Discrete Mathematics Letters Discrete Math. Lett. 7 (2021) 44–51 www.dmlett.com DOI: 10.47443/dml.2021.0053 Research Article Hybrid convolutions on Pell and Lucas polynomials Dongwei Guo1, Wenchang Chu2;∗ 1School of Mathematics and Statistics, Zhoukou Normal University, Henan, China 2Department of Mathematics and Physics, University of Salento, Lecce, Italy (Received: 25 May 2021. Received in revised form: 25 June 2021. Accepted: 5 July 2021. Published online: 7 July 2021.) c 2021 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract By means of the generating function approach, three classes of convolution sums between the numbers of Bernoulli, Genoc- chi, Euler and the polynomials of Pell and Lucas are evaluated in closed form. Several identities concerning Fibonacci and Lucas numbers are deduced as consequences. One of them gives a solution to the problem proposed recently by Frontczak [Advanced problem H-860, Fibonacci Quart. 58 (2020) 281]. Keywords: generating function; Pell polynomial; Lucas polynomial; Fibonacci number; Lucas number. 2020 Mathematics Subject Classification: 11B39, 05A19, 11B68. 1. Introduction and motivation There exist numerous summation formulae concerning Bernoulli and Euler numbers (cf. [1,4,9]) as well as Fibonacci and Lucas numbers (cf. [2,3, 11]). When Bernoulli and Euler numbers are replaced by the corresponding Bernoulli and Euler polynomials, the related finite convolution identities can be found in the recent papers by Frontczak [6] and Frontczak– Goy [8]. Denote by Bn and Ln the usual Bernoulli and Lucas numbers, respectively. Frontczak [7] proposed, in a recent issue of ‘Fibonacci Quarterly’, a problem demanding to prove that n+2 X n−k n Bn−k+2 k 2 Ln+2 − 2 5 2 f2 L − 2g = − 1; (1) k n − k + 2 k 5(n + 1)(n + 2) 0≤k≤n k≡2n where i ≡m j stands for “i is congruent to j modulo m” provided that m 2 N and i; j 2 Z. During the course of resolving this problem, we find that the related results can be generalized from Fibonacci and Lucas numbers to Pell and Lucas polynomials, that motivates the authors to write the present paper. To our knowledge, Pell and Lucas polynomials were introduced by Horadam and Mahon [10], and can be defined equiv- alently by the recurrence relations Pn(x) = 2xPn−1(x) + Pn−2(x), Qn(x) = 2xQn−1(x) + Qn−2(x); with different initial conditions P0(x) = 0 and P1(x) = 1; Q0(x) = 2 and Q1(x) = 2x. They reduce to the following four well–known numbers: 1 • Fibonacci number Fn = Pn( 2 ): Fn = Fn−1 + Fn−2 with F0 = 0 and F1 = 1. 1 • Lucas number Ln = Qn( 2 ): Ln = Ln−1 + Ln−2 with L0 = 2 and L1 = 1. • Pell number Pn = Pn(1): Pn = 2Pn−1 + Pn−2 with P0 = 0 and P1 = 1. • Pell–Lucas number Qn = Qn(1): Qn = 2Qn−1 + Qn−2 with Q0 = 2 and Q1 = 2. n n Both polynomials admit the explicit formulae of Binet forms P (x) = α −β and Q (x) = αn + βn, where for brevity, we p pn α−β n employ the following two symbols α := x+ x2 + 1 and β := x− x2 + 1. Consequently, we have the exponential generating P1 zk ezα−ezβ P1 zk zα zβ functions k=0 Pk(x) k! = α−β and k=0 Qk(x) k! = e + e : Furthermore, for a natural number λ, it is routine to check the following relations: p L F −1 p n−1 F P 2λ−1 = 2λn−n ; P L = −1 2λn : (2) n 2 F n 2 2λ F 2λ−1 p 2λ L −1 p n Q 2λ−1 = L ; Q L = −1 L : (3) n 2 2λn−n n 2 2λ 2λn ∗Corresponding author ([email protected]). D. Guo and W. Chu / Discrete Math. Lett. 7 (2021) 44–51 45 8 p 8 F p p F2λn p 2λn−n > 5; n ≡2 0; > 5; n ≡2 0; 5 < L2λ −5 p n−1 < L2λ−1 Pn F2λ = Pn F2λ−1 = −1 (4) 2 2 L > L2λn > 2λn−n : ; n ≡2 1; : ; n ≡2 1: L2λ L2λ−1 p 8 p 8 5 <L2λn; n ≡2 0; −5 p n <L2λn−n; n ≡2 0; Q F = Q F = −1 (5) n 2 2λ p n 2 2λ−1 p :F2λn 5; n ≡2 1; :F2λn−n 5; n ≡2 1: They will frequently be utilized as ‘bridges’, throughout the paper, to pass from summation formulae about Pell and Lucas polynomials to identities