In particular, we prove a relation between a generalized χ version of the power sum polynomials and this unified class of polynomials

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SYMMETRY PROPERTIES FOR A UNIFIED CLASS OF POLYNOMIALS ATTACHED TO χ

S. GABOURY^∗, R. TREMBLAY AND J. FUG `ERE

Abstract. In this paper, we obtain some generalized symmetry identities involving a unified class of polynomials related to the generalized Bernoulli, Euler and Genocchi polynomials of higher-order attached to a Dirichlet character. In particular, we prove a relation between a generalized χ version of the power sum polynomials and this unified class of polynomials.

AMS Mathematics Subject Classification : 11B68, 11S80.

Key words and phrases : Bernoulli polynomials, Euler polynomials, Genoc- chi polynomials, symmetry, Dirichlet character.

1. Introduction, Definitions and Notations

In the last years, Q.-M. Luo and Srivastava [10, 11] introduced the generalized Apostol-Bernoulli polynomialsBn^(α)(x) of order α, Q.-M. Luo [9] investigated the generalized Apostol-Euler polynomials En^(α)(x) of order α and the generalized Apostol-Genocchi polynomialsG^n(α)(x) of order α.

The generalized Apostol-Bernoulli polynomialsBn^(α)(x; λ) of order α∈ C, the generalized Apostol-Euler polynomialsEn^(α)(x; λ) of order α∈ C, the generalized Apostol-Genocchi polynomialsG^n(α)(x; λ) of order α∈ C are defined respectively by the following generating functions

( t

λe^t− 1 )α

e^xt=

∑∞ n=0

B^(α)n (x; λ)tⁿ

n! (|t + log λ| < 2π; 1^α:= 1) (1)

( 2

λe^t+ 1 )α

e^xt=

∑∞ n=0

En^(α)(x; λ)tⁿ

n! (|t + log λ| < π; 1^α:= 1) (2)

Received January 24, 2012. Revised September 4, 2012. Accepted September 5, 2012.

∗Corresponding author.

⃝ 2013 Korean SIGCAM and KSCAMc . 119

and ( 2t λe^t+ 1

)α

e^xt=

∑∞ n=0

Gn^(α)(x; λ)tⁿ

n! (|t + log λ| < π; 1^α:= 1). (3) It is obvious that the case λ = 1 and α = 1 in the above relations gives respec- tively the classical Bernoulli polynomials B_n(x), the classical Euler polynomials En(x) and the classical Genocchi polynomials Gn(x).

Recently, Ozden [13, 14] and Ozden et al. [15] provides an interesting unifica- tion of the Apostol-Bernoulli, Euler and Genocchi polynomials respectively (1), (2) and (3). Explicitly, Ozden studied the following generating function:

( 2^1−kt^k β^be^t− a^b

)α

e^xt=

∑∞ n=0

Yn, β^(α)(x; k, a, b)tⁿ

n! (4)

(t + blog( β a

)  < 2π, x ∈ R;1^α:= 1; k∈ N0; a, b∈ R \ {0}; α, β ∈ C )

. Shorthly after ¨Ozarslan [12] gives the following precise conditions of convergence of the series involved in (4)

(i) if a^b> 0 and k∈ N, thent+b log (β

a

)  < 2π; 1^α:= 1, x∈ R, α, β ∈ C.

(ii) if a^b> 0 and k = 0, then 0 < Im (

t + b log (β

a

))

< 2π; 1^α:= 1, x∈ R, α, β ∈ C.

(iii) if a^b< 0 and k∈ N0, thent+b log(

β a

)  < π; 1^α:= 1, x∈ R, α, β ∈ C.

This family of polynomials includes the above mentioned well known polynomi- als. We can see that

Yn, λ^(α)(x; 1, 1, 1) =Bn^(α)(x; λ), Yn, λ^(α)(x; 0,−1, 1) = En^(α)(x; λ) and

Y_{n, λ}^(α) (x; 1,−1, 1) = 1

2Gn^(α)(x; λ).

Moreover, Ozden et al. in [15] have extended and investigated the generating function (4) in terms of a Dirichlet character χ of conductor f ∈ N since these polynomials are very important in several fields of mathematics and physics.

They proposed the following χ-extension of the generating function for the gen- eralized Apostol-Bernoulli, Euler and Genocchi polynomials. Let χ be a Dirichlet character of conductor f ∈ N then

2^1−kt^k

f−1

∑

j=0

χ(j) (β

a

)bj

e^jt β^bfe^{f t}− a^bf e^xt=

∑∞ n=0

Yn, χ, β(x; k, a, b)tⁿ

n! (5)

(ft + bf log( β a

)  < 2π, x ∈ R; k ∈ N0; f∈ N, a, b ∈ R \ {0}; β ∈ C )

.

