Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 4 (2012), Pages 76-90.
SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF
APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS
RELATED TO GENERALIZED POWER AND ALTERNATING
SUMS
(COMMUNICATED BY R.K. RAINA)
B.-J. FUGÈRE, S. GABOURY, R. TREMBLAY
Abstract. The purpose of this paper is to obtain several series identities involving some classes of generalized Apostol-Bernoulli and Apostol-Euler polynomials introduced lately by Srivastava et al. in [16, 17] as well as the generalized sum of integer powers, the generalized alternating sum and the analogues
of the expansions of the hyperbolic tangent and the hyperbolic cotangent.
The method used is that of generating functions. It can be found that many
identities recently obtained are special cases of our results.
1. Introduction, Definitions and Notations
(α)
The generalized Bernoulli polynomials Bn (x) of order α ∈ C, the generalized
(α)
Euler polynomials En (x) of order α ∈ C and the generalized Genocchi polynomials
(α)
Gn (x) of order α ∈ C, each of degree n as well as in α, are defined respectively by
the following generating functions (see,[4, vol.3, p.253 et seq.], [8, Section 2.8] and
[10]):
α
∞
X
t
tk
(α)
xt
·
e
=
Bk (x)
(|t| < 2π; 1α := 1),
(1.1)
t
e −1
k!
k=0
α
∞
X
tk
2
(α)
xt
·
e
=
E
(x)
(|t| < π; 1α := 1)
(1.2)
k
et + 1
k!
k=0
and
2t
et + 1
α
· ext =
∞
X
(α)
Gk (x)
k=0
tk
k!
(|t| < π; 1α := 1).
(1.3)
The literature contains a large number of interesting properties and relationships
involving these polynomials [1, 2, 3, 4, 5, 15]. Q.-M. Luo and Srivastava ([12,
(α)
14]) introduced the generalized Apostol-Bernoulli polynomials Bn (x) of order α,
1991 Mathematics Subject Classification. Primary 11B68; Secondary 11S80.
Key words and phrases. Bernoulli numbers and polynomials, Euler numbers and polynomials,
Genocchi numbers and polynomials, Apostol-Bernoulli polynomials, Apostol-Euler polynomials,
Apostol-Genocchi polynomials, Generalized power sums, Generalized alternating sums.
Submitted June 19, 2012. Published November 6, 2012.
76
77
(α)
Q.-M. Luo [9] investigated the generalized Apostol-Euler polynomials En (x) of
(α)
order α and the generalized Apostol-Genocchi polynomials Gn (x) of order α (see
also,[10, 11, 13]).
(α)
The generalized Apostol-Bernoulli polynomials Bn (x; λ) of order α ∈ C, the
(α)
generalized Apostol-Euler polynomials En (x; λ) of order α ∈ C, the generalized
(α)
Apostol-Genocchi polynomials Gn (x; λ) of order α ∈ C are defined respectively
by the following generating functions
α
∞
X
tk
t
(α)
xt
·
e
=
B
(x;
λ)
(|t + ln λ| < 2π; 1α := 1)
(1.4)
k
λet − 1
k!
k=0
2
λet + 1
α
·e
xt
=
∞
X
(α)
Ek (x; λ)
k=0
tk
k!
(|t + ln λ| < π; 1α := 1)
(1.5)
and
2t
t
λe + 1
α
· ext =
∞
X
(α)
Gk (x; λ)
k=0
tk
k!
(|t + ln λ| < π; 1α := 1).
(1.6)
It is easy to see that
(α)
(α)
Bn (x) = Bn (x; 1),
(α)
(α)
En (x) = En (x; 1)
(α)
(α)
and Gn (x) = Gn (x; 1).
Recently, Srivastava et al. in [16, 17] have investigated some new classes of
Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials with parameters a, b and c defined by the following generating functions.
Definition 1.1. Let a, b, c ∈ R+ , (a 6= b) and n ∈ N0 . The generalized Apostol(α)
Bernoulli polynomials Bn (x; λ; a, b, c) of order α, the generalized Apostol-Euler
(α)
polynomials En (x; λ; a, b, c) of order α and the generalized Apostol-Genocchi poly(α)
nomials Gn (x; λ; a, b, c) of order α are defined respectively by the following generating functions
α
∞
a
X
t
tn α
xt
(α)
,
t
ln
+
ln
λ
<
2π;
1
:=
1
,
c
=
B
(x;
λ;
a,
b,
c)
n
λbt − at
n!
b
n=0
(1.7)
2
t
λb + at
and
α
2t
t
λb + at
cxt =
∞
X
E(α)
n (x; λ; a, b, c)
n=0
α
cxt =
∞
X
n=0
a
tn , t ln
+ ln λ < π; 1α := 1 (1.8)
n!
b
G(α)
n (x; λ; a, b, c)
a
tn , t ln
+ ln λ < π; 1α := 1 .
n!
b
(1.9)
If we take a = 1, b = c = e in (1.7), (1.8) and (1.9) respectively, we have
(1.4), (1.5) and (1.6). Obviously, when we set λ = 1, α = 1, a = 1, b = c = e in
(1.7), (1.8) and (1.9), we have classical Bernoulli polynomials Bn (x), classical Euler
polynomials En (x) and classical Genocchi polynomials Gn (x).
