# On Power Sums Involving Lucas Functions Sequences

Authors: Stefano Barbero
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We present some general formulas related to sum of powers, also with alternating sign, involving Lucas functions sequences. In particular, our formulas give a synthesis of various identities involving sum of powers of well-known polynomial sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Chebyshev polynomials. Finally, we point out some interesting divisibility properties between polynomials arising from our results.


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## Body

## 1. Introduction

The spirit of this paper is to develop the interesting ideas of Horadam presented in [1, 2], finding some general formulas for special sum of powers (also with alternating signs) related to a wide class of functions and in particular to some important classes of well-known polynomials. We will use a slight different notation with respect to Horadam.Definition 1.
One defines the Lucas functions sequenceWn(d)(x)n=0+∞ as the sequence of functions satisfying the recurrence relation (1)W0dx=Δd-21+-1d,W1dx=Δd-2ax+-1dbx,Wn+1dx=pxWndx+qxWn-1dxn≥1,where d is an integer and p(x),q(x) are polynomials such that (2)ax+bx=px,axbx=-qx,Δ=p2x+4qx=ax-bx.One supposes that all the functions in (1) and (2) are well defined for suitable values of the variable x.Clearly from the previous definition, we have(3)Wndx=Δd-2anx+-1dbnx,ax=px+Δ2,bx=px-Δ2,and, in particular, when d is a positive integer, the sequence (1) is a polynomial sequence. It is worth noting that very important families of polynomials satisfy the recurrence (1), such as Chebyshev, Fibonacci, Lucas, and Jacobsthal polynomials as Horadam pointed out in [1]. For example, considering px=2x,qx=-1, we retrieve for d=1 the sequence Un-1xn=1+∞ and for d=2 the sequence 2Tnxn=0+∞ involving Chebyshev polynomials of second and first kind, respectively.In the next section, we present our results on power sums of the kind∑k=1n-1δkW2kl+rdx/qlkxs with δ∈0,1, l, r integers, l≠0, and n, s positive integers. Finally we discuss interesting consequences of our formulas, related to some divisibility properties for polynomials obtained generalizing the so-called Melham convolutions. These convolutions were introduced in [3] for Fibonacci and Lucas numbers and studied in many recent papers, for example, [4–9], also with their extensions to Fibonacci, Lucas, and Chebyshev polynomials. For the sake of simplicity from now on we omit the dependence on x.


