GENERALIZED BERNOULLI POLYNOMIALS REVISITED AND SOME OTHER APPELL SEQUENCES

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1 Georgian Mathematical Journal Volume 1 5), Number 4, GENERALIZED BERNOULLI POLYNOMIALS REVISITED AND SOME OTHER APPELL SEQUENCES PASCAL MARONI AND MANOUBI MEJRI Abstract. We study the problem posed by Nörlund in terms of dual sequences. We determine the functional equation fulfilled by the canonical form of any generalized Bernoulli sequence. Surprisingly these canonical forms are positive definite. Some results are given for an Euler sequence. Mathematics Subject Classification: 4C5, 33C45. Key words and phrases: Orthogonal polynomials, dual sequences, Bernoulli and Euler polynomials. Introduction The Appell polynomials and their generalization were studied by many authors [7] [1], [18] [], []. At the present time there are papers dealing with Appell polynomials related to quadrature rules [], [3], some generalizations of Appell polynomials [4], [6] and Bernoulli polynomials [5]. The present paper deals with some Appell in particular Bernoulli and Euler) sequences in terms of dual sequences. In fact we study the following problem: Let {C n } n be an Appell sequence, determine all sequences {B n } n such that ) Bn+1 x) = B n x), n, n + 1 B n+1 D 1 n + 1 with D ω is the Hahn operator defined by ) x) = C n x), n, fx + ω) fx) D ω f)x) :=, ω, f P. ω When C n x) = B n x; k) generalized Bernoulli polynomial), k, with B n x; ) = x n, this problem was studied by Nörlund [18]. In this paper we give new results concerning the dual sequence of generalized Bernoulli sequences. In particular, it is shown that the canonical form u k) of a generalized Bernoulli sequence of order k 1 satisfies a homogeneous kth-order linear equation with polynomial coefficients and is a positive definite form. Similar results are obtained when C n x) = Ĥn monic Hermite polynomial). Analogous considerations are possible if, in 1), we take T 1 instead of D 1 where T ω is defined by T ω f)x) := fx + ω) + fx), f P. ISSN X / $8. / c Heldermann Verlag 1)

2 698 P. MARONI AND M. MEJRI In this case, for C n x) = E n x; k) generalized Euler polynomial), k with E n x; ) = x n, we show that the canonical form e k) of the generalized Euler sequence of order k is not regular for k. Section 1 contains the material of a preliminary and introductory character. In Section we study the Bernoulli sequence. Section 3 deals with the general case. I Section 4 we study the Nörlund case in terms of dual sequences and the case where C n x) = Ĥnx). In Section 5 we study the Euler sequence and we determine the canonical form e k) of a generalized Euler sequence of order k. 1. Preliminaries and the Notation Let P be the vector space of polynomials with coefficients in C, and P be its dual. We denote by u, f the action of u P on f P. In particular, we denote by u) n := u, x n, n, the moments of u. For any form u, any polynomial h, we let hu and u be the forms defined by the duality u, f := u, f, hu, f := u, hf, f P. Let {P n } n be a sequence of monic polynomials, deg P n = n, n monic polynomial sequence: MPS) and {u n } n be its dual sequence, u n P defined by u n, P m := δ n,m, n, m. The dual sequence {u [1] n } n of {P n [1] } n where P n [1] x) := n + 1) 1 P n+1x), n, is given by [14], [15], [17]. u [1] n ) = n + 1)u n+1, n. 1.1) Similarly, the dual sequence {ũ n } n of { P n } n with P n x) := a n P n ax + b), n, a is given by [14], [15], [16] where ũ n = a n h a 1 τ b )u n, n, 1.) τ b u, f := u, τ b f = u, fx b), u P, f P, b C, h a u, f := u, h a f = u, fax), u P, f P, a C {}. A form u is called regular if we can associate with it a polynomial sequence {P n } n such that u, P m P n = r n δ n,m, n, m ; r n, n. This sequence is orthogonal with respect to u. Since we have u = λu, λ, and deg P n = n, n, it is always possible to suppose that {P n } n is a MPS); {P n } n is unique and satisfies the recurrence relation { P x) = 1, P 1 x) = x β, 1.3) P n+ x) = x β n+1 )P n+1 γ n+1 P n x), γ n+1, n. Let φ and ψ be two monic polynomials, deg φ = t, deg ψ = p 1. Writing ψ = a p x p + we say the pair φ, ψ) is admissible; in the case p = t 1, a p is not a positive integer.