78
For each k ∈ N0 , Sk (n) defined by
Sk (n) =
n
X
jk
(1.10)
j=0
is called the sum of integer powers. The exponential generating function for Sk (n)
is given by [19]
∞
X
e(n+1)t − 1
tk
.
(1.11)
Sk (n) =
k!
et − 1
k=0
We now define the generalized sum of integer powers as follows.
Definition 1.2. For an arbitrary real or complex λ, the generalized sum of integers
powers Sk (n; λ) is defined by the generating relation
∞
X
Sk (n; λ)
k=0
λe(n+1)t − 1
tk
=
.
k!
λet − 1
(1.12)
It is obvious that
Sk (n; 1) = Sk (1).
(1.13)
For k ∈ N0 and n ∈ N, Tk (n) defined by
Tk (n) =
n−1
X
(−1)k nk
(1.14)
k=0
is called the alternating sum. The exponential generating function for Tk (n) is
given by
∞
X
tk
1 − (−1)n ent
Tk (n) =
.
(1.15)
k!
1 + et
k=0
The generalized alternating sum of order α is defined in [7] as follows.
Definition 1.3. For any arbitrary real or complex parameter λ, the generalized
(α)
alternating sum of order α, Tk (n; λ) is defined by the following generating function:
α
∞
X
tk
1 − λ(−1)n ent
(α)
Tk (n; λ) =
.
(1.16)
k!
1 + λet
k=0
It is easy to observe that
(1)
Tk (n; 1) = Tk (n).
(1.17)
In this paper, we present several series identities involving the generalized ApostolBernoulli and the generalized Apostol-Euler polynomials defined respectively by
(1.7) and (1.8). In Section 2, we obtain several symmetry identities for the generalized Apostol-Bernoulli polynomials a relation between the these polynomials
and the generalized sum of integer powers (1.12). In Section 3, we prove several
identities involving the generalized Apostol-Euler, the generalized alternating sum
and the analogues of the expansions of the hyperbolic tangent and the hyperbolic
cotangent. Some identities are also obtained by using the relationships between the
generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials.
79
2. Symmetry identities for the generalized Apostol-Bernoulli
polynomials
In this section, we establish some symmetry identities involving the generalized
(α)
Apostol-Bernoulli polynomials Bn (x; λ; a, b, c) defined in the first section and the
generalized sum of integer powers defined by (1.12). This is done by using the
method of generating functions. These results provides generalization of known
identities [18, 21, 22, 23]
Theorem 2.1. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0, for a, b, c ∈ R+
(a 6= b) and for λ ∈ C, we have the following identity:
!
(µ − 1)ν ln a
n
(α)
; λ; a, b, c µn−k ν k+1
Bn−k νx +
µ ln c
k
k=0
!
k−i
k
X k
b
(α−1)
(µy; λ; a, b, c) ln
Sk−i (µ − 1, λ) Bi
×
i
a
i=0
!
n
X
(ν − 1)µ ln a
n
(α)
=
Bn−k µx +
; λ; a, b, c ν n−k µk+1
k
ν ln c
k=0
!
k−i
k
X
b
k
(α−1)
(νy; λ; a, b, c) ln
.
×
Sk−i (ν − 1, λ) Bi
a
i
i=0
n
X
(2.1)
t2α−1 c µνxt (λbµνt − aµνt )c µνyt
. We have to expand the
(λbµt − aµt )α (λbνt − aνt )α
last function into series in two way to prove the theorem. We have
Proof. Considering g(t) =
t2α−1 c µνxt (λbµνt − aµνt )c µνyt
(λbµt − aµt )α (λbνt − aνt )α
α
µνt
α−1
µt
λb
− aµνt
νt
1
νxµt
= α α−1
c
c µνyt
µ ν
λbµt − aµt
λbνt − aνt
λbνt − aνt
 µt

bν
(µ−1)νt ln a α
α−1
λ
−
1
ln c
aν
c
µt
νt

νxµt 
=
c
c µνyt


ν t
µα ν α−1
λbµt − aµt
λbνt − aνt
λ ab ν − 1
!
∞
X
(µ − 1)ν ln a
(µt)n
1
(α)
= α α−1
Bn
νx +
; λ; a, b, c
µ ν
µ ln c
n!
n=0
!
!
∞
∞
k
X
X
(νt)i
b
tk
(α−1)
×
Sk (µ − 1, λ) ν ln
Bi
(µy; λ; a, b, c)
a
k!
i!
i=0
k=0
" n
!