## 2. Power Sums

First of all, we give some straightforward calculation rules related to the functionsWn(d), which we will use along our proofs.Proposition 2.
For the functions defined by the recurrence (1), the following rules hold:(4)W-nd=-1d+nq-nWnd,(5)Wnd=Δd-eWne,e≡dmod⁡2,(6)Δ2WmhWnk=Wm+nh+k+-1k+nqnWm-nh+k,where d, e, h, k, m, n are integers.Proof.
These rules are direct consequences of relations (3) and easy calculations. Some of them are also listed in the paper of Horadam [2].We start with two useful lemmas which will enable us to obtain our general formulas.Lemma 3.
The following equality holds:(7)W2λ+rdqλs=Δsd-2+2-α2∑h=0ssh-1d+rqrhW2λ+rs-2hαqλs-2h,where d, λ, r are integers, s is positive integer, and α∈1,2 with α≡dsmod⁡2.Proof.
Using the binomial theorem, we have from relations (3) (8)W2λ+rds=Δsd-2∑h=0ssh-1ds-ha2λ+rhb2λ+rs-hor equivalently (9)W2λ+rds=Δsd-2∑h=0ssh-1dha2λ+rs-hb2λ+rhand if we sum (8) and (9) picking up (ab)(2λ+r)h=(-q)(2λ+r)h we have (10)W2λ+rdqλs=Δsd-22∑h=0ssh-1d+rhq-λs-2h+rha2λ+rs-2h+-1dsb2λ+rs-2hfrom which we easily find (7).Now, from identity (7), it is straightforward to obtain expressions for the powers W2λ+r(d)/qλs in the cases s odd or s even. We only need to apply some little calculations and rule (4) to change sign in the subscripts. After a suitable rearrangement of the summations involved, if s=2m+1 we get(11)Δ2md-2+d-α∑h=0m2m+1m-h-1d+rqrm-hW2λ+r2h+1αqλ2h+1;otherwise, if s=2m, we find(12)Δ2md-2∑h=1m2mm-h-1d+rqrm-hW22λ+rh2q2λh+2mm-1d+rqrΔ2d-2m.Lemma 4.
Let one considerδ∈{0,1}, α∈1,2, integers l, r with l≠0, and positive integers N, n. Then one has (13)∑k=1n-1δkW2lk+rNαqklN=Δ-2-1δnWl2n+1+rNα+β+-1nδ+1qnlNWl+rNα+βqnlNWlNβ,where β∈1,2 and β≡δ+lN+1mod⁡2.Proof.
From rule (6), we have(14)W2lk+rNαWlNδ+lN+1=Δ-2Wl2k+1+rNδ+α+lN+1+-1δ+1qlNWl2k-1+rNδ+α+lN+1and we observe that (15)WlNδ+lN+1∑k=1n-1δkW2lk+rNαqlkN=Δ-2∑k=1n-1δkWl2k+1+rNδ+α+lN+1qklN--1δk-1Wl2k-1+rNδ+α+lN+1qk-1lN=Δ-2-1δnWl2n+1+rNδ+α+lN+1qnlN-Wl+rNδ+α+lN+1since the sum telescopes. Thus we easily obtain (13) using rule (5).Thanks to rule (6), we point out that the right member of (13) could also be expressed in the following equivalent form:(16)-1δnΔ-2Δ2Wln+1+rNαWnlNβqnlNWlNβ+-1δn+1--1β+nlNWl+rNα+βWlNβ,which becomes(17)-1δnWln+1+rNαWnlN1qnlNWlN1when δ≡lNmod⁡2, since in this case β=1, or(18)-1δnWnlNαWnlNβqnlNWlNβwhen Wl+rNα+β=0, in other words, by definition (1), when r=-l and α+β=3.Now we have all what we need to find out our general formulas.Theorem 5.
For all integersd, l, r, m, n, δ, with l≠0, m≥0, n≥1, and δ∈0,1, the following identities hold:(19)∑k=1n-1δkW2lk+rdqkl2m+1=-1δnΔ2m+1d-2-α∑h=0m2m+1m-h-1d+rqrm-hW2h+1, where (20)W2h+1=Wl2n+1+r2h+1α+β+-1nδ+1qnl2h+1Wl+r2h+1α+βqnl2h+1Wl2h+1β,α,β∈1,2, α≡dmod⁡2, β≡δ+l+1mod⁡2, and (21)∑k=1n-1δkW2lk+rdqkl2m=-1δnΔ2md-2∑h=1m2mm-h-1d+rqrm-hW2h+2mm-1d+rqrΔ2d-2mUδ,n,where(22)W2h=W2l2n+1+rhδ+1+-1nδ+1q2nlhW2l+rhδ+1q2nlhW2lhδ+1,(23)Uδ,n=U0,n=nU1,n=0ifnisevenU1,n=-1ifnisodd.Proof.
