POWER SUMS OF PELL AND PELL LUCAS POLYNOMIALS

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1 POWER SUMS OF PELL AND PELL LUCAS POLYNOMIALS WENCHANG CHU AND NADIA N LI Abstract Power sums of Pell and Pell Lucas polynomials are examined Twelve summation formulas are derived which contain the identities of Ozeki (2009 and Prodinger (2009 as special cases Furthermore, two general formulas are shown for odd power sums of the unified Pell and Pell Lucas polynomials 1 Introduction and Preliminaries Extending the classical Fibonacci and Lucas numbers, Horadam and Mahon [6] introduced Pell and Pell Lucas polynomials They are defined respectively by the recurrence relation with different initial conditions P n (x 2xP n 1 (xp n 2 (x, Q n (x 2xQ n 1 (xq n 2 (x; P 0 (x 0 and P 1 (x 1, Q 0 (x 2 and Q 1 (x 2x They will be shortened as P n P n (x and Q n Q n (x, which reduce to P n (1/2 F n and Q n (1/2 L n, the classical Fibonacci and Lucas numbers The corresponding generating functions read as P k (xy k y 1 2xy y 2 1 (α β(1 yα 1 (α β(1 yβ, k0 k0 Q k (xy k 2 2xy 1 2xy y yα 1 1 yβ ; which lead to the explicit formulas of Binet forms P n (x αn β n and Q n (x α n β n α β where for brevity, we employ the following two symbols α : x x 2 1 and β : x x 2 1 It is classically well known that the mth power sum of the first n natural numbers results in a polynomial of n Recently, attention has been turned to the similar problem of polynomial representation for power sums of Fibonacci and Lucas numbers, proposed by Melham (cf [3] and [9] Ozeki [5, Theorem 2] found the following interesting formula MAY

2 THE FIBONACCI QUARTERLY F 12m 2k F 1 12n ji ( 12m ( 1 ( 12m F12j 5 m L 12j ( ij 1 (12j( 5 i m 1 (1ijL 12j Prodinger [7] derived 8 formulas for the following power sums of Fibonacci and Lucas numbers F δ2m ε2k and L δ2m ε2k where ε,δ 0,1} Four of them for odd power sums have recently been unifiedby Chu and Li [2] via the inversion technique, where two additional parameters are introduced Reading carefully Prodinger s paper, we find that his approach can further be employed to investigate the power sums of Pell and Pell Lucas polynomials P δ2m ε2kλ and Q δ2m ε2kλ where λ N We shall establish 12 summation formulas, which express the power sums just displayed as polynomials of P n and Q n Six of them may be considered as polynomial extensions of Prodinger s results, while the other remaining six seem new even when considering their reduced cases of Fibonacci and Lucas numbers What is also remarkable lies in the fact that for each power sum, there exist two polynomial expressions both in P n and in Q n Throughout the paper, we shall utilize two fundamental lemmas on binomial sums, which contain, for y 1/x, n 2n 1 and y 1/x, n 2n, Prodinger s binomial relations [7, Equations 21 and 32] as particular cases Lemma 1 (Comtet [4, Section 49] x n y n ( 1 k n ( n k (xy k (xy n 2k n k k 0 k n/2 Lemma 2 (Carlitz [1, page 23] x n y n x y 0 k<n/2 ( 1 k( n k 1 (xy k k (xy n 2k 1 Both lemmas can also be found in Swamy [8, Equations 1 and 2] and considered as reduced relations of symmetric functions with two variables In fact, the second relation can be deduced from the first one as follows Rewrite the fraction as a finite sum x n y n n 1 x y x i y n i 1 0 i<n/2 x n 1 y n 1 1χ(1 n (xyi where χ is the logical function defined by χ(true 1 and χ(false 0 Applying Lemma 1, we get the double sum expression x n y n x y ( 1 j n 1 ( n j 1 (xy n j 1 j ij (xy n 2j 1 0 ij<n/2 140 VOLUME 49, NUMBER 2

