terms of some specific special polynomials and to apply those to discovering some interesting identities.



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