The identities (19) and (21) are direct consequences of (11) and (12), combined with (13) of Lemma 4. In the case N=2h+1, we have β≡δ+l+1mod⁡2; on the other hand, if N=2h, we have β=δ+1. Clearly in the latter, we also have to take in account that α=2, so, applying rule (5), we obtain(24)-1δnW2h=W2l2n+1+rhδ+1+-1nδ+1q2nlhW2l+rhδ+1q2nlhW2lhδ+1.Finally, when we consider the sum related to the term 2mm(-1)d+rqrΔ2(d-2)m in (12), we observe that ∑k=1n‍(-1)δk=U(δ,n).In order to highlight in the next section some interesting divisibility properties between polynomials, we end with some results which made us able to rewrite equalities (19) and (21). For our purposes, we recall two important formulas due to Girard and Waring.Lemma 6 (Girard–Waring formulas).
For all nonnegative integersn and real numbers X, Y, the following identities hold: (25)Xn+Yn=∑k=0n/2-1knn-kn-kkX+Yn-2kXYk,(26)Xn+1-Yn+1X-Y=∑k=0n/2-1kn-kkX+Yn-2kXYk.Clearly these two formulas have a long history and should be widely known, so we only refer the reader to the original books of Girard [10] and Waring [11]. We also mention the paper of Gould [12], in which the reader will find some interesting remarks about the history and the use of these formulas and their generalizations. We observe that formula (26) also holds in the case X=Y; indeed (27)limY→XXn+1-Yn+1X-Y=Xnn+1,and Xnn+1 corresponds to the right member of (26) via the identity (28)n+1=∑k=0n/2-1kn-kk2n-2k.Applying these formulas, we can find alternative expressions for W(2A+B)N(D)+(-1)γqANWBN(D) in the cases N odd or N even, under the convention that ratios of the kind WBND/WBD when B=0 and D odd will take the value N, according to the result of the limit limB→0⁡aBN-bBN/aB-bB.Proposition 7.
Let one consider integersD, γ, A, B, C, and nonnegative integer h. Then the following equality holds:(29)W2A+B2h+1D+-1γqA2h+1WB2h+1D=W2A+BD+-1γqAWBD2h+1q2AhWB2h+1D+1WBD+1+W2A+BD+-1γqAWBD2RhW2A+BD+-1γqAWBD,-1γ+1qAWBD,where(30)Rhu,v=∑t=0h-1Cth,q,Δ∑j=12h-2t2h+1-2tj+1uj-1v2h-2t-j,(31)Cth,q,Δ=2h+12h+1-t2h+1-tt-1D+B+1q2A+BtΔ-D-22h-2t.Proof.
Thanks to the Girard–Waring formula (25) of Lemma 6, we can rewrite (32)W2A+B2h+1D=ΔD-2a2A+B2h+1+-1Db2A+B2h+1,qA2h+1WB2h+1D=ΔD-2qAaB2h+1+-1DqAbB2h+1obtaining (33)W2A+B2h+1D+-1γqA2h+1WB2h+1D=∑t=0hCth,q,ΔW2A+BD2h+1-2t+-1γqAWBD2h+1-2t=Chh,q,ΔW2A+BD+-1γqAWBD+∑t=0h-1Cth,q,ΔW2A+BD2h+1-2t+-1γqAWBD2h+1-2t.Now, since (34)W2A+BD2h+1-2t+-1γqAWBD2h+1-2t=∑j=12h+1-2t2h+1-2tjW2A+BD+-1γqAWBDj-1γ+1qAWBD2h+1-2t-j=W2A+BD+-1γqAWBD2h+1-2tqAWBD2h-2t+∑j=12h-2t2h+1-2tj+1W2A+BD+-1γqAWBDj+1-1γ+1qAWBD2h-2t-jthe last member of (33) is equal to (35)W2A+BD+-1γqAWBDChh,q,Δ+∑t=0h-1Cth,q,Δ2h+1-2tqAWBD2h-2t+W2A+BD+-1γqAWBD2RhW2A+BD+-1γqAWBD,-1γ+1qAWBD,where, by definition (31) of Ct(h,q,Δ) and by the Girard–Waring formula (26) of Lemma 6, we have with simple calculations (36)Chh,q,Δ+∑t=0h-1Cth,q,Δ2h+1-2tqAWBD2h-2t=2h+1∑t=0h-1t2h-tt-1D+Bq2A+BtΔ-D-22h-2tqAWBD2h-2t=2h+1q2AhWB2h+1D+1WBD+1.Proposition 8.
Let one consider integersD, γ, A, B, C, and nonnegative integer h. Then, if D is odd, one has (37)W22A+BhD+-1γq2AhW2BhD=q2Ah-1W22A+BD+-1γq2AW2BDW2BhDW2BD+W22A+BDW22A+B2-q2AW2B2ShW22A+B2-q2AW2B2,q2AW2B2;else, if D is even, one gets (38)W22A+BhD+-1γq2AhW2BhD=W22A+BD-q2AW2BDThW22A+B2-q2AW2B2,q2AW2B2+1+-1γq2AhW2BhD,where(39)Shu,v=∑t=0h-1/2Ktq∑j=0h-2-2th-1-2tj+1ujvh-2-2t-j,Ktq=h-1-tt-q4A+2Bt,Thu,v=∑t=0h/2Htq∑j=0h-1-2th-2tj+1ujvh-1-2t-j,Htq=hh-th-tt-q4A+2Bt.Proof.
