Applications of Riordan matrix functions to Bernoulli and Euler polynomials

Transcription

1 Illinois Wesleyan University From the SelectedWors of Tian-Xiao He 06 Applications of Riordan matrix functions to Bernoulli and Euler polynomials Tian-Xiao He Available at:

2 Applications of Riordan Matrix Functions to Bernoulli and Euler Polynomials Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL , USA February 9, 06 Abstract We define Riordan matrix functions associated with Riordan arrays and study their algebraic properties We also give their applications in the construction of new classes of Bernoulli and Euler polynomials and Bernoulli and Euler numbers, referred to as the duals and conjugate Bernoulli and Euler polynomials and dual and conjugate Bernoulli and Euler numbers, respectively AMS Subject Classification: 05A05, 05A5, 5B36, 5A09, 05A30, 05A0, 05A9 Key Words and Phrases: Riordan array, Riordan group, Pascal matrix, Bernoulli polynomials and numbers, Euler polynomials and numbers, conjugate Bernoulli and Euler polynomials Introduction Riordan arrays are infinite, lower triangular matrices defined by the generating function of their columns They form a group, called the Riordan group (see Shapiro, Getu, Woan and Woodson ] More formally, let us consider the set of formal power series (fps F Rt]; the order of f(t F, f(t 0 f t (f R, is the minimal number r N such that f r 0; F r is the set of formal power series of order r Let d(t F 0 and h(t F ; the pair (d(t, h(t defines the (proper Riordan array D (d n, n, N (d(t, h(t having d n, t n ]d(th(t ( or, in other words, having d(th(t as the generating function whose coefficients mae-up the entries of column From the fundamental theorem of Riordan arrays (see Shapiro 9, 0], it is immediate to show that the usual row-by-column product of two Riordan arrays is also a Riordan array: (d (t, h (t(d (t, h (t (d (td (h (t, h (h (t (

3 T X He The Riordan array I (, t acts as an identity for this product From the product, the inverse of (d(t, h(t is ( (d(t, h(t d(h (t, h(t (3 where h(t is the compositional inverse of h(t In this way, the set R of proper Riordan arrays is a group In spited by Pascal matrix discussed in Aceto and Trigiante ], Barry 3], Call and Velleman 4], and Edelman and Strang 9], we defined Riordan matrix functions as follows Definition Let d(t F 0 and h(t F, and let the pair of {d(t, h(t} define the (proper Riordan array D (d n, n, N (d(t, h(t, where d n, t n ]d(th(t Then the corresponding Riordan matrix function R(x (d(t, h(t(x, x R, is defined by R(x (d(t, h(t(x : d n, x n x n t n ]d(th(t (4 From the definition of Riordan matrix functions shown in Definition, we may have Theorem The collection of all Riordan matrix functions form a group, denoted by {R(x} and called Riordan function group When x, {R(} {R}, which is the Riordan group As an example, we consider the function P x] defined in 4]: { ( x n n P x] n 0 otherwise, where P x] maps R into R(x, ie, P : x (/( t, t/( t(x ( ( n (x because of (/( t, t/( t (( n Homomorphically, n 0 (5 P x + y] P x]p y] (6 for any x, y R, which is also shown in Zhang and Liu 5] In fact, the (i, j entry of the right-hand side of (6 can be written as i j while the (i, j entry of the left-hand side of (6 is ( i (x + y i j j ( i i j j 0 ( ( i x i y j, (7 j ( i j which is exactly (7, where we use the identity ( ( i j y x i j ( i j i j ( i j j ( ( i i j x i y j, j j Pascal matrix function P x] is studied in a numerous papers including, 3, 4, 6, 9, 3], etc We now present a Riordan array representation of R(x