3 POWER SUMS OF PELL AND PELL-LUCAS POLYNOMIALS The last double sum can be reformulated by letting ij k x n y n x y 0 k<n/2 k (xy k (xy n 2k 1 ( 1 jn 2k 2j 1 n 2k j 1 ( n 2k j 1 j This leads to the right member of the equation displayed in Lemma 2 because the binomial sum with respect to j results in the closed form k ( 1 jn 2k 2j 1 n 2k j 1 ( n 2k j 1 j ( 1 k( n k 1 k which is justified by the telescoping method and the binomial relation n 2k 2j 1 ( n 2k j 1 ( n 2k j 1 n 2k j 1 j j ( n 2kj 2 j 1 The rest of the paper will be organized as follows In the next section, six summation formulas will be derived for power sums of Pell polynomials Then six similar theorems will be shown for power sums of Pell Lucas polynomials in the third section Finally in the fourth section, the paper will end up with illustrating two further formulas for odd powers of G n (a,c,x, the unified polynomials extending those named by Pell and Pell Lucas with two extra parameters a and c 2 Power Sums for Pell Polynomials In this section, we will investigate power sums of Pell polynomials According to the binomial theorem, it is trivial to check that α P δ2m ε2kλ β ε2kλ } δ2m ε2kλ α β ( 1 ( δ 2m (α δmj β ε2kλ (α β δ2m j δ m Writing the summation domain as δ m j m} δ m j δ} 1 δ j m} and then inverting the summation order, by j δ j, for the first one, we get the equality P δ2m 2m ( 1 m(1ε ε2kλ χ(δ 0( m m (α β 2m ( 1 ( δ 2m j1 δ (α δmj β ε2kλ (α β δ2m ( 1 δ(α β δmj ε2kλ } (α β δ2m which can further be simplified into the equation ( P δ2m 2m ( 1 m(1ε m ε2kλ χ(δ 0 m (α β 2m ( δ 2m ( 1 ((1ε (α β δ2m j1 δ α (δ2j(ε2kλ ( 1 δ β (δ2j(ε2kλ} MAY

4 THE FIBONACCI QUARTERLY Summing over 1 k n for the expression displayed in the last line α (δ2j(ε2kλ ( 1 δ β (δ2j(ε2kλ} α (δ2jεα2(δ2jλ α 2(δ2j(n1λ 1 α 2(δ2jλ ( 1 δ β (δ2jεβ2(δ2jλ β 2(δ2j(n1λ 1 β 2(δ2jλ α (δ2j(ε λα2(δ2j(n1λ α 2(δ2jλ α (δ2jλ ( 1 λδ β (δ2jλ ( 1 δλδ β (δ2j(ε λβ2(δ2jλ β 2(δ2j(n1λ α (δ2jλ ( 1 λδ β (δ2jλ α(δ2j(ελ2nλ ( 1 δλδ β (δ2j(ελ2nλ α (δ2jλ ( 1 λδ β (δ2jλ α(δ2j(ελ ( 1 δλδ β (δ2j(ελ α (δ2jλ ( 1 λδ β (δ2jλ we derive the following formula for power sums of Pell polynomials Proposition 3 (Reduction formula P δ2m ε2kλ n( 1m(1ε ( 2m (α β 2m χ(δ 0 m j1 δ ( 1 ((1ε ( δ 2m (α β δ2m } α (δ2j(ελ ( 1 δλδ β (δ2j(ελ α(δ2j(ελ2nλ ( 1 δλδ β (δ2j(ελ2nλ α (δ2jλ ( 1 λδ β (δ2jλ α (δ2jλ ( 1 λδ β (δ2jλ Based on this proposition, three cases corresponding to δ 0, δ 1, λ 2 1 and δ 1, λ 2 0 will be examined, where λ 2 ε stands for the congruence relation λ ε (mod 2 21 δ 0 In this case, the equation displayed in Proposition 3 becomes ( Pε2kλ 2m 2m n( 1 m(1ε m (α β 2m ( 2m ( 1 ((1ε P2j(ελ (α β 2m P } 2j(ελ2nλ j1 According to Lemma 2, we get the equality (α β P 2j(ελ2nλ α2j(ελ2nλ β 2j(ελ2nλ Q ελ2nλ (α ελ2nλ β ελ2nλ j 1 ( 1 i(ελ( 2j i 1 i (α ελ2nλ β ελ2nλ 2j 1 ( 1 (j i(ελ( ij 1 (α 1 ελ2nλ β ελ2nλ 1 i1 142 VOLUME 49, NUMBER 2 (1