3 GENERALIZED BERNOULLI POLYNOMIALS 699 Definition 1.1 [14], [17]). A form u is called semi-classical when it is regular and satisfies the equation φu) + ψu =, 1.4) where the pair φ, ψ) is admissible. {P n } n is called semi-classical. The corresponding orthogonal sequence Remark 1. [14], [16], [17]). When deg φ and deg ψ = 1, u is called a classical form. Lemma 1.3. Consider the sequence { P n } n obtained by shifting P n, i.e., P n x) = a n P n ax + b), n, a. When u satisfies 1.4), then ũ = h a 1 τ b )u fulfils the following equation φũ + ψũ =, 1.5) where φx) = a degφ) φax + b), ψx) = a 1 deg φ ψax + b). Definition 1.4. A sequence {P n } n is called an Appell sequence when P [1] n x) = P n x), n. Lemma 1.5 [16]). Let {P n } n be a MPS and let {u n } n be its dual sequence, then {P n } n is an Appell sequence if and only if where u n) is the nth derivative of u. u n = 1)n u n), n, 1.6) n! Lemma 1.6 [7], [16], []). If {P n } n is an Appell sequence, then {P n } n is orthogonal if and only if P n is obtained by shifting Ĥn, where {Ĥn} n is a monic Hermite sequence. Finally, we introduce the Hahn s operator Dω f ) x) := fx + ω) fx), f P, ω. ω We have D ω = 1 ω τ ω I P ) where I P is the identity operator in P. The transposed t D ω of D ω is t D ω = 1 ω τ ω I P ) = D ω.. Connection between the Legendre Sequence and the Bernoulli Sequence Let {B n } n be the Bernoulli sequence and {u n } n be its dual sequence. Definition.1 [9], [18], [19]). The Bernoulli sequence {B n } n is defined by the generating function te tx e t 1 = n B n x) tn n!..1)

4 7 P. MARONI AND M. MEJRI Lemma.. The following relations define the Bernoulli sequence B [1] n x) = B n x), n,.) D 1 B n+1 n + 1 x) = xn, n..3) Proof. By virtue of the relation.1) we get relations.).3) and vice versa. Lemma.3. The canonical form u of Bernoulli sequence satisfies u ) = D 1 δ = δ δ 1..4) Proof. On account of relation.), by Lemma 1.6 we get u n = 1)n u n), n..5) n! Let {w n } n be the dual sequence of {x n } n. We have w n = 1)n δ n), n..6) n! Taking relation.3) and the definition of {w n } n into account, we obtain Therefore By virtue of Lemma 1.1 w n, D 1 B m+1 = m + 1)δ n,m, n, m. D 1 w n, B m =, m n +, D 1 w n, B n+1 = n + 1). n+1 D 1 w n = λ n, u, n. But D 1 w n, B µ = λ n,µ, µ n + 1 and λ n,µ =, µ n, λ n,n = n + 1), n. Hence D 1 w n = n + 1)u n+1, n..7) Setting n = in this equation, we get D 1 w = u 1. Now from.5) and.6) we can deduce.4). Proposition.4. The form u satisfies the following properties φu ) + ψu =,.8) with φx) = xx 1), ψx) = x 1). where L is the Legendre form. u, f = u = τ 1 1 h 1 ) L,.9) fx)dx, f P..1)