4 Dual and Conjugate Bernoulli and Euler Polynomials 3 Theorem 3 Every element R(x (d(t, h(t(x in R(x can be written as a Riordan array (d(xt, h(xt/x Particularly, the identity of R(x is (, t(x, that has Riordan ( representation (, t The inverse of (d(t, h(t(x has the Riordan representation d( h(xt,, where h is the compositional inverse of h Hence, the collection of (d(xt, h(xt/x h(xt x forms a group which is isomorphic to R Proof The results can be derived directly from the definition of Riordan group The detailed proof is omitted here By means of Theorem 3 and the notation P x] (/( t, t/( t(x, we may give an alternative expression of (6 as ( ( ( xt, t xt yt, t yt (x + yt, t, (8 (x + yt which is obviously true due to the multiplication rule of Riordan arrays (8 also gives a much easier proof of (6 It should be mentioned that (d(xt, h(xt/x is not Sheffer matrix (see, for example, ] In other wards, (d(xt, h(xt/x, where vector (,, T, is not a Sheffer polynomial coefficient vector As noted in Shapiro 9], any element of Riordan group R with integer entries having finite order must have order or We call the element of order in R a Riordan involution In combinatorial situations, a Riordan array often has nonnegative integer entries and hence it can not have order Therefore, we define (see, for example, Cheon and Kim 7] and Cheon, Kim, and Shapiro 8] an element R R to have pseudo-order if RM has order, where M (, t Those R are called pseudo-riordan involutions or briefly pseudoinvolutions (see Cameron and Nwanta 5] and 9] We say two matrices A and B are similar associated with a Riordan type array R if there holds the relationship A RBR We now present an alternative expression of Riordan matrix functions From the multiplication rule of Riordan arrays, we have Proposition 4 Every element R(x (d(t, h(t(x in R(x is similar to (d(t, h(t R associated with (, xt, ie, (d(t, h(t(x (, xt(d(t, h(t(, xt (9 Furthermore, (d(t, h(t is a pseudo-involution in R if and only if (d(t, h(t (d( t, h( t (0 Corollary 5 (d(t, h(t is a pseudo-involution in R if and only if h(t h( t and d( td( h(t, ( where h(t is the compositional inverse of h(t The last equation is equivalent to d(td( h(t (

5 4 T X He Proof Comparing (0 and (d(t, h(t (/d( h(t, h(t, where h(t is the compositional inverse of h(t, we obtain the sufficient and necessary conditions ( Then, ( follows If R(x (d(t, h(t(x R(x has order for every real x, then, R(x is called an involution of R(x Several subgroups of R(x are defined below based on Theorems and 3 Appell subgroup of R(x is the set of Riordan arrays D (d(xt, h(xt/x for which h(xt xt; the associated subgroup of R(x, that is the set of Riordan arrays D (d(xt, h(xt/x for which d(xt ; The Bell subgroup of of R(x is the set of Riordan arrays D (d(xt, h(xt/x for which h(xt xtd(xt, ie, D (d(xt, td(xt; The checerboard subgroup of R(x is the set of Riordan arrays D (d(xt, h(xt/x for which d(xt is an even function and h(xt is an odd function in terms of t; The stochastic subgroup of R(x is the set of Riordan arrays D (d(xt, h(xt/x whose row weighted sums with weight (, x, x, T are x n ; The hitting-time subgroup of R(x is the set of Riordan arrays D (d(xt, h(xt/x for which d(xt t(d/dth(xt/h(xt, ie, D (xth (xt/h(xt, h(xt/x The derivative subgroup of R(x is the set of Riordan arrays D ((d/dth(xt/x, h(xt (h (xt, h(xt/x Similar to Theorems and of 4], we have the following sequence characterizations of Riordan matrix function R(x R(x Theorem 6 Let R(x : (d n, x n n, N0 be a lower triangular matrix function Then R(x is a Riordan matrix function if and only if there exist an A(x and a Z(x sequences of functions, A(x (a 0 (x, a (x,, and Z(x (z 0 (x, z (x,, such that and or equivalently, d n+,+ x n a j (xd n,+j x n j (3 j0 d n+,0 z j (xd n,j x n j, (4 j0 xt h(xta(xt and Z(xt d(xt d 0,0 (5 td(xt In next section, we will demonstrate some applications of Pascal array functions to construct the duals of Bernoulli and Euler numbers and polynomials In Section 3, we discuss the construction of conjugates of Bernoulli and Euler numbers and polynomials as well as their duals by using Riordan matrix functions