∞
(µ − 1)ν ln a
1 X X n
(α)
= α α
Bn−k νx +
; λ; a, b, c µn−k ν k+1
µ ν n=0
k
µ ln c
k=0
!
k−i # n
k
X
k
b
t
(α−1)
×
Sk−i (µ − 1, λ) Bi
(µy; λ; a, b, c) ln
.
i
a
n!
i=0
g(t) =
(2.2)
80
By expanding in a different way, we have
" n
!
∞
(ν − 1)µ ln a
1 X X n
(α)
g(t) = α α
; λ; a, b, c ν n−k µk+1
Bn−k µx +
ν µ n=0
ν ln c
k
k=0
!
k−i # n
k
X
k
b
t
(α−1)
×
Sk−i (ν − 1, λ) Bi
.
(νy; λ; a, b, c) ln
i
a
n!
i=0
(2.3)
By setting a = 1, b = c = e in Theorem 1, we obtain one of the results exhibited
by Zhang and Yang [23, Eq. 8]:
Corollary 2.2. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0 and for λ ∈ C, we
have
n k X
X
n
k
(α)
(α−1)
Bn−k (νx; λ) µn−k ν k+1
Sk−i (µ − 1, λ) Bi
(µy; λ)
k
i
i=0
k=0
=
n k X
X
n
k
(α)
(α−1)
Bn−k (µx; λ) ν n−k µk+1
Sk−i (ν − 1, λ) Bi
(νy; λ). (2.4)
k
i
i=0
k=0
Putting x = 0, y = 0 and α = 1 in Theorem 1, we have:
Corollary 2.3. For all integers µ > 0, ν > 0, n ≥ 0 and for a, b, c ∈ R+ (a 6= b)
and λ ∈ C, we have the following relation:
!
n−k
(µ − 1)ν ln a
b
n
k−1 n−k
; λ; a, b, c µ
ν
Sn−k (µ − 1, λ) ln
Bk
µ ln c
a
k
k=0
!
n−k (2.5)
n
X
(ν − 1)µ ln a
n
b
=
Bk
; λ; a, b, c ν k−1 µn−k Sn−k (ν − 1, λ) ln
.
k
ν ln c
a
n
X
k=0
Finally, substituting λ = 1, a = 1, b = c = e in (2.5), we find
n n X
X
n k−1 n−k
n k−1 n−k
µ
ν
Bk Sn−k (µ − 1) =
ν
µ
Bk Sn−k (ν − 1)
k
k
k=0
(2.6)
k=0
a result given by Tuenter [18].
Theorem 2.4. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0, for a, b, c ∈ R+ ,
(a 6= b) and for λ ∈ C, we have the following identity:
! µ−1 ν−1
!
n
X
iν ln ab + (2µν − ν − µ) ln a
n X X i+j n−k k (α)
λ µ
ν Bn−k νx +
; λ; a, b, c
k i=0 j=0
µ ln c
k=0
!
jµ ln ab
(α)
× Bk
µy +
; λ; a, b, c
ν ln c
! ν−1 µ−1
!
n
X
iµ ln ab + (2µν − ν − µ) ln a
n X X n−k k (α)
=
ν
µ Bn−k µx +
; λ; a, b, c
k i=0 j=0
ν ln c
k=0
!
jν ln ab
(α)
× λi+j Bk
νy +
; λ; a, b, c .
µ ln c
(2.7)
81
Proof. Let the function h(t) be given by
h(t) =
t2α c µνxt (λµ bµνt − aµνt )(λν bµνt − aµνt )c µνyt
,
(λbµt − aµt )α+1 (λbνt − aνt )α+1
(2.8)
which can be expanded as follows:
α
µ µνt
α
µt
λ b
− aµνt
νt
1
µνxt
c
(µν)α λbµt − aµt
λbνt − aνt
λbνt − aνt
ν µνt
λ b
− aµνt
×
c µνyt
µt
λb − aµt
!
α
α
b
λµ e µνt ln( a ) − 1
µt
a2µνt−νt−µt
νt
µνxt
c
=
b
(µν)α
λbµt − aµt
λbνt − aνt
λe νt ln( a ) − 1
!
b
λν e νµt ln( a ) − 1
×
c µνyt
b
λe µt ln( a ) − 1
α
µ−1
X
(2µν−ν−µ)t ln a
b
1
µt
µνxt
ln
c
=
c
c
λi e iνt ln( a )
(µν)α
λbµt − aµt
i=0
α ν−1
X
b
νt
×
λj e jµt ln( a ) c µνyt
νt
νt
λb − a
j=0
iνt ln ab
(2µν − ν − µ)t ln a
α
µ−1
X
+
µνxt +
1
µt
i
ln
c
ln c
=
λ
c
(µν)α i=0
λbµt − aµt
jµt ln ab
ν−1
α
X
µνyt +
νt
ln c
×
λj
c
νt
νt
λb − a
j=0
!