In order to prove (37) if D is odd, we observe that (40)W22A+BhD=ΔD-2a22A+Bh-b22A+Bh,q2AhW2BhD=ΔD-2q2Aa2Bh-q2Ab2Bh.Thus, applying Girard–Waring formula (26), we have (41)W22A+BhD+-1γq2AhW2BhD=∑t=0h-1/2KtqW22A+B2h-1-2tW22A+BD+-1γq2AW2B2h-1-2tq2AW2BD.Since (42)W22A+B2h-1-2tW22A+BD+-1γq2AW2B2h-1-2tq2AW2BD=q2AW2B2h-1-2tW22A+BD+-1γq2AW2BD+W22A+BD∑j=1h-1-2th-1-2tjW22A+B2-q2AW2B2jq2AW2B2h-1-2t-j,the right member of (41) becomes (43)W22A+BD+-1γq2AW2BD∑t=0h-1/2Ktqq2AW2B2h-1-2t+W22A+BDW22A+B2-q2AW2B2ShW22A+B2-q2AW2B2,q2AW2B2,where, again by Girard–Waring formula (26), we obtain (44)∑t=0h-1/2Ktqq2AW2B2h-1-2t=q2Ah-1W2hBDW2BD.On the other hand, by means of Girard–Waring formula (25), when D is even, we have (45)W22A+BhD=ΔD-2a22A+Bh+b22A+Bh=ΔD-2∑t=0h/2HtqW22A+B2h-2tq2AhW2BhD=ΔD-2q2Aa2Bh+q2Ab2Bh=ΔD-2∑t=0h/2Htqq2AW2B2h-2t;thus (46)W22A+BhD+-1γq2AhW2BhD=W22A+BhD-q2AhW2BhD+1+-1γq2AhW2BhD=ΔD-2∑t=0h/2HtqW22A+B2h-2t-q2AW2B2h-2t+1+-1γq2AhW2BhD.Now substituting the identity (47)W22A+B2h-2t=∑j=0h-2th-2tjW22A+B2-q2AW2B2jq2AW2B2h-2t-jin the last member of (46), with some simple calculations we can finally find (38).As straightforward consequences of identities (29), (37), and (38) applied in equalities (19) and (21), we state the following two corollaries of Theorem 5.Corollary 9.
Under the same hypotheses of Theorem5, one has (48)∑k=1n-1δkW2lk+rdqkl2m+1=Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βqnlWl+rα+β+1Σ1,2m+1+Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β2Σ2,2m+1,where (49)Σ1,2m+1=-1δnΔ2m+1d-2-α∑h=0m2m+1m-h-1d+rqrm-h2h+1Wl+r2h+1α+β+1Wl2h+1β,Σ2,2m+1=-1δnΔ2m+1d-2-α∑h=1m2m+1m-h-1d+rqrm-hRhqnl2h+1Wl2h+1β,Rh=RhWl2n+1+rα+β+-1δn+1qnlWl+rα+β,-1δn+1qnlWl+rα+β;moreover we have (50)Δ2∣Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β.Proof.
We only need to consider the numerator ofW2h+1 in (20) of Theorem 5 and use equality (29) of Proposition 7 with A=nl, B=l+r, γ=δn+1, and D=α+β. Finally equality (50) is obvious when α+β=3,4, thanks to rule (5), and if α+β=2, that is, α=β=1, we have from rule (6)(51)Δ2Wln+1+r1Wln1=Wl2n+1+r2+-1nl+1qnlWl+r2=Wl2n+1+r2+-1δn+1qnlWl+r2since β=1 corresponds to δ≡lmod⁡2.Corollary 10.
Under the same hypotheses of Theorem5, if δ=0, for all positive integers n, one has (52)∑k=1nW2lk+rdqkl2m=W2l2n+1+r1-q2nlW2l+r1q2nlW2l+r1Σ1,2m+W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2Σ2,2m+n2mm-1d+rqrΔ2d-2mwith (53)Σ1,2m=Δ2md-2∑h=1m2mm-h-1d+rqrm-hW2l+rh1W2lh1,(54)Σ2,2m=Δ2md-2∑h=1m2mm-h-1d+rqrm-hShq2nlhW2lh1,Sh=ShW2l2n+1+r2-q2nlW2l+r2,q2nlW2l+r2.Moreover, if δ=1 and n is even, one gets (55)∑k=1n-1kW2lk+rdqkl2m=W2l2n+1+r2-q2nlW2l+r2Σ3,2mwith (56)Σ3,2m=Δ2md-2∑h=1m2mm-h-1d+rqrm-hThq2nlhW2lh2,Th=ThW2l2n+1+r2-q2nlW2l+r2,q2nlW2l+r2,Δ2∣W2l2n+1+r2-q2nlW2l+r2.Finally, if δ=1 and n is odd, one obtains(57)∑k=1n-1kW2lk+rdqkl2m=-Σ4,2m-W2l2n+1+r2-q2nlW2l+r2Σ3,2m-2mm-1d+rqrΔ2d-2mwith (58)Σ4,2m=2Δ2md-2∑h=1m2mm-h-1d+rqrm-hW2l+rh2W2lh2.Proof.
According to the parity ofδ+1, the numerator W2h of (22) in the identity (21) of Theorem 5 corresponds to (37) or (38) of Proposition 8 with D=δ+1, A=nl, B=l+r, and γ=nδ+1. In particular, taking into account the values of U defined in (23), we easily find identity (52) and the remaining identities (55) and (57), by means of equalities (37) and (38), respectively. Finally, from rule (6), we obviously have Δ2W2ln+1+2r1W2ln1=W2(l(2n+1)+r)(2)-q2nlW2(l+r)(2).