6 Dual and Conjugate Bernoulli and Euler Polynomials 5 Applications to Bernoulli and Euler polynomials Bernoulli polynomials B n (x and Euler polynomials E n (x for n 0,, are defined by (see, for example, Apostol ] and Olver, Lozier, Boisvert, and Clar 6] te xt e t n 0 B n (x tn n! ext and e t + E n (x tn n! (6 n 0 Bernoulli numbers B n and Euler numbers E n for n 0,, are defined by ( B n B n (0 and E n n E n (7 A large literature scatters widely in boos and journals on Bernoulli numbers B n and Bernoulli polynomials B n (x and Euler numbers E n and Euler polynomials E n (x They can be studied by means of the binomial expressions connecting them: B n (x E n (x ( n B x n, n 0, (8 0 0 ( n ( x n E, n 0, (9 where E E (/ The study brings consistent attention of researchers woring in combinatorics, number theory, applied analysis, etc We now give some applications of a typically useful Riordan array function, the Pascal matrix function P x] (/( xt, t/( xt (see 4], to the Bernoulli and Euler polynomials Since there exists P x]p x] P x]p x] P 0] I, (0 P x] P x] Thus, the well-nown formula about Bernoulli polynomials B n (x and Bernoulli numbers B n B n (0 shown in (8 implies the following inverse relationship between B(x (B 0 (x, B (x, T and B (B, B, T (see, for example, 3]: More precisely, we have B(x P x]b ( Theorem 3] Let B(x (B 0 (x, B (x, T and B (B 0, B, T, and let P x] be defined as above Then there exists a pair of inverse relationships And the latter can be presented as B(x P x]b B P x]b(x (

7 6 T X He B 0 B B n which implies ( 0 ( ( 0 ( x 0 0 ( n ( 0 ( x n n ( ( x n n n 0 B n 0 B 0 (x B (x B n (x, (3 ( n ( x n B (x (4 for n 0,, Similarly, Euler polynomials E n (x can be presented in terms of Euler numbers shown in (9 Denote E(x (E 0 (x, E (x, T and E (E 0, E, T By maing use of the Pascal matrix, we may write (9 as a matrix form where E(x P x ] D ] E P Dt] diag, t, t, ] x ] E (, (5 A numerous properties of Bernoulli and Euler polynomials are studied by using a unified approach in 3] based on the matrix representation In this section, we will use Pascal matrix function to study the duals of Bernoulli and Euler numbers Let {a n } n 0 be a sequence of complex numbers We refer to {a n} n 0 defined by a n 0 ( n ( a (6 as the dual sequence of {a n } n 0 (see, for example, Graham, Knuth, and Patashni 0] If a n a n, then {a n } n 0 is called the self-dual sequence (see Sun ] It is easy to see that Because a n 0 j0 ( n 0 ( j ( n ( a (7 ( +j δ n,j, where δ n,j is a Kronecer symbol, which is when j n and zero otherwise, the pair of {a n } n 0 and {a n} n 0 is a pair of inverse sequences Hence, (a n a n It is well nown that {( n B n } n 0 is a self-dual sequence (see Sun 3] We now study the dual sequences of {B n } n 0 and {E n } n 0, denoted by {B n} n 0 and {E n} n 0, respectively, and dual Bernoulli polynomials and dual Euler polynomials, denoted by {B n(x} n 0 and {E n(x} n 0, respectively Here,

8 Dual and Conjugate Bernoulli and Euler Polynomials 7 B n 0 ( n ( B and En 0 ( n ( E (8 Inspired by (8 and (9 we define the dual Bernoulli polynomials and the dual Euler polynomials below: Definition Denote B (x : (B0(x, B(x,, B : (B0, B,, E (x : (E0(x, E(x,, and E : (E0, E, Let Dt] : diag(, t, t, Then B (x P x]b and E (x P x ] ] D E, (9 which imply where E E (/ B n(x E n(x ( n ( ( n 0 0 B x n, n 0, and (30 x n E, n 0, (3 The relationships between Bernoulli polynomials, Euler polynomials and their duals can be presented below Theorem 3 Let B (x, B, E (x, E, and Dt] be defined before Then ( B (x D ]B( x and E (x D ]E x +, (3 which imply Furthermore, ( Bn(x ( n B n ( x and En(x ( n E n x + (33 which imply B D ]B( and E D ]E (0, (34 Bn ( n B n + n and En ( n E n (0 n E n ( (35 Proof From (9 we have B D ]P ]B and E D ]P ]E (36 Using (36 and (9, we obtain B (x P x]d ]P ]B D ]P x]p ]B D ]P x ]B (37