5 GENERALIZED BERNOULLI POLYNOMIALS 71 Proof. We need the following results [13], [16]. The Legendre form L satisfies x 1)L) xl =,.11) L, f = 1 From Lemma.3 we have u ) = δ δ 1. Hence 1 1 fx)dx, f P..1) xx 1)u ) =,.13) from which we get.8). Let ũ = h a 1 τ b u. By virtue of Lemma 1.3, equation.8) becomes φũ) + ψũ =, where φx) = x+a 1 b)x+a 1 b a 1 ), ψx) = x+a 1 b a 1 ). Choosing a = b = 1 we get ũ = L. Hence follows.9). Relations.9) and.1) imply.1). Remark.5. Denote ) x + 1 ˆB n x) = n B n, n..14) Then { ˆB n } n is the symmetric Bernoulli sequence satisfying We have te tx sinh t = n ˆB n x) tn n!. û = h τ 1/ u = L..15) Proposition.6 c.f. []). Let {L n } n be the monic orthogonal sequence of Legendre. The following formula holds: π Γn + ) L n+1 x) = n+1 n + 1)Γn + 1) n 1 + 1) +n Γn + + ) Γ + 1)Γ + ) Γn + ) ˆB +1 x), n..16) Proof. We need the formulas [13], [16] x 1)L n + 1) n+1x) = n + 1)xL n+1 x) n + 1 L nx), n,.17) x 1)L n+1x) + xl n+1x) n + 1)n + )L n+1 x) =, n..18) On account of.18) we can prove by induction the relation x 1)L m+) n+1 x) + m + 1)xL m+1) n+1 x) + m + m n + 1)n + ))L m) n+1x) =, n, m,.19)

6 7 P. MARONI AND M. MEJRI where P k) is the kth derivative of P P. We can write n+1 L n+1 x) = λ n, ˆB x), n,.) where λ n,m = û m, L n+1, m n + 1. On account of.15) we get Putting m = into.1), we get λ n,m = 1 L, Lm) m! n+1, m n ) λ n, = L, L n+1 =, n..) Taking.1) into account,.1) can be written as 1 { } λ n,m+1 = L m) m + 1)! n+11) L m) n+1 1), m n..3) Let ε = ±1, from.17) we get hence L n+1 ε) = L n+1 ε) L n ε) By virtue of relation.19) we have which implies L m+1) ε) L m) ε) L m) n+1ε) = = On account of.4) we obtain L m) n+1ε) = = 1 n + 1 ε n + 1, n, π Γn + ) ε) n+1 n + 1)Γn + 1 n..4) ), m n 1)m + n + ), m n + 1, εm + 1) 1 Γn + + m) ε) m Γm + 1) Γn + m) L n+1ε), m n + 1. π Γn + ) n + 1)ε) m+n+1 Γm + 1) Γn + + m) Γn + m)γn + 1 ), Finally, from relations.) and.5) we can deduce.16). 3. General Results Let {B n } n be a MPS and {u n } n be its dual sequence. Lemma 3.1. Let {v n } n be the dual sequence of { B D n+1 } 1 n+1 have i) The dual sequences {u n } n and {v n } n satisfy the relation m n ) n. Then we D 1 v n = n + 1)u n+1, n. 3.1)

7 GENERALIZED BERNOULLI POLYNOMIALS 73 ii) If {B n } n is an Appell sequence, then { B D n+1 } 1 n+1 we have the functional relation n is an Appell one and D 1 v = u ). 3.) Proof. i) The proof of 3.1) is similar to that of.7). B ii) Indeed, from the definition D n+ 1 x) = B n+x+1) B n+ x), n, differentiating its both sides we obtain n+ n+ ) B n+ D 1 x) = B n+x + 1) B n+x), n, n + n + but {B n } n is an Appell sequence and thus we have ) B n+ D 1 x) = B n+1x + 1) B n+1x), n, n + which implies ) [1] B n+ B n+1 D 1 x) = D 1 n + n + 1, n. Therefore { B D n+1 } 1 is an Appell sequence. n+1 n Since { B D n+1 } 1 and {B n+1 n n} n are two Appell sequences, by virtue of 1.6) we get v n = 1)n v n), u n = 1)n u n), n. n! n! On account of 3.1) we infer that ) D 1 v n) = u n+1), n. Setting n = in the latter relation we obtain 3.). Proposition 3.. If {C n } n is an Appell sequence, than there exists a unique sequence, then {B n } n satisfying B [1] n x) = B n x), n, 3.3) D 1 B n+1 n + 1 x) = C nx), n. 3.4) Proof. Let {B n } n and {Q n } n be two sequences satisfying 3.3) and 3.4). The sequence {R n } n defined by R n x) = B n x) Q n x), n, satisfies R x) =, 3.5) R n+1 n + 1 x) = R nx), n, 3.6) R n+1 x + 1) = R n+1 x), n. 3.7) We will prove by induction that R n x) =, n. Suppose R x) = for n. From 3.6) we can deduce that R n+1 x) = a n+1.