5 and derive the following double sum expression POWER SUMS OF PELL AND PELL-LUCAS POLYNOMIALS ( Pε2kλ 2m 2m n( 1 m(1ε m m 4 m (1x 2 m ( 2m ( 1 ((1ε 4 m (1x 2 m j1 P2j(ελ Q ελ2nλ i1 ( 1 (j i(ελ( ij 1 1 P 1 } ελ2nλ 4 1 i (1x 2 1 i This gives rise to the following summation theorem, whose particular case corresponding to ε λ 1 has been treated by Prodinger [7] Theorem 4 (Representation in Pell polynomials ( Pε2kλ 2m 2m n( 1 m(1ε m m 4 m (1x 2 m P ( 2j(ελ 2m ( 1 ((1ε P j1 2jλ 4 m (1x 2 m i1 P 1 ελ2nλ ji ( 2m Observe that we also have the equality P 2j(ελ2nλ P ελ2nλ j 1 ( ij 1 1 α2j(ελ2nλ β 2j(ελ2nλ α ελ2nλ β ελ2nλ ( 1 i(1ελ( 2j i 1 i ( 1 ((1ε(j i(ελ Q ελ2nλ 4 m i1 (1x 2 m i1 (α ελ2nλ β ελ2nλ 2j 1 ( 1 (j i(1ελ( ij 1 (α ελ2nλ β ελ2nλ 1 1 i1 and consequently an alternative expression ( Pε2kλ 2m 2m n( 1 m(1ε m m 4 m (1x 2 m ( 2m ( 1 ((1ε 4 m (1x 2 m j1 P2j(ελ P ελ2nλ i1 ( 1 (j i(1ελ( ij 1 1 } Q 1 ελ2nλ It can further be stated as another summation formula, whose special case ε 0 and λ 1 can be found in Prodinger [7] Theorem 5 (Representation in Pell Lucas polynomials ( Pε2kλ 2m 2m n( 1 m(1ε m m 4 m (1x 2 m P ( 2j(ελ 2m ( 1 ((1ε P j1 2jλ 4 m (1x 2 m i1 Q 1 ελ2nλ ji ( 2m ( ij 1 1 ( 1 (m i(1ε(j iλ P ελ2nλ 4 m (1x 2 m MAY

6 THE FIBONACCI QUARTERLY 22 δ 1 and λ 2 1 Under the replacement λ 1 2λ, the corresponding equation displayed in Proposition 3 reads as m P 12m ε2k4kλ ( 1 ((1ε ( 12m P (12j(1ε2λ2n4nλ (α β 2m P } (12j(1ε2λ (2 Q (12j(12λ Q (12j(12λ According to Lemma 1, we can rewrite the difference P (12j(1ε2λ2n4nλ α(12j(1ε2λ2n4nλ β (12j(1ε2λ2n4nλ α β ( 1 (j i(1ε12j ( ij (α β 1 P 1 ε(12n(12λ which leads to the following summation theorem Theorem 6 (Representation in Pell polynomials m P 2m1 ε2k4kλ P ( (12j(1ε2λ 12m ( 1 ((1ε Q (12j(12λ 4 m (1x 2 m P 1 1ε2λ2n4nλ 4 m i (1x 2 m i ji ( 12m 12j 1 ( ij ( 1 (m i(1ε Q (12j(12λ This theorem contains two known particular cases First, Ozeki [5] got the formula corresponding to ε λ 0 Subsequently, Prodinger [7] derived the case corresponding to ε 1 and λ 0 besides Ozeki s one Similarly, by invoking Lemma 2, we also have the equality P (12j(1ε2λ2n4nλ P 1ε2λ2n4nλ α(12j(1ε2λ2n4nλ β (12j(1ε2λ2n4nλ α 1ε2λ2n4nλ β 1ε2λ2n4nλ ( 1 (j iε( ij Q 1ε2λ2n4nλ which results consequently in another summation formula Theorem 7 (Representation in Pell Lucas polynomials m P 2m1 ε2k4kλ P ( (12j(1ε2λ 12m ( 1 ((1ε Q (12j(12λ 4 m (1x 2 m P 1ε2λ 2n4nλ 4 m (1x 2 m Q 1ε2λ2n4nλ ji ( 12m ( ij ( 1 (m iε Q (12j(12λ 23 δ 1 and λ 2 0 Under the replacement λ 2λ, the corresponding equation displayed in Proposition 3 can be stated as m P 2m1 ε4kλ ( 1 ((1ε ( 12m Q (12j(ε2λ4nλ (α β 22m Q } (12j(ε2λ (3 P 2(12jλ P 2(12jλ 144 VOLUME 49, NUMBER 2