µ−1
∞
iν ln ab + (2µν − ν − µ) ln a
1 X i X (α)
(µt)n
λ
B
νx
+
;
λ;
a,
b,
c
=
(µν)α i=0 n=0 n
µ ln c
n!
!
ν−1
∞
X X
jµ ln ab
(νt)k
(α)
×
λj
Bk
µy +
; λ; a, b, c
ν ln c
k!
j=0
k=0
" n µ−1 ν−1
!
∞
jµ ln ab
1 X X n X X i+j n−k k (α)
; λ; a, b, c
=
λ µ
ν Bk
µy +
(µν)α n=0
k i=0 j=0
ν ln c
k=0
!#
iν ln ab + (2µν − ν − µ) ln a
tn
(α)
× Bn−k νx +
; λ; a, b, c
.
µ ln c
n!
h(t) =
(2.9)
Since h(t) is symmetric in µ and ν, we can also expand h(t) as follows:
! ν−1 µ−1
" n
!
∞
jν ln ab
1 X X n X X i+j n−k k (α)
h(t) =
λ ν
µ Bk
νy +
; λ; a, b, c
k i=0 j=0
(µν)α n=0
µ ln c
k=0
(2.10)
!#
iµ ln ab + (2µν − ν − µ) ln a
tn
(α)
× Bn−k µx +
; λ; a, b, c
.
ν ln c
n!
82
n
By equating the coefficient of tn! on the right-hand sides of these last two (2.9) and
(2.10), we get the identity (2.7).
Setting a = 1, b = c = e in Theorem 2 yields a result given recently by Zhang
and Yang [23, Eq. 18] :
Corollary 2.5. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0 and for λ ∈ C, we
have
! µ−1 ν−1
n
X
jµ
n X X i+j n−k k (α)
iν
(α)
µy +
;λ
λ
µ
ν Bn−k νx + ; λ Bk
µ
ν
k i=0 j=0
k=0
!
ν−1 µ−1
n
X
jν
n X X i+j n−k k (α)
iµ
(α)
=
νy +
;λ .
λ
ν
µ Bn−k µx + ; λ Bk
ν
µ
k i=0 j=0
(2.11)
k=0
Putting ν = 1 and y = 0 in Theorem 2 gives the next corollary:
Corollary 2.6. For all integers µ > 0, α ≥ 1, n ≥ 0, for a, b, c ∈ R+ , (a 6= b) and
for λ ∈ C, we have the following identity:
!
! µ−1
n
X
i ln ab + (µ − 1) ln a
n X i n−k (α)
(α)
; λ; a, b, c Bk (0; λ; a, b, c)
λµ
Bn−k x +
µ ln c
k i=0
k=0
! µ−1
!
n
X
j ln ab
(µ − 1) ln a
n X j k (α)
(α)
λ µ Bn−k µx +
=
; λ; a, b, c Bk
; λ; a, b, c .
k j=0
ν ln c
µ ln c
k=0
(2.12)
Theorem 2.7. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0, for a, b, c ∈ R+ ,
(a 6= b) and for λ ∈ C, we have the following identity:
! µ−1 ν−1
n
X
n X X i+j n−k k (α)
λ
µ
ν Bk (µy; λ; a, b, c)
k i=0 j=0
k=0
iν ln ab + (2µν − ν − µ) ln a + jµ ln
(α)
× Bn−k νx +
µ ln c
!
µ−1
ν−1
n
X
n X X i+j n−k k (α)
=
λ
ν
µ Bk (νy; λ; a, b, c)
k i=0 j=0
k=0
iµ ln ab + (2µν − ν − µ) ln a + jν ln
(α)
× Bn−k µx +
ν ln c
b
a
!
; λ; a, b, c
(2.13)
b
a
!
; λ; a, b, c .
Proof. The proof of Theorem 3 is similar to that of Theorems 1 and 2. In the
proof of Theorem 3, we first make use of (1.7) in order to expand the function h(t)
defined by (2.8) and then apply the symmetry of h(t) in µ and ν to obtain a second
expansion of h(t). The details involved are straightforward and we leave them as
an exercise.
If we set a = 1, b = c = e in Theorem 3, we recover a result given recently by
Zhang and Yang [23, Eq. 23] :
83
Corollary 2.8. For all integers µ > 0, ν > 0, α ≥ 1, n ≥ 0 and for λ ∈ C, we
have
! µ−1 ν−1
n
X
n X X i+j n−k k (α)
ν
(α)
λ
µ
ν Bn−k νx + i + j; λ Bk (µy; λ)
k i=0 j=0
µ
k=0
! ν−1 µ−1
n
X
n X X i+j n−k k (α) µ
(α)
=
λ
ν
µ Bn−k µx + i + j; λ Bk (νy; λ) .