8 74 P. MARONI AND M. MEJRI So R n+ n+ x) = R n+1x) = a n+1, then On account of 3.7) we get R n+ x) = n + )a n+1 x + a n+. n + )a n+1 x + 1) + a n+ = n + )a n+1 x + a n+. Now we obtain a n+1 =, which implies R n+1 x) =. The uniqueness is proved. For the existence we introduce the sequence {B n } n defined by the recurrence formulas B x) = 1, 3.8) B n x) = n 1 B n 1 t)dt n B n 1 t)dt dx + C n ), n ) From relations 3.8) and 3.9) we can deduce that {B n } n satisfies 3.3). We will prove by induction that {B n } n satisfies 3.4). From 3.9) we get B 1 x) = x 1 +C 1), which implies D 1 B 1 x) = 1 = C x). Suppose By relation 3.9) we have D 1 B n+1 n + 1 x) = x D 1 B n x) = nc n 1 x). 3.1) B n t)dt B n t)dt = B n t)dt = x+1 1 D 1 B n ) t)dt + B n t)dt 1 B n t)dt. B n t)dt Using 3.1) we get D 1 B n t) = nc n 1 t), and on account of 3.9) we obtain 1 B n t)dt = C n ), whence B x n+1 D 1 n + 1 x) = n C n 1 t)dt + C n ), but {C n } n is an Appell sequence, so C nt) = nc n 1 t) and B x n+1 D 1 n + 1 x) = Hence follows the desired result. C nt)dt + C n ) = C n x).

9 GENERALIZED BERNOULLI POLYNOMIALS 75 Proposition 3.3. If {C n } n is an Appell sequence defined by the generating function Gx, t) = Ct)e tx = n C n x) tn n!, 3.11) then the sequence {B n } n defined by 3.3) 3.4) has the following generating function F x, t) = tct) e t 1 etx = B n x) tn n!. 3.1) n Proof. Let We have F x, t) = At)e tx = n B n x) tn n!. F x + 1, t) F x, t) = At)e t 1)e tx = t n On account of 3.4) this equation can be written as At)e tx e t 1) = t n C n x) tn n!. D 1 B n+1 n + 1 x)tn n!. By relation 3.11) we can deduce that At)e tx e t 1) = tct)e tx. Hence follows the desired result. 4. The Particular Cases A 1. Nörlund case. We denote by {B n.; 1)} n the Bernoulli sequence and assume that {u n 1)} n is its dual sequence. Definition 4.1 [18]). The polynomial sequence {B n x; )} n defined by B n [1] x; ) = B n x; ), n, 4.1) B n+1 x; ) D 1 = B n x; 1), n + 1 n, 4.) is called the generalized Bernoulli sequence of order. Lemma 4.. The polynomial sequence {B n x; )} n has the following generating function t ) e xt = B e t n x; ) tn 1 n!. 4.3) Proof. Taking relations 4.1) 4.) and.1) into account, Proposition 3.3 implies 4.3). n Denote by {u n )} n the dual sequence of {B n.; )} n.