## 3. Applications and Divisibility Properties

From now on we suppose thatd≥1 and l>0, r≥-2l, in order to ensure that the functions W2kl+rd are polynomials for all positive integers k. Clearly assigning different suitable values to the parameters δ, d, l, r in our formulas, we can easily obtain many interesting relations for polynomial sequences satisfying (1). In particular, for well-known sequences as Fibonacci, Lucas, Pell, and Chebyshev polynomials, having q(x)=1 or q(x)=-1, our general formulas (19) and (21) resume many identities related to their power sums, also with alternating sign. Obviously, these identities enclose also the similar identities on Fibonacci, Lucas, and Pell numbers when we evaluate the corresponding polynomial sequences in x=1. For instance, we easily retrieve the results presented in [4, 13] or the ones in [5], respectively, related to the sums of alternating sign powers of Fibonacci and Lucas numbers, to odd powers of Chebyshev polynomials, and to sum of powers of Fibonacci and Lucas polynomials. On the other hand, for polynomials with q(x)≠±1, such as Jacobsthal Jnx and Jacobsthal–Lucas polynomials jnx, which have px=1, qx=2x and d=1 or d=2, we obtain identities concerning the sums of powers of the rational functions J2kl+rx/2xlk and j2kl+rx/2xlk. We leave to the reader the exploration of the numerous variants arising from general formulas (19) and (21) by conveniently changing the involved parameters.Now, we point out some divisibility properties for polynomials obtained by multiplying our sum of powers with other suitable polynomials, generalizing the results on the so-called Melham sums, or Melham convolutions, introduced in [3] and studied in [5–9]. At the same time, we will show that the divisibility properties conjectured in [3] and proved in [5, 9] are only special cases of analogous properties for polynomials defined by recurrence (1) and simple consequences of Corollary 9.


### 3.1. Odd Powers

From (19) and (48), we recognize that the polynomial(59)qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1is equal to(60)qnl2m+1∏j=0mWl2j+1β-1δnΔ2m+1d-2-α∑h=0m2m+1m-h-1d+rqrm-hW2h+1,or equivalently to the product(61)qnl2m+1∏j=0mWl2j+1βWl2n+1+rα+β+-1nδ+1qnlWl+rα+βF,where(62)F=Σ1,2m+1qnlWl+rα+β+1+Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΣ2,2m+1.We need to pay attention in order to state the correct divisibility property involving the polynomial Wl(2n+1)+r(α+β)+(-1)nδ+1qnlW(l+r)(α+β). Indeed, if d≥3, the factor Δ2m+1d-2-α in (60) is a polynomial because 2m+1d-2-α is a positive even integer (we recall that, by definition, α∈1,2 and α≡dmod⁡2), so we may state that(63)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,but when d=2 or d=1, the term Δ2m+1d-2-α is not a polynomial since it has a negative even exponent. Thus, in these two cases, we have to take into account the fact that the presence of this term is balanced by the even powers of Δ nested in the sum on the right member of (19). In particular, from Corollary 9, we know (64)Δ2∣Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βand it is easy to verify that Δ2 is the greatest positive even power of Δ which is a polynomial factor of Wl(2n+1)+r(α+β)+(-1)nδ+1qnlW(l+r)(α+β). Therefore, for d∈1,2, the correct statement is(65)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΔ2∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1in order to preserve in the factor F the correct amount of positive even powers of Δ balancing the term Δ2m+1d-2-α.Moreover we observe that whenr=0 and d odd, that is, α=1, the following binomial identity (66)∑h=0m2m+1m-h-1dm-h2h+1=-1md21+-1d2m+12mmensures that Σ1,2m+1=0 since Wl+r(2h+1)(α+β+1)/Wl(2h+1)(β)=Wl(2h+1)(β+2)/Wl(2h+1)(β)=Δ2. Thus, with a similar reasoning as before we have for ≥3, d odd,(67)Wl2n+1β+1+-1nδ+1qnlWlβ+12∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lkdqlk2m+1and for d=1(68)Wl2n+1β+1+-1nδ+1qnlWlβ+12Δ4∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk1qlk2m+1.We resume all these results in the following theorem.Theorem 11.
For integersd, l, r, m, t, n, δ, with d≥1, l>0, r≥-2l, m≥0, t,n≥1, and δ∈0,1, considering a polynomial sequence Widxi=0+∞ satisfying recurrence (1), we have the following divisibility relations:(69)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,∀d≥3(70)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΔ2∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,ifd∈1,2where α,β∈1,2 and α≡dmod⁡2, β≡δ+l+1mod⁡2.
Furthermore, whenr=0(71)Wl2n+1β+1+-1nδ+1qnlWlβ+12∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk2t+1qlk2m+1,Wl2n+1β+1+-1nδ+1qnlWlβ+12Δ4∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk1qlk2m+1.We observe that the previous theorem allows us to give a refinement for Corollary7 in [13]. Indeed, if we consider Chebyshev polynomials, that is, p(x)=2x, q(x)=-1, Δ2=4x2-1, and Wi1xi=0+∞=Ui-1xi=0+∞, Wi2xi=0+∞=2Tixi=0+∞, with the choices δ=1, l=1, we have β=1 and, thanks to rule (6),(72)2T2n+1x-2T1x2Δ4=Δ2UnxUn-1x2Δ4=Un2xUn-12x.So, omitting the trivial factor -1n=qn2m+1, relation (71) becomes(73)Un2xUn-12x∣∏j=1mU2jx∑k=1nU2k-1x2m+1or equivalently (since 2∣U2k-1x for all k≥1 and T1x=x) (74)T2n+1x-x2∣Δ4∏j=1mU2jx∑k=1nU2k-1x2m+1.Finally we point out that, unfortunately, the statement of Corollary 7 in [13] is wrong. It asserts (in our notations) that for integers m≥0, n≥1(75)∏j=0mU2jx∑k=1nU2k-1x2m+1=T2n+1x-xQ2mx,T2n+1,where Q2mx,y is an integer coefficients polynomial of two variables with degree 2m in y. But a simple calculation when m=0, n=1 shows that the left member of (75) is U0xU1x=2x, which obviously can not be divided by the factor T3x-x=4x3-4x in the right member. Equality (75) also fails in the case ≥1. For instance, when m=1 and n=1, we get(76)8x34x2-1=U0xU2xU13x=T3x-xQ2x,T3x=4xx2-1Q2x,4x3-xand clearly 4x(x2-1)∤8x34x2-1. The correct statement is the one given by (70), which reveals that when m, n are positive integers, we need Δ2 as extra factor in the left member of (75) in order to obtain a real equality.