ν
k i=0 j=0
(2.14)
k=0
Putting ν = 1 and y = 0 in Theorem 3 gives the next corollary:
Corollary 2.9. For all integers µ > 0, α ≥ 1, n ≥ 0, for a, b, c ∈ R+ , (a 6= b) and
for λ ∈ C, we have the following identity:
!
! µ−1
n
X
i ln ab + (µ − 1) ln a
n X i n−k (α)
(α)
; λ; a, b, c Bk (0; λ; a, b, c)
λµ
Bn−k x +
µ ln c
k i=0
k=0
! µ−1
!
n
X
(µ − 1) ln a + j ln ab
n X j k (α)
(α)
λ µ Bn−k µx +
=
; λ; a, b, c Bk (0; λ; a, b, c) .
k j=0
ln c
k=0
(2.15)
3. Some identities related to generalized Apostol-Euler polynomials
In this section, we derive some identities concerning the generalized Apostol(α)
Euler polynomials En (x; λ; a, b, c) defined by (1.8), the generalized alternating
sum (1.16) and the analogues of the expansions of hyperbolic cotangent and hyperbolic tangent introduced in [20]. These results extend some known formulas
[6, 7, 21]. We conclude this section by giving some identities based on relation(α)
ships between the generalized Apostol-Euler polynomials En (x; λ; a, b, c) and the
(α)
generalized Apostol-Bernoulli polynomials Bn (x; λ; a, b, c) and between the gener(α)
alized Apostol-Bernoulli polynomials Bn (x; λ; a, b, c) and the generalized Apostol(α)
Genocchi polynomials Gn (x; λ; a, b, c) defined by (1.9).
Theorem 3.1. For n ∈ N0 , µ, ν ∈ N, α ≥ 1, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C.
If µ and ν have the same parity, then the following identity holds true:
!
k
n
X
n n−k k
b
αν ln a
(α)
(α)
µ
ν
ln
En−k νx −
; λ; a, b, c Tk (µ; λ)
k
a
µ ln c
k=0
!
k
n
X
n n−k k
b
αµ ln a
(α)
(α)
=
ν
µ ln
En−k µx −
; λ; a, b, c Tk (ν; λ).
k
a
ν ln c
(3.1)
k=0
Proof. Let the function g(t) be given by
α
b
c µνxt 1 − λ(−1)µ eµνt ln( a )
g(t) =
α
(λbµt + aµt ) (λbνt + aνt )
α
(3.2)
84
Making use of (1.8) and (1.16) to expand g(t) to obtain first:
!α
b
1 − λ(−1)µ eµνt ln( a )
c
b
λeνt ln( a ) + 1
k
∞
∞
X
νt ln ab
(µt)n X (α)
αν ln a
(α)
νx −
; λ; a, b, c
En
Tk (µ; λ)
µ ln c
n!
k!
n=0
k=0
!
∞ X
n
k
X
tn
b
αν ln a
n n−k k
(α)
(α)
; λ; a, b, c Tk (µ; λ) .
En−k νx −
µ
ν
ln
a
µ ln c
n!
k
n=0
1 −ανt ln a
g(t) = α c ln c
2
=
1
2α
=
1
2α
2
λbµt + aµt
α
νxµt
k=0
(3.3)
Now, since µ and ν have the same parity, then the function g(t) is symmetric in µ
and ν. Therefore, we can expand g(t) as follows:
!
k
∞
n
1 X X n n−k k
tn
b
αµ ln a
(α)
(α)
g(t) = α
En−k µx −
; λ; a, b, c Tk (ν; λ) .
ν
µ ln
2 n=0
a
ν ln c
n!
k
k=0
(3.4)
n
By equating the coefficient of tn! on the right-hand side of the last two equations
(3.3) and (3.4), we thus recover the identity (3.1) asserted by Theorem 4.
As a special case, if we set a = 1, b = c = e in Theorem 4, we obtain the following
corollary given recently by Lu and Srivastava in [7, Eq. 30].
Corollary 3.2. For n ∈ N0 , µ, ν ∈ N, α ≥ 1 and for λ ∈ C. If µ and ν have the
same parity, then the following identity holds true:
!
!
n
n
X
X
n n−k k (α)
n n−k k (α)
(α)
(α)
µ
ν En−k (νx ; λ) Tk (µ; λ) =
ν
µ En−k (µx; λ) Tk (ν; λ).
k
k
k=0
k=0
(3.5)
Now, letting α = λ = 1, we recover the result given by Yang and Qiao in [21,
Eq. 18]:
Corollary 3.3. For n ∈ N0 and µ, ν ∈ N. If µ and ν have the same parity, then
we have
!
!
n
n
X
X
n n−k k
n n−k k
µ
ν En−k (νx) Tk (µ) =
ν
µ En−k (µx) Tk (ν).
k
k
(3.6)
k=0
k=0
Theorem 3.4. For n ∈ N0 , µ, ν ∈ N, α ≥ 1, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C.