10 76 P. MARONI AND M. MEJRI Proposition 4.3. Form u ) satisfies the properties φu ) ) ) + ψu ) + χu ) =, 4.4) with φx) = xx 1)x ), ψx) = 3x 6x + ), χx) = 6x 1), u ), f = 1 xfx)dx + 1 x)fx)dx, f P. 4.5) Proof. On account of 4.1) and 4.), relation 3.) of Lemma 3.1 implies u ) ) = D 1 u 1) = u 1) τ 1 u 1). 4.6) Differentiating this equation we obtain u ) ) = u 1) ) τ1 u 1) ). From relation.4) we have u 1) = δ δ 1, then we get u ) ) = δ δ1 + δ. This yields We can evidently deduce 4.4). Let xx 1)x ) u )) =. 4.7) u ), f = β α Ux)fx)dx, f P. Taking relations 4.6) and.1) into account we can deduce that β α U x)fx)dx [Ux)fx)] β α = 1 fx)dx 1 fx)dx, f P. Let us choose α =, β =, then this condition can be written as U x) W x))fx)dx = Ux)fx)], f P, with Setting we get W x) = { 1, < x < 1, 1, 1 < x <. U)f) U)f) =, f P, 4.8) U x) W x) =,

11 this yields GENERALIZED BERNOULLI POLYNOMIALS 77 W x) = { x + a, x 1, x + b, 1 x, and 4.8) gives a =, b =. Consequently U1 ) = U1 + ) and 4.5) holds. Corollary 4.4. Let û ) = τ 1 u ), û ) satisfy ) ˆφû ) + ˆψû ) ) + ˆχû ) = 4.9) with ˆφx) = xx 1), ˆψx) = 3x + 5), ˆχx) = 6x, û ), f = x )fx)dx, f P, 4.1) û ) ) = 1 n n + 1)n + 1), û ) ) =, n. 4.11) n+1 Proof. The proof is evident from Proposition 4.3 and the definition of û ). Remark The form û ) is symmetric and positive definite.. The study of the sequences {B n.; 1)} n, {B n.; )} n suggests the study of the Nörlund problem [18] in terms of dual sequences. Definition 4.6 [1], [11], [18], [19]). The polynomial sequence {B n.; k)} n, k defined by B n [1] x; k + 1) = B n x; k + 1), n, k, 4.1) B n+1 x; k + 1) D 1 = B n x; k), n + 1 n, k, 4.13) with B n x; ) = x n, n, is called the generalized Bernoulli sequence of order k. Lemma 4.7. The polynomials of the sequence {B n.; k)} n have the following generating function ) k t e xt = B e t n x; k) tn, k. 4.14) 1 n! n Proof. On account of relations 4.1), 4.13) and Proposition 3.3, we can deduce the desired result by induction with respect to k. Denote by {u n k)} n the dual sequence of {B n.; k)} n, k, so u ) = δ.

12 78 P. MARONI AND M. MEJRI Proposition 4.8. The forms {u k)} k 1 satisfy the relations u k) ) = D 1 u k 1), k ) φk+1 u k) ) k ) k) k k ) φ ) k+1 u k)) =, k 1, 4.16) with φ k+1 x) = k x ). =1 Proof. By virtue of relations 4.1), 4.13) and 3.) of Lemma 3.1 we obtain 4.15). Taking relation 4.15) into account, we can prove by induction with respect to m that u k) m) = D m 1u k m), 1 m k. For m = k, u k) k) = D k 1u ) = D k 1δ, and, obviously, D 1δ k = k k ) 1) δ [1], which implies k x )u k) k) =, therefore we can deduce 4.16). with Proposition 4.9. For any k, the form u k + 1) is positive definite. First we will prove a lemma. Lemma 4.1. There exist polynomials Ω k, k, such that k +1 u k + 1), f = Ω kx)fx)dx, f P, k, 4.17) Ω x) = 1, x Ω k x) = Ω ξ)dξ + Ω 1 ξ)dξ, x 1 x + 1, k 1, Ω 1 x) =, k 1, k Ω k k x) = Ω ξ)dξ, 1 k x k + 1. x 1 Proof. For k =, 4.17) is true. For k 1, we have from 4.15) therefore u k + 1)) = u k) τ 1 u k), u k + 1)), f = u k), f u k), fx + 1). 4.18)