involving Fibonacci and Lucas numbers. By means of the generating function approach, we shall prove, in the next section, three main theorems that evaluate in closed form the convolution sums between Bernoulli numbers, and Pell and Lucas polynomials, including Frontczak’s identity (1) as a particular case. Then in Section 3, three summation formulae will be established about Genocchi numbers and Lucas polynomials. Finally, the paper will end up with section 4, where two interesting identities invoving Euler numbers and Lucas polynomials will be presented. Throughout the paper, the coefficient of zm in the formal power series φ(x) will be denoted by [zm]φ(x), and χ will stand for the logical function defined by χ(true) = 1 and χ(false) = 0. 2. Convolutions with Bernoulli numbers x P1 xn Recall that the Bernoulli numbers (cf. [5, x1.14]) have the generating function ex−1 = n=0 Bn n! . Their convolutions with Pell and Lucas polynomials Pk(x) and Qk(x) will be examined. (α−β)z P1 k zk ezα−ezβ P1 zk x2.1. Rewriting the generating function e(α−β)z −1 = k=0(α − β) Bk k! and multiplying it with α−β = k=0 Pk(x) k! , we can express the following binomial convolution as m X m (α − β)z ezα − ezβ (α − β)m−k B P (x) = m![zm] × = m![zm−1]ezβ: k m−k k e(α−β)z − 1 α − β k=0 This consequently leads us to the following identity. m X m Theorem 2.1 (m 2 ). (α − β)m−k B P (x) = mβm−1: N k m−k k k=0 Because B2k+1 = 0 except for B1 = −1=2, the last identity can equivalently be stated as X m m (α − β)m−k B P (x) = Q (x): (6) k m−k k 2 m−1 0≤k≤m k≡2m Proposition 2.1 (m; λ 2 N). X p m−km m (a) F 5 B F = F L : λ k m−k kλ 2 λ mλ−λ 0≤k≤m k≡2m X m m (b) Lm−k B F = L F : m ≡ 0 λ k m−k kλ 2 λ mλ−λ 2 0≤k≤m k≡20 X m m (c) Lm−k B L = L L : m ≡ 1 λ k m−k kλ 2 λ mλ−λ 2 0≤k≤m k≡21 Proof. The first identity (a) has been obtained by Frontczak and Goy [8], which can be deduced from (6) by letting p L −1 x = λ for odd λ and x = L for even λ (7) 2 2 λ and subsequently applying (2) and (3) for simplifications. Similarly, for the two remaining identities (b) and (c), we can derive them from (6) by taking p p 5 −5 x = F for even λ and x = F for odd λ (8) 2 λ 2 λ and then making use of (4) and (5). When λ = 1, the above identity (a) reduces to the first formula below, which is due to Frontczak [6]. 45 D. Guo and W. Chu / Discrete Math. Lett. 7 (2021) 44–51 46 Corollary 2.1 (m 2 N). X m−k m m (a) 5 2 B F = L : k m−k k 2 m−1 0≤k≤m k≡2m X m−k m m (b) 8 2 B P = Q : k m−k k 2 m−1 0≤k≤m k≡2m X m 3m (c) 3m−k B F = F : m ≡ 0 k m−k 2k 2 2m−2 2 0≤k≤m k≡20 X m 3m (d) 3m−k B L = L : m ≡ 1 k m−k 2k 2 2m−2 2 0≤k≤m k≡21 Proof. The first identity (a) can also be derived by letting x = 1=2 in Theorem 2.1. Instead, letting x = 1 in the same theorem gives the second identity (b). The two remaining identities (c) and (d) are the λ = 2 cases of (b) and (c) in Proposition 2.1. (α−β)z (α−β)z P1 2k z2k x2.2. The generating function of Bernoulli numbers can be rewritten as 2 + e(α−β)z −1 = k=0(α − β) B2k (2k)! . 2 zα zβ P1 n k ko zk m Multiplying it with e − e = k=0 2 Qk(x) − 2(2x) k! and then extracting the coefficient of x , we find the following convolution formula. X m n o Theorem 2.2 (m 2 ). (α − β)m−k B 2kQ (x) − 2(2x)k = 2m−2m(α − β)2P (x): N k m−k k m−1 0≤k≤m k≡2m Analogous to the derivation for Proposition 2.1, by specifying in the above theorem by (7) and (8), and then appealing to the transformations (2–5), we can prove the following three summation formulae. Proposition 2.2 (m; λ 2 N). X p m−km n o (a) F 5 B 2kL − 2Lk = 2m−2(5m)F F : λ k m−k kλ λ λ mλ−λ 0≤k≤m k≡2m X m n p o (b) Lm−k B 2kL − 2(F 5)k = 2m−2mL L : m ≡ 0 λ k m−k kλ λ λ mλ−λ 2 0≤k≤m k≡20 X m n p p o p (c) Lm−k B 2kF 5 − 2(F 5)k = 2m−2mL F 5: m ≡ 1 λ k m−k kλ λ λ mλ−λ 2 0≤k≤m k≡21 The above identities contain further the following interesting particular cases.