7 In view of Lemma 2, we have the expression Q (12j(ε2λ4nλ Q ε2λ4nλ POWER SUMS OF PELL AND PELL-LUCAS POLYNOMIALS α(12j(ε2λ4nλ β (12j(ε2λ4nλ α ε2λ4nλ β ε2λ4nλ ( 1 (j iε( ij (α β Pε2λ4nλ which leads consequently to the summation theorem Theorem 8 (Representation in Pell polynomials m P 2m1 ε4kλ Q ( (12j(ε2λ 12m ( 1 ((1ε P 2(12jλ 4 1m (1x 2 1m P ε2λ4nλ ji ( 12m Alternatively, applying Lemma 1 gives the expression ( ij ( 1 (m iε Q ε2λ4nλ 4(1x 2 } 1m i P 2(12jλ Q (12j(ε2λ4nλ α (12j(ε2λ4nλ β (12j(ε2λ4nλ ( 1 (j i(1ε12j 1 from which we derive another summation formula ( ij Q 1 ε2λ4nλ Theorem 9 (Representation in Pell Lucas polynomials m P 2m1 ε4kλ Q ( (12j(ε2λ 12m ( 1 ((1ε P 2(12jλ 4 1m (1x 2 1m Q 1 ε2λ4nλ 4 m1 (1x 2 m1 ji ( 12m 12j 1 ( ij ( 1 (m i(1ε P 2(12jλ 3 Power Sums for Pell Lucas Polynomials Analogously, the power sums of Pell Lucas polynomials can be considered First, it is not difficult to check that Q δ2m ε2kλ (α ε2kλ β ε2kλ δ2m ( δ 2m (α δmj β ε2kλ j δ m Similarly as for P δ2m ε2kλ, the last sum can be reformulated as Q δ2m ε2kλ χ(δ 0 ( 2m m ( ( 1 mε δ 2m j1 δ α (δ2j(ε2kλ β (δ2j(ε2kλ} ( 1 (ε MAY

8 THE FIBONACCI QUARTERLY Summing over 1 k n for the expression displayed in the last line α (δ2j(ε2kλ β (δ2j(ε2kλ} α (δ2jεα2(δ2jλ α 2(δ2j(n1λ 1 α 2(δ2jλ β (δ2jεβ2(δ2jλ β 2(δ2j(n1λ 1 β 2(δ2jλ α(δ2j(ελ2nλ ( 1 λδ β (δ2j(ελ2nλ α (δ2jλ ( 1 λδ β (δ2jλ α(δ2j(ελ ( 1 λδ β (δ2j(ελ α (δ2jλ ( 1 λδ β (δ2jλ we find the following formula for power sums of Pell Lucas polynomials Proposition 10 (Reduction formula Q δ2m ε2kλ n( 1mε( 2m χ(δ 0 ( 1 m (ε( δ 2m j1 δ } α (δ2j(ελ ( 1 λδ β (δ2j(ελ α(δ2j(ελ2nλ ( 1 λδ β (δ2j(ελ2nλ α (δ2jλ ( 1 λδ β (δ2jλ α (δ2jλ ( 1 λδ β (δ2jλ Now we are ready to examine three cases of the last power sum 31 δ 0 In this case, the equation displayed in Proposition 10 reads as Q 2m ε2kλ n( 1mε( 2m ( 1 m (ε( 2m P 2j(ελ j1 By means of Lemma 2, the last fraction can be written as (α β P 2j(ελ2nλ α2j(ελ2nλ β 2j(ελ2nλ Q ελ2nλ α ελ2nλ β ελ2nλ j 1 ( 1 i(ελ( 2j i 1 i (α ελ2nλ β ελ2nλ 2j 1 ( 1 (j i(ελ( ij 1 (α 1 ελ2nλ β ελ2nλ 1 i1 which leads to the following double sum expression ( Q 2m 2m ( ε2kλ n( 1 mε 2m m P2j(ελ Q ελ2nλ i1 j1 ( 1 (ε P 2j(ελ2nλ ( 1 (j i(ελ ( } ij i (1x 2 1 i P 1 1 ελ2nλ } (4 Interchanging the summation order gives rise to the following theorem which reduces, for ε 1 and λ 1, to a formula due to Prodinger [7] 146 VOLUME 49, NUMBER 2

