On Power Sums Involving Lucas Functions Sequences

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, andChebyshev polynomials. Finally, we point out some interesting divisibility properties between polynomials arising from our results.

In the next section, we present our results on power sums of the kind ∑  =1 (−1)  ( () 2+ ()/  ())  with  ∈ {0, 1}, ,  integers,  ̸ = 0, and ,  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][5][6][7][8][9], also with their extensions to Fibonacci, Lucas, and Chebyshev polynomials.For the sake of simplicity from now on we omit the dependence on .

Power Sums
First of all, we give some straightforward calculation rules related to the functions  ()   , which we will use along our proofs.
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.
Proof.From rule (6), we have and we observe that since 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: which becomes when  ≡  mod 2, since in this case  = 1, or when  (+) (+) = 0, in other words, by definition (1), when  = − and  +  = 3.
Now we have all what we need to find out our general formulas.
Lemma 6 (Girard-Waring formulas).For all nonnegative integers  and real numbers , , the following identities hold: 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  = ; indeed lim and   (+1) corresponds to the right member of (26) via the identity Applying these formulas, we can find alternative expressions for  () (2+) + (−1)     ()  in the cases  odd or  even, under the convention that ratios of the kind  ()   / ()  when  = 0 and  odd will take the value , according to the result of the limit lim →0 (  −   )/(  −   ).
Proposition 7. Let one consider integers , , , , , and nonnegative integer ℎ.Then the following equality holds: where Proof.Thanks to the Girard-Waring formula (25) of Lemma 6, we can rewrite obtaining Now, since the last member of (33) is equal to where, by definition (31) of   (ℎ, , Δ) and by the Girard-Waring formula (26) of Lemma 6, we have with simple calculations Proposition 8. Let one consider integers , , , , , and nonnegative integer ℎ.Then, if  is odd, one has else, if  is even, one gets where Proof.In order to prove (37) if  is odd, we observe that Thus, applying Girard-Waring formula (26), we have Since the right member of (41) becomes where, again by Girard-Waring formula (26), we obtain On the other hand, by means of Girard-Waring formula (25), when  is even, we have thus Mathematical Problems in Engineering Now substituting the identity in the last member of (46), with some simple calculations we can finally find (38).

Applications and Divisibility Properties
From now on we suppose that  ≥ 1 and  > 0,  ≥ −2, in order to ensure that the functions  () 2+ are polynomials for all positive integers .Clearly assigning different suitable values to the parameters , , ,  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 () = 1 or () = −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  = 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 () ̸ = ±1, such as Jacobsthal   () and Jacobsthal-Lucas polynomials   (), which have () = 1, () = 2 and  = 1 or  = 2, we obtain identities concerning the sums of powers of the rational functions  2+ ()/(2)  and  2+ ()/(2)  .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][6][7][8][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 is equal to or equivalently to the product where We need to pay attention in order to state the correct divisibility property involving the polynomial  60) is a polynomial because (2+1)(−2)− is a positive even integer (we recall that, by definition,  ∈ {1, 2} and  ≡  mod 2), so we may state that ( but when  = 2 or  = 1, the term Δ (2+1)(−2)− 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 and it is easy to verify that Δ 2 is the greatest positive even power of Δ which is a polynomial factor of We resume all these results in the following theorem.

2 Mathematical
Problems in Engineering