If µ and ν have the same parity, then the following identity holds true:
! µ−1 ν−1
n
X
n XX
(α)
(−λ)i+j µn−k ν k En−k
k i=0 j=0
νx +
(iν + jµ) ln
k=0
!
− (µ + ν) ln a
; λ; a, b, c
µ ln c
b
a
(α)
× Ek (µy; λ; a, b, c)
! ν−1 µ−1
n
X
n XX
(α)
=
(−λ)i+j ν n−k µk En−k
k i=0 j=0
k=0
µx +
(iµ + jν) ln
!
− (µ + ν) ln a
; λ; a, b, c
ν ln c
b
a
(α)
× Ek (νy; λ; a, b, c).
(3.7)
85
Proof. Let the function g(t) be given by
h
νt µ i h
µt ν i
cµνxt cµνyt 1 − −λ ab
1 − −λ ab
g(t) =
α+1
α+1
(λbµt + aµt )
(λbνt + aνt )
(3.8)
which can be expanded, with the help of (1.8), as follows:
µ 
b νt
1
−
−λ
a
1
2
1

g(t) = 2α
cµνxt νt 
νt
2
λbµt + aµt
a
λ ab
+1

ν  b µt
α
1  1 − −λ a
2

× µt
cµνyt
µt
a
λbνt + aνt
λ ab
+1
iνt !
α
µ−1
X
a−(µ+ν)t
b
2
i
µνxt
=
(−λ)
c
22α
λbµt + aµt
a
i=0
α
jµt ! ν−1
X
2
b
cµνyt
×
(−λ)j
a
λbνt + aνt
j=0
jµt ln ab
iνt ln ab
(µ + ν)t ln a
α
µ−1 ν−1
X
X
+
−
µνxt
+
1
2
ln
c
ln
c
ln c
(λ)i+j
c
= 2α
2 i=0 j=0
λbµt + aµt
α
2
×
cµνyt
λbνt + aνt
!
µ−1 ν−1
∞
(iν + µj) ln ab − (µ + ν) ln a
(µt)n
1 X X i+j X (α)
= 2α
(λ)
En
νx +
; λ; a, b, c
2 i=0 j=0
µ ln c
n!
n=0
×
∞
X
(α)
Ek (µy; λ; a, b, c)
k=0
=

α
(νt)k
k!
" n
! µ−1 ν−1
∞
1 X X n XX
(α)
(−λ)i+j µn−k ν k Ek (µy; λ; a, b, c)
22α n=0
k i=0 j=0
k=0
!#
(iν + µj) ln ab − (µ + ν) ln a
tn
(α)
; λ; a, b, c
.
× En−k νx +
µ ln c
n!
Using the fact that g(t) is symmetric since µ and ν have the same parity, we can
also expand g(t) in the following the way:
" n
! ν−1 µ−1
∞
1 X X n XX
(α)
g(t) = 2α
(−λ)i+j ν n−k µk Ek (νy; λ; a, b, c)
2 n=0
k i=0 j=0
k=0
!#
(iµ + νj) ln ab − (µ + ν) ln a
tn
(α)
×En−k µx +
; λ; a, b, c
.
ν ln c
n!
(3.9)
n
Equating coefficients of tn! in the right-hand side of the last two equations gives the
identity of the Theorem 5.
Letting a = 1, b = c = e in Theorem 5, we find the following corollary given
recently by Lu and Srivastava in [7, Eq. 43].
86
Corollary 3.5. For n ∈ N0 , µ, ν ∈ N, α ≥ 1 and for λ ∈ C. If µ and ν have the
same parity, then the following identity holds true:
! µ−1 ν−1
n
X
n XX
ν
(α)
(α)
(−λ)i+j µn−k ν k En−k νx + i + j; λ Ek (µy; λ)
k i=0 j=0
µ
k=0
! ν−1 µ−1
n
X
n XX
µ
(α)
(α)
=
(−λ)i+j ν n−k µk En−k µx + i + j; λ Ek (νy; λ).
ν
k i=0 j=0
(3.10)
k=0
According to [20], we have the following analogues of the expansions of hyperbolic
cotangent and hyperbolic tangent, respectively:
∞
X
Bn (0; λ) + λBn (1; λ)
λe2z + 1
=
(2z)n−1 ,
λe2z − 1
n!
n=0
(3.11)
∞
X
(2z)n
λe2z − 1
=
−
+ 1.
E
(0;
λ)
n
λe2z + 1
n!
n=0
(3.12)
From (3.11), we can obtain the following theorem.
Theorem 3.6. For n ∈ N0 , µ, ν ∈ N, α ≥ 1, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C.
Let δi,j denotes the Kronecker delta defined by δi,i = 1 and δi,j = 0 for i 6= j. If µ
is odd and ν is even then the following identity holds true:
!