### 3.2. Even Powers

When we deal with sums involving even powers, considering identity (21) and the results of Corollary 10, from (52), we easily find that for the difference(77)Dd=∑k=1nW2lk+rdqkl2m-W2l2n+1+r1-q2nlW2l+r1q2nlW2l+r1Σ1,2m+n2mm-1d+rqrΔ2d-2mthe following divisibility relation holds for all d≥2:(78)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj1Dd,and if d=1(79)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj1D1,since in this case we need to balance the factor Δ-2m in D1.Whenδ=1 and n is even, from (55) with d≥2, we have(80)W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+rdqkl2m.In the case d=1, we have to consider the presence of the factor Δ-2m in the term Σ3,2m, so we apply the same reasoning we have done in the previous subsection, in order to preserve the positive even powers of Δ nested in W2(l(2n+1)+r)(2)-q2nlW2(l+r)(2)Σ3,2m. Thus, since from Corollary 10 we know that Δ2∣W2(l(2n+1)+r)(2)-q2nlW2(l+r)(2), we find the divisibility relation for n even(81)W2l2n+1+r2-q2nlW2l+r2Δ2=W2ln+1+r1W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+r1qkl2m.In the last case δ=1, n odd, corresponding to equality (57), if we consider the sum(82)Sd=∑k=1n-1kW2lk+rdqkl2m+Σ4,2m+2mm-1d+rqrΔ2d-2m,we have(83)W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2Sd,W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj2S1,since we need to balance the factor Δ-2m in S1.Whenr=0, taking into account the following binomial identity,(84)∑h=1m2mm-h-1dm-h=121+-1d2m--1md2mm,we also obtain further simplifications of the previous relations.Indeed, whend is odd, from (84), (53), and (58), we get (85)Σ1,2m=-1m+1Δ2md-222mm,Σ4,2m=-1m+1Δ2md-22mm.In particular, the right members of equalities (55) and (57) differ only by a minus sign; thus we have for all positive integers n and t(86)W2l2n+12-q2nlW2l2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk2t+1qkl2m,W2l2n+12-q2nlW2l2Δ2=W2ln+11W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk1qkl2m.We resume all these results in the following theorem.Theorem 12.
For integersd, l, r, m, t, n, δ, with d≥1, l>0, r≥-2l, m≥0, t,n≥1, and δ∈0,1, considering a polynomial sequence Widxi=0+∞ satisfying recurrence (1), one has the following divisibility relations:(87)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj1Dd,W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+rdqkl2m,nodd,W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2Sd,neven,W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj1D1,W2l2n+1+r2-q2nlW2l+r2Δ2=W2ln+1+r1W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+r1qkl2m,noddW2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj2S1,neven,∀d≥2,where(88)Dd=∑k=1nW2lk+rdqkl2m-W2l2n+1+r1-q2nlW2l+r1q2nlW2l+r1Σ1,2m+n2mm-1d+rqrΔ2d-2m,Sd=∑k=1n-1kW2lk+rdqkl2m+Σ4,2m+2mm-1d+rqrΔ2d-2m.Furthermore, if r=0(89)W2l2n+12-q2nlW2l2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk2t+1qkl2m,W2l2n+12-q2nlW2l2Δ2=W2ln+11W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk1qkl2m.