13 GENERALIZED BERNOULLI POLYNOMIALS 79 Supposing we get k u k), f = V k x)fx)dx, fx)v k+1 x)] k+1 + Putting = we obtain 1 = with W k+1 x) = +1 k+1 Ω x)fx)dx + Ω Ω Ω V k+1x)fx)dx Ω x)fx)dx = Ω x)fx + 1)dx { Ω x) Ω 1 x 1) }fx)dx k+1 k Ω x 1)fx)dx. V k+1 k + 1)fk + 1) V k+1 )f) =, 4.19) V k+1x) W k+1 x) =, a.e., x), < x < 1, x) Ω 1 x 1), < x < + 1, 1 k 1, x 1), k < x < k + 1. Therefore Ω ξ)dξ + a k), x 1, V k+1 x) = {Ω ξ) Ω 1 ξ 1)}dξ + a k), k x + 1, 1 k 1, Ω ξ 1)dξ + a kk), k x k + 1.

14 71 P. MARONI AND M. MEJRI But 4.19) implies V k+1 ) =, V k+1 k + 1) =, consequently a k) = and k+1 k a k k) = Ω ξ 1)dξ = Ω ξ)dξ. k The continuity conditions V k+1 ) = V k+1 + ), 1 k 1 k ) yield +1 { Ω ξ) Ω 1 ξ 1) }dξ + a k) = a +1 k), k 1, 4.) with Ω 1 x) =. It can readily be seen on using 4.) that Therefore V k+1 x) = k x 1 Hence follows 4.18). a k) = Ω ξ)dξ + 1 x 1 Ω ξ)dξ, k x k + 1. Ω 1 ξ)dξ, k. Ω 1 ξ)dξ, x + 1, k 1, Proof of Proposition 4.9. It is sufficient to see that Ω k x) for k. This is evident by induction from 4.18), since Ω x) = 1. A. The Case where C n x) = Ĥnx), n. Let us recall that {Ĥn} n is the monic Hermite sequence and let {v n } n be its dual sequence. Let {B n } n the polynomial sequence defined by Let {u n } n be its dual sequence. B [1] n x) = B n x), n, 4.1) D 1 B n+1 n + 1 x) = Ĥnx), n. 4.) Lemma The sequence {B n } n has the following generating function te xt t 4 e t 1 = B n x) tn n!. 4.3) n

15 GENERALIZED BERNOULLI POLYNOMIALS 711 Proof. By virtue of 4.1) 4.) and on account of the generating function of {Ĥn} n [13] we obtain e xt t 4 = Ĥ n x) tn n!. n The desired result can be deduced from Proposition 3.3. Proposition 4.1. The form u satisfies u ) + x 1)u ) + 4x 4x 6)u ) 4x 1)u =. 4.4) 4n + )u ) n+1 4n + 1)u ) n nn + 1)u ) n 1 + nn 1)n )u ) n 3 =, n 3, 4.5) with u ) = 1, u ) 1 = 1, u ) = 5, u 6 ) 3 = 7. 8 Moreover, + u, f = Ux)fx)dx, f P, 4.6) where for any λ C, Ux) = 1 π x e t dt + λ St)dt, x R, 4.7) Sx) = x 1 {, x, exp x 1/4 ) sinx 1/4 ), x >, 4.8) in such a way that u is positive definite. Proof. We need the following relation satisfied by the Hermite form [16] v ) + xv =. 4.9) On account of 4.4) 4.5), relation 3.) in Lemma 3.1 implies Differentiating the latter relation we get u ) = D 1 v = v τ 1 v. 4.3) u ) = v ) τ 1 v ) = xv τ 1 xv ) from 4.9)) = xv + x 1)τ 1 v. By 4.3), τ 1 v = v u ), which implies u ) = v x 1)u ). This equation is equivalent to Differentiating both sides of 4.31) we obtain v = x 1)u ) 1 u ). 4.31) v ) = u ) x 1)u ) 1 u ) 3).