!
k
n + 1 [δn+1−k,1 + 2Bn+1−k (0; λ)] n−k X k
µi ν n−i
−
µ
i
k
n+1
i=0
k=0
! X
i
n−k+i−j
µ ln a
i
b
(α)
(α−1)
(1)
×Ek−i µx −
(νy; λ; a, b, c)
; λ; a, b, c
ln
Ti−j (ν; λ)Ej
ν ln c
j
a
j=0
! !
X
n
k
k−i
X
n n−k k (α)
ν ln a
k
b
(1)
=
µ
ν En−k νx −
; λ; a, b, c
ln
Tk−i (µ; λ)
k
µ ln c
i
a
i=0
n+1
X
k=0
(α−1)
× Ei
(µy; λ; a, b, c).
(3.13)
Proof. When µ is odd and ν is even, the function g(t), given below, is not symmetric
in µ and ν, so we have on the one hand
α
2
g(t) = 2α−1
cµνxt
2
λbνt + aνt
α−1
2
×
cµνyt
λbµt + aµt
1
b
1 − λ(−1)ν eµνt ln( a )
λbµt + aµt
!
!
µt ln a
α
b
µνxt −
1 − λ(−1)µ eµνt ln( a )
2
ln c
= 2α−1
c
b
2
λbνt + aνt
1 − λ(−1)ν eµνt ln( a )
!
α−1
b
1 − λ(−1)ν eµνt ln( a )
2
×
cµνyt
b
λbµt + aµt
λeµt ln( a ) + 1
n−1 !
∞
X
Bn (0; λ) + λBn (1; λ)
−1
b
= 2α−1
µνt ln
2
n!
a
n=0
!
∞
X
(νt)k
µ ln a
(α)
×
Ek
µx −
; λ; a, b, c
ν ln c
k!
1
k=0
b
1 − λ(−1)µ eµνt ln( a )
b
1 − λ(−1)ν eµνt ln( a )
!
87
!
i ! ∞
X (α−1)
µt ln ab
(µt)j
×
; λ)
Ej
(νy; λ; a, b, c)
i!
j!
i=0
j=0
"
!
∞
n
−1 X X n
= 2α−1
[Bn−k (0; λ) + λBn−k (1; λ)] µn−1−k
2
k
n=0 k=0
!
k
X
k i n−1−i (α)
µ ln a
; λ; a, b, c
×
µν
Ek−i µx −
ν ln c
i
i=0
! #
i
n−1−k+i−j
X i
b
tn−1
(1)
(α−1)
×
ln
Ti−j (ν; λ)Ej
.
(νy; λ; a, b, c)
j
a
n!
j=0
∞
X
(1)
Ti (ν
(3.14)
On the other other hand, we can also expand g(t) as follows:
!
α
b
1 − λ(−1)µ eµνt ln( a )
2
µνxt
c
g(t) = 2α−1
2
λbµt + aµt
λbνt + aνt
α−1
2
×
c µνyt
λbνt + aνt
!
νt ln a
α
b
µνxt −
1 − λ(−1)µ eµνt ln( a )
1
2
ln c
= 2α−1
c
b
2
λbµt + aµt
1 + λeνt ln( a )
α−1
2
c µνyt
×
λbνt + aνt
! ∞
k !
∞
X
X (1)
νt ln ab
(µt)n
1
ν ln a
(α)
= 2α−1
En
νx −
; λ; a, b, c
Tk (µ; λ)
2
µ ln c
n!
k!
n=0
k=0
!
∞
X (α−1)
(νt)i
Ei
(µy; λ; a, b, c)
×
i!
i=0
" n
!
∞
1 X X n n−k k (α)
ν ln a
; λ; a, b, c
= 2α−1
µ
ν En−k νx −
2
µ ln c
k
n=0 k=0
!
#
k−i
k
X
k
b
tn
(1)
(α−1)
ln
(µy; λ; a, b, c)
×
Tk−i (µ; λ)Ei
.
i
a
n!
i=0
1
Since λBn (x + 1; λ) − Bn (x; λ) = nxn−1 (see [14]) then
Bn (0; λ) + λBn (1; λ) = δn,1 + 2Bn (0; λ).
(3.15)
Making use of this last relation (3.15) involving the Apostol-Bernoulli polynomials
n
and numbers and equating the coefficients of tn! in (3.14) and (3.15), we obtain
(3.13).
Substituting a = 1, b = c = e in Theorem 6 furnishes the following corollary.
Corollary 3.7. For n ∈ N0 , µ, ν ∈ N, α ≥ 1 and for λ ∈ C. Let δi,j denotes the
Kronecker delta defined by δi,i = 1 and δi,j = 0 for i 6= j. If µ is odd and ν is even
then the following identity holds true:
!
!
k
n + 1 [δn+1−k,1 + 2Bn+1−k (0; λ)] n−k X k
(α)
−
µ
µi ν n−i Ek−i (µx; λ)
n+1
i
k
i=0
k=0
!
i
X
i
(1)
(α−1)
×
T (ν; λ)Ej
(νy; λ)
j i−j
j=0
n+1
X
88
!