## 3.1. Odd Powers

From (19) and (48), we recognize that the polynomial(59)qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1is equal to(60)qnl2m+1∏j=0mWl2j+1β-1δnΔ2m+1d-2-α∑h=0m2m+1m-h-1d+rqrm-hW2h+1,or equivalently to the product(61)qnl2m+1∏j=0mWl2j+1βWl2n+1+rα+β+-1nδ+1qnlWl+rα+βF,where(62)F=Σ1,2m+1qnlWl+rα+β+1+Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΣ2,2m+1.We need to pay attention in order to state the correct divisibility property involving the polynomial Wl(2n+1)+r(α+β)+(-1)nδ+1qnlW(l+r)(α+β). Indeed, if d≥3, the factor Δ2m+1d-2-α in (60) is a polynomial because 2m+1d-2-α is a positive even integer (we recall that, by definition, α∈1,2 and α≡dmod⁡2), so we may state that(63)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,but when d=2 or d=1, the term Δ2m+1d-2-α is not a polynomial since it has a negative even exponent. Thus, in these two cases, we have to take into account the fact that the presence of this term is balanced by the even powers of Δ nested in the sum on the right member of (19). In particular, from Corollary 9, we know (64)Δ2∣Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βand it is easy to verify that Δ2 is the greatest positive even power of Δ which is a polynomial factor of Wl(2n+1)+r(α+β)+(-1)nδ+1qnlW(l+r)(α+β). Therefore, for d∈1,2, the correct statement is(65)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΔ2∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1in order to preserve in the factor F the correct amount of positive even powers of Δ balancing the term Δ2m+1d-2-α.Moreover we observe that whenr=0 and d odd, that is, α=1, the following binomial identity (66)∑h=0m2m+1m-h-1dm-h2h+1=-1md21+-1d2m+12mmensures that Σ1,2m+1=0 since Wl+r(2h+1)(α+β+1)/Wl(2h+1)(β)=Wl(2h+1)(β+2)/Wl(2h+1)(β)=Δ2. Thus, with a similar reasoning as before we have for ≥3, d odd,(67)Wl2n+1β+1+-1nδ+1qnlWlβ+12∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lkdqlk2m+1and for d=1(68)Wl2n+1β+1+-1nδ+1qnlWlβ+12Δ4∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk1qlk2m+1.We resume all these results in the following theorem.Theorem 11.
For integersd, l, r, m, t, n, δ, with d≥1, l>0, r≥-2l, m≥0, t,n≥1, and δ∈0,1, considering a polynomial sequence Widxi=0+∞ satisfying recurrence (1), we have the following divisibility relations:(69)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+β∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,∀d≥3(70)Wl2n+1+rα+β+-1nδ+1qnlWl+rα+βΔ2∣qnl2m+1∏j=0mWl2j+1β∑k=1n-1δkW2lk+rdqlk2m+1,ifd∈1,2where α,β∈1,2 and α≡dmod⁡2, β≡δ+l+1mod⁡2.
Furthermore, whenr=0(71)Wl2n+1β+1+-1nδ+1qnlWlβ+12∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk2t+1qlk2m+1,Wl2n+1β+1+-1nδ+1qnlWlβ+12Δ4∣qnl2m+1∏j=1mWl2j+1β∑k=1n-1δkW2lk1qlk2m+1.We observe that the previous theorem allows us to give a refinement for Corollary7 in [13]. Indeed, if we consider Chebyshev polynomials, that is, p(x)=2x, q(x)=-1, Δ2=4x2-1, and Wi1xi=0+∞=Ui-1xi=0+∞, Wi2xi=0+∞=2Tixi=0+∞, with the choices δ=1, l=1, we have β=1 and, thanks to rule (6),(72)2T2n+1x-2T1x2Δ4=Δ2UnxUn-1x2Δ4=Un2xUn-12x.So, omitting the trivial factor -1n=qn2m+1, relation (71) becomes(73)Un2xUn-12x∣∏j=1mU2jx∑k=1nU2k-1x2m+1or equivalently (since 2∣U2k-1x for all k≥1 and T1x=x) (74)T2n+1x-x2∣Δ4∏j=1mU2jx∑k=1nU2k-1x2m+1.Finally we point out that, unfortunately, the statement of Corollary 7 in [13] is wrong. It asserts (in our notations) that for integers m≥0, n≥1(75)∏j=0mU2jx∑k=1nU2k-1x2m+1=T2n+1x-xQ2mx,T2n+1,where Q2mx,y is an integer coefficients polynomial of two variables with degree 2m in y. But a simple calculation when m=0, n=1 shows that the left member of (75) is U0xU1x=2x, which obviously can not be divided by the factor T3x-x=4x3-4x in the right member. Equality (75) also fails in the case ≥1. For instance, when m=1 and n=1, we get(76)8x34x2-1=U0xU2xU13x=T3x-xQ2x,T3x=4xx2-1Q2x,4x3-xand clearly 4x(x2-1)∤8x34x2-1. The correct statement is the one given by (70), which reveals that when m, n are positive integers, we need Δ2 as extra factor in the left member of (75) in order to obtain a real equality.