In this paper, we study the generating function for the χ-extended unified polynomials of higher order denoted byY_{n, χ, β}^(α) (x; k, a, b) and defined as follows:

Definition 1.1. Let χ be a Dirichlet character of conductor f ∈ N. Let k ∈ N0, a, b∈ R \ {0} and α, β ∈ C. Then the χ-extended unified polynomials of higher orderY_{n, χ, β}^(α) (x; k, a, b) are given by



2^1−kt^k

f∑−1 j=0

χ(j) (β

a

)bj

e^jt β^bfe^{f t}− a^bf





α

e^xt =

∑∞ n=0

Y_{n, χ, β}^(α) (x; k, a, b)tⁿ

n! (6)

(ft + bf log( β a

)  < 2π, x ∈ R; k ∈ N0; f∈ N, a, b ∈ R \ {0}; β ∈ C )

.

The case x = 0,Yn, χ, β^(α) (0; k, a, b) =Yn, χ, β^(α) (k, a, b) gives the χ-extended unified numbers of higher order.

Remark 1.1. If we set χ≡ 1 in (6), we obtain the generating function (4). The χ-extended versions B^{(α)n, χ}(x, β) of the Apostol-Bernoulli polynomials of higher order,E^{n, χ(α)}(x, β) of the Apostol-Euler polynomials of higher order andG^{n, χ(α)}(x, β) of the Apostol-Genocchi polynomials of higher order are given respectively by

Bn, χ^(α)(x, β) =Yn, χ, β^(α) (x; 1, 1, 1) (7) En, χ^(α)(x, β) =Yn, χ, β^(α) (x; 0,−1, 1) (8) Gn, χ^(α)(x, β) = 2Yn, χ, β^(α) (x; 1,−1, 1). (9) Moreover, if we put α = 1 and β = 1 in (7)-(9), we get the following χ-extended version of the classical Bernoulli, Euler and Genocchi polynomials [18, 19]:

Bn, χ(x) =Yn, χ, 1⁽¹⁾ (x; 1, 1, 1) (10) En, χ(x) =Yn, χ, 1⁽¹⁾ (x; 0,−1, 1) (11) G_{n, χ}(x) = 2Yn, χ, 1⁽¹⁾ (x; 1,−1, 1) (12) respectively.

Now, making use of this definition and the Cauchy product, we find the next theorem.

Theorem 1.2. Let k∈ N0, a, b∈ R \ {0} and α, β ∈ C. Then Yn, χ, β^(α) (x; k, a, b) =

∑n j=0

(n j )

x^n−j Yn, χ, β^(α) (k, a, b). (13)

Recently, many authors have studied the symmetry properties for the Bernoulli, Euler and Genocchi polynomials [1, 2, 3, 4, 6, 7, 16, 17]. The purpose of this

paper is to prove some identities of symmetry for this χ-extended unified poly- nomials of higher order. It points out the recurrence relation and multiplication theorem for the unified polynomials of higher order attached to χ.

2. Symmetry identities related to the generalized unified polynomials of higher order

In this section, we aim to obtain several symmetry identities involving the χ-extended unified polynomials of higher order defined by (6). In particular, we obtain an identity related to a generalized χ version of the power sum. Some spe- cial cases are also given for each identities. At this point, we need the following definition for the generalized χ version of the power sum.

Definition 2.1. Let χ be a Dirichlet character of conductor f ∈ N. Let k ∈ N0; l, n∈ N; a, b ∈ R \ {0} and α, β ∈ C, then we set

Tel, χ(n; k, a, β, b) = 2^1−kk!

( l k

) _f−1

∑

j=0 n−1

∑

i=0

χ(j) (β^b

a^b )j+if

(j + if )^l−k (l≥ k). (14)

The next theorem holds for the χ-extended unified polynomialsY_{n, χ, β}⁽¹⁾ (x; k, a, b):

Theorem 2.2. Let χ be a Dirichlet character of conductor f ∈ N. Let k ∈ N0; l (l≥ k), n ∈ N; a, b ∈ R \ {0} and β ∈ C, then

(β^b a^b

)f n

Yl, χ, β⁽¹⁾ (nf ; k, a, b)− Yl, χ, β⁽¹⁾ (0; k, a, b) = Te_{l, χ}(n; k, a, β, b)

a^bf . (15)

Proof. Consider the generating function (6) with α = 1, we have

∑∞ l=0

[(β^b a^b

)f n

Y_{l, χ, β}⁽¹⁾ (nf ; k, a, b)− Y_{l, χ, β}⁽¹⁾ (0; k, a, b) ]

t^l l!