16 71 P. MARONI AND M. MEJRI Indeed, by the latter relation and 4.31), relation 4.9) implies u ) 3) + x 1)u ) + x x + 1)u ) =. Hence we obtain 4.4). The recurrence relation 4.5) is equivalent to 4.4) and is easily obtained. Next, suppose + u, f = Ux)fx)dx, f P. Taking into account the result [13], [16] v, f = 1 + e x fx)dx, f P, π and by virtue of 4.3), we get + U x)fx)dx = 1 + ) e x e x 1) fx)dx. π Consequently, for any λ C U x) = 1 π e x e x 1)) + λsx), where the function S defined by 4.8) is an integral representation of the null form [1]. Hence we obtain 4.7). Since + x x n ) St)dt dx = + x x n ) St)dt dx we have = n + 1) 1 lim x + xn+1 u, f = 1 + π x ) e t dt fx)dx > St)dt =, n, for any polynomial fx). Thus u is positive definite. In particular u, 1 = 1 + x ) e t dt dx = 1, π x 1 as it is easy to see. x 1

17 GENERALIZED BERNOULLI POLYNOMIALS Some Results on the Euler Sequence Let us introduce the operator ) fx + ω) + fx) T ω f x) :=, f P, ω C. 5.1) We have T ω = τ ω+i P, where I P is the identity operator in P. The transposed t T ω of T ω is t T ω = T ω. In this section we denote by {E n } n the monic Euler polynomial sequence and by {e n } n its dual sequence. Definition 5.1 [8], [11]). The Euler sequence {E n } n is defined by the generating function e xt e t + 1 = n E n x) tn n!. 5.) Lemma 5.. The following relations define the Euler sequence E n [1] x) = E n x), n, 5.3) T 1 E n )x) = x n, n. 5.4) Proof. By virtue of 5.) we obtain 5.3) 5.4) and vice versa. Lemma 5.3. We have e = T 1 δ = δ + δ ) Proof. The proof is analogous to that of Lemma.3, where D 1 is replaced by T 1. Remark 5.4. We denote by Ê n x) = n E n x + 1 Then {Ên} n is the symmetric Euler sequence satisfying We have with e tx cosh t = n ), n. 5.6) Ê n x) tn n! ê = h τ 1/ )e = δ 1 + δ ) Proposition 5.5. Let {P n } be a MPS, then we have n P n x) = λ n, Ê x), n, 5.8) λ n,m = 1 { Γm + 1) P m) n } 1) + P n m) 1), m n.

18 714 P. MARONI AND M. MEJRI Proof. Writing P n x) = n λ n, Ê x), n, we have λ n,m = ê m, P n, m n. Then λ n,m = 1 m! ê, P m) n, m n, by 5.7) we get the desired result. Corollary 5.6 c.f. []). Let {L n } n be a Legendre sequence, then the following equality holds: π Γn + 1) L n x) = n+1 Γn + 1) n 1 + 1) n+ Γn ) Γ + 1)Γn + 1 )Êx) n. 5.9) Proof. Taking Proposition 5.5 and relation.5) into account we get 5.9). Proposition 5.7. Let {C n } n be an Appell sequence and {σ n } n be its dual sequence, then there exists a unique sequence {P n } n satisfying P [1] n x) = P n x), n, 5.1) T 1 P n ) x) = C n x), n. 5.11) If u is the canonical form of {P n } n, then we have T 1 σ = u. 5.1) Proof. The proposition is proved by using 3.3) 3.4) and 3.), where D 1 is replaced by T 1. Definition 5.8 [1]). The polynomial sequence {E n.; k)} n, k defined by E n [1] x; k + 1) = E n x; k), n, k, 5.13) T 1 E n x; k + 1) = E n x; k), n, k, 5.14) with E n x; ) = x n, n, is called the generalized Euler sequence of order k. Let us denote by {e n k)} n the dual sequence of {E n.; k)} n, then e )=δ. Proposition 5.9. We have e k) = k k ) k δ. 5.15) Proof. Using 5.13) 5.14) and relation 5.1) in Proposition 5.7, we get e k) = T 1 e k 1), k 1. With the latter relation we can prove by induction with respect to m that e k) = T m 1e k m), 1 m k.