9 8 T X He Then the first formula of (3 follows from the rightmost expression and ( The second formulas of (9 and (36 can be applied to derive E (x P x ] ] D D ]P ]E P x ] ] ] P D ]D E D ]P x + ] ] ( D E D ]E x +, completing the proof of the second formula of (3 Thus, (33 follows immediately Substituting x 0 and x / into the first and the second formulas of (3, respectively, yield (34 Hence, and give (35 B n B n(0 ( n B n ( B n ( + n ( n B n + n E n n E n ( ( n E n (0 The first few elements of dual Bernoulli numbers and Euler numbers are {B n} n 0 {, 3, 3 6 } 9 53, 3,, 5, 30 4,, and {E n} n 0 {,, 0,, 0, 6, 0, }, respectively, where we notice that starting from the fourth term, the even-indexed dual Euler numbers are zero while the odd-indexed (classical Euler numbers are zero The first few terms of the dual Bernoulli polynomial sequence {B n} n 0 are B 0(x, B (x x + 3, B (x x + 3x + 3 6, B 3(x x x + 3 x + 3, B 4(x x 4 + 6x 3 + 3x + x , B5(x x x x3 + 30x + 9 x + 5, etc, 6

10 Dual and Conjugate Bernoulli and Euler Polynomials 9 while the first few terms of the dual Euler polynomial sequence {E n} n 0 are E 0(x, E (x x, E (x x 4, E 3(x x x, E 4(x x 4 3 x + 5 6, E5(x x 5 5 x3 + 5 x, etc 6 The generating function of dual Bernoulli polynomial sequence B n(x is n 0 Bn(x tn n! ( B n ( x tn n! n 0 te(x+t e t e(x+t, + t e t t where the rightmost expression will be used to define the conjugate polynomial sequence of {B n(x} n 0 in next section Similarly, the generating function of dual Euler polynomial sequence E n(x is n 0 En(x tn n! ( ( n E n x + t n n! n 0 e(x /t e t + e(x /t + e t, where the rightmost expression will be used to define the conjugate polynomial sequence of {E n(x} n 0 in next section We survey those results below and leave more discussion about conjugate polynomial sequences in Section 3 Theorem 4 Let B n(x and E n(x be dulas of Bernoulli and Euler polynomials defined before Then their generating functions are respectively e (x+t + t e t t e (x /t + e t n 0 n 0 B n(x tn n! (38 E n(x tn n! (39

11 0 T X He Remar 3 By using the same argument, the self dual sequence {( n B n } n 0 induces the corresponding dual polynomial sequence {B n (x + } n 0, which has the generating function te (x+t e t n 0 B n (x + tn n! 3 Conjugate polynomials and their duals Let (d(xt, h(t be a Riordan arrays in terms of real variable t Then we will define a pair of polynomial sequences, denoted by {F n (x} n 0 and { F n (x} n 0, respectively, where { F n (x} n 0 is called the conjugate polynomial sequence of {F n (x} n 0, which will be also defined as follows: Definition 3 Let (d(xt, h(t be a Riordan arrays in terms of real variable t Then the polynomial sequence {F n (x} n 0 and its conjugate polynomial sequence { F n (x} n 0 are defined by their exponential generating functions (see, for example, Wilf 4] as d(xt h(t F n (x tn n! and d(xt + h(t F n (x tn n! (40 n 0 n 0 Here, d(xt/( h(t and d(xt/( + h(t are called the sum (generating function and alternating sum (generating function of the Riordan array (d(xt, h(t, which are defined in Shapiro and the author 5] Particularly, if d(xt e xt, then we have Proposition 3 Let (d(xt, h(t be a Riordan arrays in terms of variable t with d(xt e xt, and let {F n (x} n 0 and { F n (x} n 0 be the polynomial sequence and its conjugate defined in Definition 3 Then for a real α there hold and F (x P x α]f (α and F (x P x α] F (α (4 F (α P x + α]f (x and F (α P x + α] F (x, (4 where F (x : (F 0 (x, F (x, F (x, T, F (α : (F 0 (α, F (α, F (α, T, F (x : ( F 0 (x, F (x, F (x, T, and F (α : ( F 0 (α, F (α, F (α, T Proof From the definitions of F n (x and F n (x, we have e αt h(t n0 F n (α tn n! and e αt + h(t Hence, denoted by f n (x either F n (x and F n (x, we obtain n0 F n (α tn n!