= (β^b

a^b

)f n

2^1−kt^k∑_f−1

j=0χ(j)(_β

a

)bj

e^jte^{nf t} a^bf[(β

a

)bf

e^{f t}− 1] −2^1−kt^k∑f−1 j=0χ(j)(_β

a

)bj

e^jt a^bf[(β

a

)bf

e^{f t}− 1]

= 2^1−kt^k∑f−1 j=0χ(j)(_β

a

)bj

e^jt a^bf



 (β^b

a^b

)f n

e^{nf t}− 1 (_β

a

)bf

e^{f t}− 1





= 2^1−kt^k∑f−1 j=0χ(j)(_β

a

)bj

e^jt a^bf

n∑−1 i=0

(β a

)bf i

e^{if t}

= 2^1−kt^k a^bf

f∑−1 j=0

n∑−1 i=0

χ(j) (β

a )bf i+bj

e^{(j+if )t}

=

∑∞ l=0

2^1−k a^bf

f−1

∑

j=0 n−1

∑

i=0

χ(j) (β

a )bf i+bj

(j + if )^l t^l+k l! .

(16)

Substituting l → l − k (l ≥ k) and comparing the coefficient of ^t_l!^l yields the

desired result. 

Remark 2.1. If we set k = 0, a =−1, β = b = 1 and f ≡ 1 (mod 2) in (15), we recover a relation for the χ-extended version of the classical Euler polynomials [1]

(−1)ⁿ⁻¹E_{l, χ}(nf ) + E_{l, χ}(0) = 2 eT_{l, χ}(n; 0,−1, 1, 1). (17) Remark 2.2. Putting k = 1, a = b = 1 in (15), we recover a relation for the χ-extended version of the Apostol-Bernoulli polynomials [4]

β^{f n} Bl, χ(f n, β)− Bl, χ(0, β) = eT_{l, χ}(n; 1, 1, β, 1). (18) Theorem 2.3. Let χ be a Dirichlet character of conductor f ∈ N. Let k ∈ N0; l (l≥ k), m, n ∈ N; a, b ∈ R \ {0} and β ∈ C. Let w1 and w2 ∈ N. Then, we have

a^{bw2f (w1−1)}

∑n s=0

(n s )

wⁿ₁^−sw₂^s+kY_n^(m)_{−s, χ, β}w1(w2x; k, a^w1, b)

×

∑s l=0

(s l

)Tes−l, χ(w1; 0, a^w2, β^w2, b)Yl, χ, β^(m−1)w2(w1y; k, a^w2, b)

= a^{bw1f (w2−1)}

∑n s=0

(n s )

w^s+k₁ wⁿ₂^−sYn^(m)−s, χ, β^w2(w1x; k, a^w2, b)

×

∑s l=0

(s l

)Te_{s−k, χ}(w₂; 0, a^w1, β^w1, b)Y_{l, χ, β}^(m−1)w1(w₂y; k, a^w1, b).

(19)

Proof. Let

G(t) :=

2^{(1−k)(2m−1)}t^2km−k (∑_f−1

j=0χ(j) (β^w1

aw1

)bj

e^jw1t )m

e^w1w2xt (β^w1bfe^{w1f t}− a^w1bf)^m

×

(β^bw1w2fe^{w1w2f t}− a^bw1w2f) (∑_f−1

j=0χ(j) (β^w2

a^w2

)bj

e^jw2t )m

e^w1w2yt (β^w2bfe^{w2f t}− a^w2bf)^m .

(20)

Expanding G(t) into a series, we get

G(t) = 1

w^km₁ w^k(m₂ ⁻¹⁾



2^1−kw^k1t^k

f−1

∑

j=0

χ(j) (β^w1

aw1

)bj

e^jw1t β^w1bfe^{w1f t}− a^w1bf





m

e^w1w2xt

×

(β^bw1w2fe^{w1w2f t}− a^bw1w2f β^w2bfe^{w2f t}− a^w2bf

) (f∑−1 j=0

χ(j) (β^w2

a^w2 )bj

e^jw2t )

× (

2^1−kw^k₂t^k

f∑−1 j=0

χ(j)(_β

a

)bw₂j

e^jw2t β^w2bfe^{w2f t}− a^w2bf

)m−1

e^w1w2yt

= a^{bw2f (w1−1)} w^km₁ w^k(m₂ ⁻¹⁾

∑∞ n=0

Y_{n, χ, β}^(m) w1(w2x; k, a^w1, b)(w1t)ⁿ n!

f−1∑

j=0

χ(j) (β^w2

a^w2 )bj

e^jw2t

×

w∑1−1 i=0

(β^w2 a^w2

)bf i

e^{w2if t}

∑∞ l=0

Y_{l, χ, β}^(m−1)w2(w1y; k, a^w2, b)(w2t)^l l!