19 GENERALIZED BERNOULLI POLYNOMIALS 715 Then we obtain e k) = T k 1e ) = k τ 1 + I P ) k δ = k = k k ) δ. Hence follows the desired result. Remark 5.1. Relation 5.15) shows that e k) is not regular for any integer k. Acknowledgement We thank the referee for his valuable reviewing of our paper and for bringing certain references to our attention. References 1. F. Abdelkarim and P. Maroni, The D ω -classical orthogonal polynomials. Results Math ), No. 1-, G. Bretti and P. E. Ricci, Euler polynomials and the related quadrature rule. Georgian Math. J. 81), No. 3, G. Bretti, M. X. He, and P. E. Ricci, On quadrature rules associated with Appell polynomials. Int. J. Appl. Math. 11), No. 1, G. Bretti and P. E. Ricci, Multidimentional extensions of Bernoulli and Appell polynomials. Taiwanese J. Math. 84), No. 3, P. Natalini and A. Bernardini, A generalization of the Bernoulli polynomials. J. Appl. Math. 3, no. 3, G. Bretti, C. Cesarano, and P. E. Ricci, Laguerre-type exponentials and generalized Appell polynomials. Comput. Math. Appl. 484), No. 5-6, A. Angelesco, Sur les polynômes orthogonaux en rapport avec d autres polynômes. Buletinul Societétii Stiite din Cluj. 1191), L. Carlitz, Eulerian numbers and polynomials. Math. Mag /1959), L. Carlitz, Note on the integral of the product of several Bernoulli polynomials. J. London Math. Soc ), L. Carlitz, Note on Nörlund s polynomial B z) n. Proc. Amer. Math. Soc ), L. Carlitz, Multiplication formulas for generalized Bernoulli and Euler polynomials. Duke Math. J. 7196), L. Carlitz, Eulerian numbers and polynomials of higher order. Duke Math. J. 7196), T. S. Chihara, An introduction to orthogonal polynomials. Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York London Paris, P. Maroni, Le calcul des formes linéaires et les polynômes orthogonaux semi-classiques. Orthogonal polynomials and their applications Segovia, 1986), 79 9, Lecture Notes in Math., 139, Springer, Berlin, P. Maroni, L orthogonalité et les récurrences de polyn ˆmes d ordre supérieur à deux. Ann. Fac. Sci. Toulouse Math. 5) 11989), No. 1, P. Maroni, Fonctions eulériennes. Polynômes orthogonaux classiques. Techniques de l Ingénieur A ), P. Maroni, Semi-classical character and finite-type relations between polynomial sequences. Appl. Numer. Math ), No. 3,

20 716 P. MARONI AND M. MEJRI 18. N. E. Nörlund, Mémoire sur les polynômes de Bernoulli. Acta. Math. 4319), N. E. Nörlund, Vorlesungen über Differenzenrechnung. Chelsea. Publ. Co. New-York, J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J. Math ), No. 3, T. J. Stieltjes, Recherches sur les fractions continues [Suite et fin]. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys ), No. 1, A5 A47.. Vu Kim Tuan and Ngugen Thi Tinh, Legendre, Euler and Bernoulli polynomials. C. R. Acad. Bulgare Sci ), No. 5, Authors addresses: P. Maroni Laboratoire Jacques-Louis Lions Université Pierre Marie Curie Boite courrier Paris Cedex 5 France maroni@ann.jussieu.fr Received.1.4; revised ) M. Mejri Institut Supérieur des Sciences Appliquées et de Technologie Rue Omar Ibn El Khattab Gabès 67 Tunisia manoubi.mejri@issatgb.rnu.tn