## 3.2. Even Powers

When we deal with sums involving even powers, considering identity (21) and the results of Corollary 10, from (52), we easily find that for the difference(77)Dd=∑k=1nW2lk+rdqkl2m-W2l2n+1+r1-q2nlW2l+r1q2nlW2l+r1Σ1,2m+n2mm-1d+rqrΔ2d-2mthe following divisibility relation holds for all d≥2:(78)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj1Dd,and if d=1(79)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj1D1,since in this case we need to balance the factor Δ-2m in D1.Whenδ=1 and n is even, from (55) with d≥2, we have(80)W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+rdqkl2m.In the case d=1, we have to consider the presence of the factor Δ-2m in the term Σ3,2m, so we apply the same reasoning we have done in the previous subsection, in order to preserve the positive even powers of Δ nested in W2(l(2n+1)+r)(2)-q2nlW2(l+r)(2)Σ3,2m. Thus, since from Corollary 10 we know that Δ2∣W2(l(2n+1)+r)(2)-q2nlW2(l+r)(2), we find the divisibility relation for n even(81)W2l2n+1+r2-q2nlW2l+r2Δ2=W2ln+1+r1W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+r1qkl2m.In the last case δ=1, n odd, corresponding to equality (57), if we consider the sum(82)Sd=∑k=1n-1kW2lk+rdqkl2m+Σ4,2m+2mm-1d+rqrΔ2d-2m,we have(83)W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2Sd,W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj2S1,since we need to balance the factor Δ-2m in S1.Whenr=0, taking into account the following binomial identity,(84)∑h=1m2mm-h-1dm-h=121+-1d2m--1md2mm,we also obtain further simplifications of the previous relations.Indeed, whend is odd, from (84), (53), and (58), we get (85)Σ1,2m=-1m+1Δ2md-222mm,Σ4,2m=-1m+1Δ2md-22mm.In particular, the right members of equalities (55) and (57) differ only by a minus sign; thus we have for all positive integers n and t(86)W2l2n+12-q2nlW2l2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk2t+1qkl2m,W2l2n+12-q2nlW2l2Δ2=W2ln+11W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk1qkl2m.We resume all these results in the following theorem.Theorem 12.
For integersd, l, r, m, t, n, δ, with d≥1, l>0, r≥-2l, m≥0, t,n≥1, and δ∈0,1, considering a polynomial sequence Widxi=0+∞ satisfying recurrence (1), one has the following divisibility relations:(87)W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj1Dd,W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+rdqkl2m,nodd,W2l2n+1+r2-q2nlW2l+r2∣q2mnl∏j=1mW2lj2Sd,neven,W2l2n+1+r1W2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj1D1,W2l2n+1+r2-q2nlW2l+r2Δ2=W2ln+1+r1W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk+r1qkl2m,noddW2l2n+1+r2-q2nlW2l+r2∣Δ2mq2mnl∏j=1mW2lj2S1,neven,∀d≥2,where(88)Dd=∑k=1nW2lk+rdqkl2m-W2l2n+1+r1-q2nlW2l+r1q2nlW2l+r1Σ1,2m+n2mm-1d+rqrΔ2d-2m,Sd=∑k=1n-1kW2lk+rdqkl2m+Σ4,2m+2mm-1d+rqrΔ2d-2m.Furthermore, if r=0(89)W2l2n+12-q2nlW2l2∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk2t+1qkl2m,W2l2n+12-q2nlW2l2Δ2=W2ln+11W2nl1∣q2mnl∏j=1mW2lj2∑k=1n-1kW2lk1qkl2m.


## 4. Conclusions

We have found formulas which give a clear, interesting, and comprehensive synthesis of many identities related to power sums of well-known families of polynomials and, in general, for a wide class of functions satisfying recurrence (1). We think that these formulas, beyond their intrinsic beauty, could be useful in applications. Indeed many of the polynomials satisfying recurrence (1), such as Chebyshev and Jacobsthal–Lucas polynomials, have several applications in physics and engineering, for example, concerning solutions of partial and ordinary differential equations or good approximations of functions using polynomials. In some calculations power sums may arise and our formulas give a simpler form to these sums (also with alternating sign), providing relatively easy summations avoiding powers of polynomials. Moreover divisibility relations between polynomials, which also generalize the ones considered by Melham for Fibonacci and Lucas numbers in [3], could give other additional information related to the knowledge of the factorization of the power sums and convolutions involved. Finally, thanks to Girard–Waring formulas, we have also proved the equivalents of the two conjectures presented in [3] for all the polynomial sequences defined by (1).



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*Source: 8124934-2017-10-25.xml*