12 Dual and Conjugate Bernoulli and Euler Polynomials f n (x tn n! e αt e(x αt ± h(t (x α n f (α tn+ n!! n0 n 0 0 ( n ( n (x α n t n f (α n! n 0 0 Comparing the leftmost and the rightmost sides, we obtain (4 (4 follows from (4 The Bernoulli polynomial sequence defined by (6 can be considered as the sum generating function of Riordan matrix function (e xt, ( + t e t /t: te xt e t e xt +t et t n0 B n (x tn n! Inspired by the wor shown in Shapiro and the author 5], we define the conjugate Bernoulli polynomial sequence below by using the alternating sum (generating function of (e xt, ( + t e t /t: e xt + +t et t n0 B n (x tn n! (43 The coefficient polynomials of the above expansion form a sequence denoted by { B n } n 0 and referred to as the conjugate Bernoulli polynomial sequence, which first few terms are B 0 (x, B (x x +, B (x x + x + 5 6, B 3 (x x x + 5 x +, B 4 (x x 4 + x 3 + 5x + 8x , B 5 (x x x x3 + 0x x , etc Similarly, n0 E n(x tn n! is the sum (generating function of exponential Riordan array (e xt, ( e t / The corresponding alternating sum (generating function is e xt + et n0 Ẽ n (x tn n!, (44 where the first few terms of the polynomial sequence {Ẽn} n 0, called the conjugate Euler polynomials, are

13 T X He Ẽ 0 (x, Ẽ (x x +, Ẽ (x x + x +, Ẽ 3 (x x x + 3x + 4, Ẽ 4 (x x 4 + x 3 + 6x + x + 0, Ẽ 5 (x x x4 + 0x x + 50x + 9, etc From Proposition 3 for α 0, we may now that the conjugate Bernoulli polynomials { B n (x} n 0 have expression where B n (x 0 called the conjugate Bernoulli numbers Hence, by denoting one may write ( n x n B, (45 B n : B n (0, (46 B(x ( B 0 (x, B (x,, B n (x, T and B ( B 0, B,, B n, T, B(x P x] B and B P x] B(x (47 Similarly, from Proposition 3 for α /, we learn that the conjugate Euler polynomials {Ẽn(x} n 0 satisfy where Ẽ n (x 0 ( ( n x n Ẽ, (48 Ẽ : Ẽ ( called the conjugate Euler numbers Hence, by denoting one may write, (49 Ẽ(x (Ẽ0(x, Ẽ(x,, Ẽn(x, T and Ẽ (Ẽ0, Ẽ,, Ẽn, T, Ẽ(x P x ] D ] Ẽ P x ] Ẽ ( and (50 Ẽ P x + ]D]Ẽ(x (5

14 Dual and Conjugate Bernoulli and Euler Polynomials 3 The first few terms of conjugate Bernoulli numbers are { } { Bn,, 5 9,, 6 3 0, 76 } 3, The first few terms of conjugate Euler numbers are {Ẽn } {,, 7, 38, 77, 5, } We now define duals of conjugate Bernoulli polynomial sequence and conjugate Euler number sequence similar to (6 and (9, respectively Definition 33 Let { B n (x} n 0 be the conjugate Bernoulli polynomial sequence defined by (43, and let the conjugate Bernoulli numbers be defined by (46 Then the duals of conjugate Bernoulli numbers are defined by B n : 0 and the duals of conjugate Bernoulli polynomials are defined by B (x : ( n ( B (5 0 ( n x n B (53 Denote B : ( B 0, B, T and B (x : ( B 0(x, B (x, T Then B and B (x satisfy B D ]P ] B and B (x P x] B, (54 where Dt] : diag(, t, t, Similarly, let {Ẽn(x} n 0 be the conjugate Euler polynomial sequence defined by (44, and let the conjugate Euler numbers be defined by (46 Then the duals of conjugate Euler numbers are defined by Ẽ n : 0 and the duals of conjugate Euler polynomials are defined by ( n ( Ẽ (55 Ẽ (x : 0 ( n ( ( n x n Ẽ (56 0 ( x n Ẽ ( (57 Denote Ẽ : (Ẽ 0, Ẽ, T and Ẽ (x : (Ẽ 0(x, Ẽ (x, T Then Ẽ and Ẽ (x satisfy Ẽ D ]P ]Ẽ and Ẽ (x P x ] ] D Ẽ, (58 where Dt] : diag(, t, t,