= a^{bw2f (w1−1)} 2w^km₁ w^k(m−1)₂

∑∞ n=0

Yn, χ, β^(m) w1(w2x; k, a^w1, b)(w1t)ⁿ n!

×∑^∞

s=0

2

f−1∑

j=0 w∑1−1

i=0

χ(j) (β^w2

a^w2

)b(j+f i)

(j + f i)^s(w2t)^s s!

×

∑∞ l=0

Y_{l, χ, β}^(m−1)w2(w1y; k, a^w2, b)(w2t)^l l!

= a^{bw2f (w1−1)} 2w^km₁ w^k(m₂ ⁻¹⁾

∑∞ n=0

Y_{n, χ, β}^(m) w1(w2x; k, a^w1, b)(w1t)ⁿ n!

×

∑∞ s=0

Tes, χ(w1; 0, a^w2, β^w2, b)(w2t)^s s!

∑∞ l=0

Yl, χ, β^(m−1)w2(w1y; k, a^w2, b)(w2t)^l l!

=a^{bw2f (w1−1)} 2w₁^kmw₂^km

∑∞ n=0

[∑n s=0

( n s )

w₁^n−sw^s+k₂ Y_n^(m)_{−s, χ, β}w1(w2x; k, a^w1, b)

×

∑s l=0

( s l )

Te_{s−l, χ}(w1; 0, a^w2, β^w2, b)Yl, χ, β^(m−1)w2(w1y; k, a^w2, b) ]tⁿ

n!. (21) In a similar way, we have

G(t) = 1

w^k(m₁ ⁻¹⁾w₂^km



2^1−kw^k₂t^k

f−1

∑

j=0

χ(j) (βw2

aw2

)bj

e^jw2t β^w2bfe^{w2f t}− a^w2bf





m

e^w1w2xt

×

(β^bw1w2fe^{w1w2f t}− a^bw1w2f β^w1bfe^{w1f t}− a^w1bf

) (f−1∑

j=0

χ(j) (β^w1

a^w1 )bj

e^jw1t )

×



2^1−kw^k1t^k

f−1

∑

j=0

χ(j) (β^w1

aw1

)bj

e^jw1t β^w1bfe^{w1f t}− a^w1bf





m−1

e^w1w2yt

=a^{bw1f (w2−1)} 2w₁^kmw₂^km

∑∞ n=0

[∑n s=0

( n s )

w^s+k₁ wⁿ₂^−sY_{n−s, χ, β}^(m) w2(w1x; k, a^w2, b)

×

∑s l=0

( s l )

Tes−k, χ(w2; 0, a^w1, β^w1, b)Y_{l, χ, β}^(m−1)w1(w2y; k, a^w1, b) ]t^l

l!. (22)



If we set k = 0, a =−1 and b = 1 in Theorem 2.5, we obtain a result exhibited by Kim et al. [5, Eq.(2.13)].

Corollary 2.4. Let m, n∈ N and β ∈ C. Let w1, w2 and f ∈ N with w1 ≡ 1 (mod 2), w₂≡ 1 (mod 2) and f ≡ 1 (mod 2), we have

∑n s=0

(n s )

wⁿ₁^−sw^s₂En^(m)−s, χ(w2x; β^w1)

×

∑s l=0

(s l

)Te_{s−l, χ}(w₁; 0,−1, β^w2, 1)E_{l, χ}^(m−1)(w₁y; β^w2)

=

∑n s=0

(n s )

w^s₁w₂^n−sEn^(m)−s, χ(w₁x; β^w2)

×

∑s l=0

(s l

)Tes−k, χ(w2; 0,−1, β^w1, 1)E_{l, χ}^(m−1)(w2y; β^w1).

(23)

Theorem 2.5. Let χ be a Dirichlet character of conductor f ∈ N. Let k ∈ N0, m, n∈ N; a, b ∈ R \ {0} and β ∈ C. Let w1and w₂be two natural numbers.

Then, we have

∑n r=0

( n r

)_w

1−1

∑

i=0 w∑₂−1

l=0

(β a

)bf (i+l)

wⁿ₁^−rw^r₂Yn^(m)−r, χ, β

(

w2x +w2f w1

i; k, a, b )

× Yr, χ, β^(m)

(

w1y +w1f w2

l; k, a, b )

=

∑n r=0

( n r

)_w2−1

∑

i=0 w∑1−1

l=0

(β a

)bf (i+l)

w^r1w₂^n−rYn^(m)−r, χ, β

(

w1x +w1f w2

i; k, a, b )

× Yr, χ, β^(m)

(

w2y +w2f w1

l; k, a, b )

.