9 POWER SUMS OF PELL AND PELL-LUCAS POLYNOMIALS Theorem 11 (Representation in Pell polynomials Q 2m ε2kλ ( 1mε( 2m n ( 1 (ε( 2m m i1 P 1 2nλελ ji j1 ( 1 (m iε(j iλ 4 1 i (1x 2 1 i ( 2m Alternatively, we can also reformulate the fraction as P 2j(ελ2nλ j 1 P ελ2nλ α2j(ελ2nλ β 2j(ελ2nλ α ελ2nλ β ελ2nλ ( 1 i(1ελ( 2j i 1 i P2j(ελ ( ij 1 Q2nλελ 1 (α ελ2nλ β ελ2nλ 2j 1 ( 1 (j i(1ελ( ij 1 (α ελ2nλ β ελ2nλ 1 1 i1 from which we derive another double sum expression Q 2m ε2kλ n( 1mε( 2m ( 2m m P2j(ελ P ελ2nλ i1 j1 ( 1 (ε ( 1 (j i(1ελ( ij 1 1 } Q 1 ελ2nλ This is equivalent to the following summation formula, whose special case corresponding to ε 0 and λ 1 is due to Prodinger [7] Theorem 12 (Representation in Pell Lucas polynomials Q 2m ε2kλ n( 1mε( 2m ( 1 m (ε( 2m i1 Q 1 ελ2nλ ji j1 ( 1 (j i(1λ(m iε( 2m P2j(ελ ( ij 1 Pελ2nλ 1 32 δ 1 and λ 2 1 Replacing λ by 1 2λ, we may state the corresponding equation displayed in Proposition 10 as m Q 12m ε2k4kλ ( 1 (ε( 12m Q (12j(1ε2λ2n4nλ Q } (12j(1ε2λ (5 Q (12j(12λ Q (12j(12λ In view of Lemma 2, the following equality holds Q (12j(1ε2λ2n4nλ Q 1ε2λ2n4nλ α(12j(1ε2λ2n4nλ β (12j(1ε2λ2n4nλ α 1ε2λ2n4nλ β 1ε2λ2n4nλ ( 1 (j i(1ε( ij (α β P 1ε2λ2n4nλ from which we establish the following summation theorem MAY

10 THE FIBONACCI QUARTERLY Theorem 13 (Representation in Pell polynomials Q 2m1 ε2k4kλ Q 1ε2λ2n4nλ ( 1 (m iεj i 4 i (1x 2 i Q (12j(12λ P 1ε2λ2n4nλ Instead, applying Lemma 1, we get another expression ji ( 12m ( ij ( 1 (ε( 12m Q(12j(1ε2λ Q (12j(12λ Q (12j(1ε2λ2n4nλ α (12j(1ε2λ2n4nλ β (12j(1ε2λ2n4nλ ( 1 (j iε12j ( ij Q 1 1 1ε2λ2n4nλ which leads to an alternative summation formula Theorem 14 (Representation in Pell Lucas polynomials Q 2m1 m ε2k4kλ ( 1 (ε( 12m Q 1 1ε2λ2n4nλ ji Q(12j(1ε2λ ( 1 (m iε ( 12m Q (12j(12λ Q (12j(12λ 12j 1 Prodinger [7] found the cases corresponding to ε 0, 1 and λ 0 ( ij 33 δ 1 and λ 2 0 Under the replacement λ 2λ, the corresponding equation displayed in Proposition 10 becomes m Q 2m1 ε4kλ ( 1 (ε( 12m P (12j(ε2λ4nλ P } (12j(ε2λ (6 P 2(12jλ P 2(12jλ According to Lemma 1, we have the equality P (12j(ε2λ4nλ α(12j(ε2λ4nλ β (12j(ε2λ4nλ α β ( 1 (j iε12j ( ij (α β 1 P 1 ε2λ4nλ which yields the following summation theorem Theorem 15 (Representation in Pell polynomials Q 2m1 m ε4kλ ( 1 (ε( 12m P 1 ε2λ4nλ ji P(12j(ε2λ P 2(12jλ ( 1 (m iε 4 i (1x 2 i ( 12m P 2(12jλ 12j 1 ( ij 148 VOLUME 49, NUMBER 2