15 4 T X He Clearly, the first formula of (5 implies From Definition 33, we immediately have B P ]D ] B D ]P ] B Theorem 34 Let { B n (x} n 0 and { B (x} n 0 be the conjugate Bernoulli polynomial sequence and its dual sequence defined by (43 and (5, respectively, and let B(x and B (x be the vectors of the conjugate Bernoulli polynomial sequence and its dual sequence defined before Then there hold which implies B (x D ]P x ] B D ] B( x and B(x D ]P x ] B D ] B ( x, (59 B n(x ( n Bn ( x and B n (x ( n B n ( x (60 Proof From the corresponding definitions, we have B (x P x] B P x]d ]P ] B D ]P x]p ] B D ]P x ] B, which implies the first formulas of (53 and (54, where we used the identity Dα]P x] P αx]dα] Using a similar argument, we may prove the second formulas of (53 and (54 Furthermore, we obtain Theorem 35 Let {Ẽn(x} n 0 and {Ẽ (x} n 0 be the conjugate Euler polynomial sequence and its dual sequence defined by (44 and (55, respectively, and let Ẽ(x and Ẽ (x be the vectors of the conjugate Euler polynomial sequence and its dual sequence defined before Then there hold ] Ẽ (x D ]P x]d Ẽ(x D ]P x]d ( Ẽ D ]Ẽ x + ] ( Ẽ D ]Ẽ x + and, (6 which implies ( Ẽn(x ( n Ẽ n x + and Ẽn(x ( n Ẽ n ( x + (6

16 Dual and Conjugate Bernoulli and Euler Polynomials 5 Proof From the corresponding definitions, we have Ẽ (x P x ] ] D Ẽ P x ] ] D D ]P ]Ẽ ] D ]P x]d Ẽ, which yields the first formulas of (6 and (6, and the second formulas of (6 and (6 follow immediately The first few terms of the dual of conjugate Bernoulli polynomial sequence are B 0(x, B (x x +, B (x x + x + 5 6, B 3(x x x + 5 x, B 4(x x 4 + x 3 + 5x , B 5(x x x x x 0 3, etc Hence, the first few terms of dual conjugate Bernoulli numbers are { } { B n,, 5 } 7, 0, 6 30, 0 3, The first few terms of the dual of conjugate Euler polynomial sequence are Ẽ 0(x, Ẽ (x x, Ẽ (x x x + 7 4, Ẽ 3(x x 3 3x + 4 x 9 4, Ẽ 4(x x 4 4x 3 + x 9x , Ẽ 5(x x 5 5x x3 95 x x 6 6, etc Hence, the first few terms of dual conjugate Euler numbers are

17 6 T X He } {Ẽ n {,, 4,, 60, 456, } We now establish a relationship between conjugate Bernoulli polynomials and (classical Bernoulli polynomials through the their duals Theorem 36 Let { B n (x} n 0 be the conjugate Bernoulli polynomial sequence defined by (43, and let its dual sequence, { B n(x} n 0, be defined by (30 Denote by B(x and B (x the vectors of the conjugate Bernoulli polynomial sequence and its dual sequence defined before Then the generating function of { B n (x} n 0 is e (x+t t e t t te (x+t ( te t n0 B n(x tn n! (63 Furthermore, the conjugates of the duals of the conjugate of Bernoulli polynomials are Bernoulli polynomials More precisely, we have or equivalently, e (x+t + t e t t te(x+t e t n0 B n ( x ( tn, (64 n! B n(x ( n B n ( x (65 Proof (63 is from (43 and (54, while (64 follows the definition of conjugate polynomials shown in Definition 3 and the generating function of Bernoulli polynomials Similarly, we have Theorem 37 Let {Ẽn(x} n 0 be the conjugate Euler polynomial sequence defined by (44, and let its dual sequence, {Ẽ n(x} n 0, be defined by (3 Denote by Ẽ(x and Ẽ (x the vectors of the conjugate Bernoulli polynomial sequence and its dual sequence defined above Then the generating function of {Ẽn(x} n 0 is e (x /t e t e(x+/t 3e t n0 Ẽ n(x tn n! (66 Furthermore, the conjugates of the duals of the conjugate of Euler polynomials are Euler polynomials More precisely, we have or equivalently, e (x /t + e t e(x /t e t + n0 E n ( x + / ( tn, (67 n! ( Ẽ n(x ( n E n x + (68 Comparing the results of Theorems 3 and 36 and 37, we finally obtain