(24)

Proof. Let

H(t) :=

2^2m(1−k))t^2km(∑f−1 j=0χ(j)(_β

a

)bj

e^jw1t )m

e^w1w2xt(

β^bw2fe^{w1w2f t}− a^bw2f) (β^bfe^{w1f t}− a^bf)^m+1

(β^bw1fe^{w1w2f t}− a^bw1f) (∑f−1 j=0χ(j)(_β

a

)bj

e^jw2t )m

e^w1w2yt

(β^bfe^{w2f t}− a^bf)^m+1 . (25) Expanding H(t) into a series, we get

H(t) = 1 (w1w2)^km

(

2^1−kw^k₁t^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw1t β^bfe^{w1f t}− a^bf

)m

e^w1w2xt

×

(β^bw1fe^{w1w2f t}− a^bw1f β^bfe^{w2f t}− a^bf

) (

2^1−kw^k₂t^k

f∑−1 j=0

χ(j)(_β

a

)bj

e^jw2t β^bfe^{w2f t}− a^bf

)m

× e^w1w2yt

(β^bw2fe^{w1w2f t}− a^bw2f β^bfe^{w1f t}− a^bf

)

=a^{bf (w1+w2−2)} (w1w2)^km

(

2^1−kw₁^kt^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw1t β^bfe^{w1f t}− a^bf

)m

e^w1w2xt

× (_w

1−1

∑

i=0

(β a

)bf i

e^{w2if t} ) (

2^1−kw^k₂t^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw2t β^bfe^{w2f t}− a^bf

)m

× e^w1w2yt (_w

2−1

∑

l=0

(β a

)bf l

e^{w1lf t} )

=a^{bf (w1+w2−2)} (w1w2)^km

w∑₁−1 i=0

(β a

)bf i(

2^1−kw₁^kt^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw1t β^bfe^{w1f t}− a^bf

)m

× e^w1t

( w2x+^w2f

w1i ) w∑2−1

l=0

(β a

)bf l(

2^1−kw^k₂t^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw2t β^bfe^{w2f t}− a^bf

)m

× e^w2t

( w₁y+^w1f

w2l)

=a^{bf (w1+w2−2)} (w1w2)^km

[_w

1−1

∑

i=0

(β a

)bf i∑∞ n=0

Yn, χ, β^(m)

(

w2x +w2f w1

i; k, a, b

)(w1t)ⁿ n!

]

× [_w

2−1

∑

l=0

(β a

)bf l∑∞ r=0

Y_{r, χ, β}^(m) (

w1y +w1f w2

l; k, a, b )(w2t)^r

r!

]

=a^{bf (w1+w2−2)} (w1w2)^km

∑∞ n=0

[ _n

∑

r=0

( n r

)_w

1−1

∑

i=0 w∑₂−1

l=0

(β a

)bf (i+l)

w^n−r₁ w^r₂

× Y_{n−r, χ, β}^(m) (

w2x +w2f w1

i; k, a, b )

Y_{r, χ, β}^(m) (

w1y +w1f w2

l; k, a, b ) ]tⁿ

n!. (26) Likewise,

H(t) = 1 (w1w2)^km

(

2^1−kw^k₂t^k

f−1

∑

j=0

χ(j)(_β

a

)bj

e^jw2t β^bfe^{w2f t}− a^bf

)m

e^w1w2xt

×

(β^bw2fe^{w1w2f t}− a^bw2f β^bfe^{w1f t}− a^bf

) (

2^1−kw^k1t^k

f∑−1 j=0

χ(j)(_β

a

)bj

e^jw1t β^bfe^{w1f t}− a^bf

)m

× e^w1w2yt

(β^bw1fe^{w1w2f t}− a^bw1f β^bfe^{w2f t}− a^bf

)

=a^{bf (w1+w2−2)} (w1w2)^km

∑∞ n=0

[ _n

∑

r=0

( n r

)_w

2−1

∑

i=0 w∑₁−1

l=0

(β a

)bf (i+l)

w^r1w^n−r₂

× Yn^(m)−r, χ, β

(

w1x +w1f w2

i; k, a, b )

Yr, χ, β^(m)

(

w2y +w2f w1

l; k, a, b ) ]tⁿ

n!. (27)

 Now, putting χ≡ 1 and replacing k, a, b by 1 in Theorem 2.7, we find a result of Zhang and Yang [20, Eq.18].