11 From Lemma 2, we also have the expression P (12j(ε2λ4nλ P ε2λ4nλ POWER SUMS OF PELL AND PELL-LUCAS POLYNOMIALS α(12j(ε2λ4nλ β (12j(ε2λ4nλ α ε2λ4nλ β ε2λ4nλ ( 1 (j i(1ε( ij Q ε2λ4nλ which results in another summation formula Theorem 16 (Representation in Pell Lucas polynomials Q 2m1 m ε4kλ ( 1 (ε( 12m Q ε2λ4nλ ji P(12j(ε2λ ( 1 (m iεj i( 12m P 2(12jλ ( ij Pε2λ4nλ P 2(12jλ 4 Further Formulas for Odd Power Sums The Pell and Pell Lucas polynomials can be unified by introducing two extra parameters a and c They are defined by the recurrence relation with the initial values being specified by G n (a,c,x 2xG n 1 (a,c,xg n 2 (a,c,x (7 G 0 (a,c,x a and G 1 (a,c,x c (8 By employing the usual series manipulation (cf [4, Section 113], it is not difficult to derive the generating function k0 and the explicit formula G k (a,c,xy k a(c 2axy 1 2xy y 2 c 2axaα (α β(1 yα c 2axaβ (α β(1 yβ G n (a,c,x uαn vβ n α β where u and v are two parameters determined by with n N 0 (9 u : c 2axaα and v : c 2axaβ (10 Then the Pell and Pell Lucas polynomials are the following particular cases P n (x G n (0,1,x and Q n (x G n (2,2x,x When x 1/2, the odd power sums of G n (a,c,1/2 have recently been investigated by the authors [2] Following exactly the same procedure presented in that paper, we can further establish the following summation theorems MAY

12 THE FIBONACCI QUARTERLY Theorem 17 (Unified representation formula G 2m1 G 1 2n1 2k (a,c,x (a,c,x c1 4 m i (1x 2 m i ki ( 12m m k 12k 1 ( ki (a 2 2acx c 2 m i G 2k1 (2,2x,x Theorem 18 (Unified representation formula m G 2m1 2k 1 (a,c,x G 1 2n (a,c,x a1 4 m i (1x 2 m i ki ( 12m m k 12k 1 ( ki (c 2 2acx a 2 m i G 2k1 (2,2x,x When a 0, c 1 and a 2, c 2x, these formulas agree with those displayed in Theorems 6 and 14 specified with ε 0, 1 and λ 0 However, we have failed to derive similar results for even power sums References [1] L Carlitz, A Fibonacci array, The Fibonacci Quarterly, 12 (1963, [2] W Chu and N N Li, Power sums of Fibonacci and Lucas numbers, Quaestiones Mathematicae, to appear [3] S Clary and P D Hemenway, On sums of cubes of Fibonacci numbers, Applications of Fibonacci Numbers, 5 (1993, [4] L Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and enlarged edition; D Reidel Publishing Co, Dordrecht, 1974 [5] K Ozeki, On Melham s sum, The Fibonacci Quarterly, 46/472 (2009, [6] A F Horadam and Bro J M Mahon, Pell and Pell Lucas polynomials, The Fibonacci Quarterly, 231 (1985, 7 20 [7] H Prodinger, On a sum of Melham and its variants, The Fibonacci Quarterly, 46/473 (2008/2009, [8] M N S Swamy, On certain identities involving Fibonacci and Lucas numbers, The Fibonacci Quarterly, 353 (1997, [9] M Wiemann and C Cooper, Divisibility of an F-L type convolution, Applications of Fibonacci Numbers, 9 (2004, MSC2010: 11B39, 05A15 School of Mathematical Sciences, Dalian University of Technology, Dalian , P R China address: chuwenchang@unisalentoit School of Mathematical Sciences, Dalian University of Technology, Dalian , P R China address: lina3718@163com Current address (W Chu: Dipartimento di Matematica, Università del Salento, Lecce-Arnesano P O Box 193, Lecce Italy 150 VOLUME 49, NUMBER 2