18 Dual and Conjugate Bernoulli and Euler Polynomials 7 Theorem 38 Let F, F, F and F, where F B or E, be defined as before Then F F or equivalently F ( F Hence, the operators F and F are commutative The matrix form of conjugate Bernoulli polynomials and Euler polynomials provides a unified approach in the construction of Bernoulli and Euler polynomials identities As an example, we present the following extension wor of 4] for the conjugate Bernoulli polynomials First, we need a notation of direct sum of matrices Let A and B be any m m and n n square matrices, respectively 7] use the notation for the direct sum of matrices A and B: A B A 0 0 B ] (69 Hence, we may obtain an extension to the conjugate Bernoulli polynomials in a matrix form of Lemma 3 of 7] Theorem 39 Let n be any positive integer, and let B(x and P x] be defined ] as before Denote (x (, x, x, T, D diag(0,, /,, and X] 0], ie, Then which implies X] 0 x x 0 n n xn x n n X] B(x + y P x]d B(y + X](x, (70 B (x + y x n l ( n Bl (y x n l + H n x n (7 l l for all n 0 Proof Noting that B 0 (y, the left-hand side of (70 can be written as

19 8 T X He LHS of (70 X]P x] B(y x + ( B (y x + ( x B (y + ( B (y X] x n + ( n x n B (y + + ( n n Bn (y X] x x x n 0 x ( + x 0 ( B (y + ( n xn ( x B (y + ( B (y ( n x n B (y + + ( n n Bn (y 0 ( B (y + X] ( x B (y + ( B (y ( n x n B (y + + ( n n Bn (y where the second term can be written as 0 (( + ( B (yx + ( B (y (( + ( + + n n( B (yx n + ( ( + 3 3( + + n n( B (yx n + + n( n n Bn (y 0 ( ( B (yx + B (y ( n ( B (yx n + n B (yx n + n( n n Bn (y In the last step we use the identities n( n ( n and ( n ( n + ( ( n (,

20 Dual and Conjugate Bernoulli and Euler Polynomials 9 for, n N with n Since the last matrix can be written as P x]db(y, we obtain (70 By multiplying the matrices of (70, LHS of (7 H n x n + which is the RHS of (7 l ( n Bl (y x n l l l l ( n Bl (y x n l + H n x n, l l Remar 3 Some of the results, for instance (4, in this section can be proved by using the multiplication rule of the exponential Riordan arrays Here, we present a unified elementary approach in our proofs by suing matrix functions Worth noting a recent paper ] on a generalized Riordan group, to which our main results of this paper can be extended Due to the limit of space, more properties and applications of conjugate and dual Bernoulli polynomials and Euler polynomials will presented in a subsequent paper 4 Acnowledgements We would lie to express our thanfulness to the anonymous referees for their helpful comments and remars that led to an improved/revised version of the original manuscript References ] L Aceto and D Trigiante, The matrices of Pascal and other greats, American Math Monthly, 08 (00, No 3, 3 45 ] T M Apostol, A primer on Bernoulli numbers and polynomials, Math Magazine, 8 (008, No 3, ] P Barry, On integer-sequence-based constructions of generalized Pascal triangles, J Int Seq, 9 (006, Article 064 4] G S Call and D J Velleman, Pascal s matrices, American Math Monthly, 00 (993, No 4, ] N T Cameron and A Nwanta, On some (pseudo involutions in the Riordan group, J Integer Seq, 8 (005, -6 6] G-S Cheon, A note on the Bernoulli and Euler polynomials, Appl Math Letters, 6(003, ] G-S Cheon and H Kim, Simple proofs of open problems about the structure of involutions in the Riordan group Linear Algebra Appl, 48 (008, ] G-S Cheon, H Kim, L W Shapiro, Riordan group involutions Linear Algebra Appl, 48 (008, ] A Edelman and G Strang, Pascal matrices, American Math Monthly, (004, No 3, 89 97

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