!
n
k
X
X
n n−k k (α)
k
(1)
(α−1)
=
µ
ν En−k (νx; λ)
Tk−i (µ; λ)Ei
(µy; λ).
k
i
i=0
(3.16)
k=0
Letting λ = 1 in (3.16), we get the result obtained by Liu and Wang in [6,
Theorem 2.4].
Corollary 3.8. For n ∈ N0 , µ, ν ∈ N, α ≥ 1. If µ is odd and ν is even then the
following identity holds true:
−2
n+1
X
k=0
k6=n
!
!
!
k
i
X
i
n + 1 Bn+1−k (0) n−k X k
(1)
(α−1)
i n−i (α)
µ
µν
Ek−i (µx)
Ti−j (ν)Ej
(νy)
n+1
i
j
k
i=0
j=0
!
!
n
k
X
X
n n−k k (α)
k
(1)
(α−1)
=
µ
ν En−k (νx)
Tk−i (µ)Ei
(µy).
k
i
i=0
(3.17)
k=0
We can proceed similarly to Theorem 3.6, but using this time (3.12) to establish
the next result.
Theorem 3.9. For n ≥ 1, µ, ν ∈ N, α ≥ 1, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C.
If µ is even and ν is odd then the following identity holds true:
!
!
k
n
X
X
k
µ ln a
n
(α)
n−k
; λ; a, b, c
µi ν n−i Ek−i µx −
En−k (0; λ)µ
ν ln c
i
k
i=0
k=0
!
n−k+i−j
i
X
i
b
(α−1)
(1)
(νy; λ; a, b, c)
×
ln
Ti−j (ν; λ)Ej
j
a
j=0
!
n
X
n n−k k (α)
µ ln a
ν
µ En−k µx −
−
; λ; a, b, c
k
ν ln c
k=0
!
k−i
k
X
k
b
(1)
(α−1)
×
ln
(νy; λ; a, b, c)
Tk−i (ν; λ)Ei
i
a
i=0
!
n
X
n n−k k (α)
ν ln a
µ
ν En−k νx −
=
; λ; a, b, c
k
µ ln c
k=0
!
k
k−i
X
k
b
(1)
(α−1)
×
ln
Tk−i (µ; λ)Ei
(µy; λ; a, b, c).
i
a
i=0
(3.18)
An interesting special case of Theorem 7 is obtained by setting a = 1, b = c = e
and λ = 1 in (3.18).
Corollary 3.10. For n ≥ 1, µ, ν ∈ N, α ≥ 1. If µ is even and ν is odd then the
following identity holds true:
n−1
X
k=0
!
!
!
k
i
X
X
i
n
k
(1)
(α−1)
n−k
i n−i (α)
Ti−j (ν)Ej
(νy)
En−k (0)µ
µν
Ek−i (µx)
k
i
j
i=0
j=0
!
!
n
k
X
X
n n−k k (α)
k
(1)
(α−1)
=
µ
ν En−k (νx)
Tk−i (µ)Ei
(µy)
k
i
i=0
k=0
a result given first by Liu and Wang in [6, Theorem 2.7].
(3.19)
89
Finally, we would like to mention that many other identities can be obtained from
those shown in this paper. As example, it is easy to see that the two following
relationships hold between the Apostol-Bernoulli and Apostol-Euler polynomials
and the Apostol-Bernoulli and Apostol-Genocchi polynomials. Respectively, we
have for n ∈ N0 , µ, ν ∈ N, α ∈ N, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C,
(α)
E(α)
n (x; λ; a, b, c) =
(−2)α n! Bn+α (x; −λ; a, b, c)
(n + α)!
and
(3.20)
(α)
Gn (x; −λ; a, b, c)
.
(3.21)
(−2)α
Let us see two examples of application of these two results. First, combining
(3.20) with Theorem 4 yields
B(α)
n (x; λ; a, b, c) =
Theorem 3.11. For n ∈ N0 , µ, ν ∈ N, α ∈ N, a, b, c ∈ R+ , (a 6= b) and for λ ∈ C.
If µ and ν have the same parity, then the following identity holds true:
!
αν ln a
k (n − k)! B(α)
n
X
n−k+α νx − µ ln c ; −λ; a, b, c
n n−k k
b
(α)
µ
ν
ln
Tk (µ; λ)
k
a
(n − k + α)!
k=0
!
k
(α)
n
ln a
X
(n − k)! Bn−k+α µx − αµ
; −λ; a, b, c (α)
n n−k k
b
ν ln c
Tk (ν; λ).
=
ν
µ ln
a
(n − k + α)!
k
k=0
(3.22)
Next, considering (3.21) with Theorem 1 gives
Theorem 3.12. For all integers µ > 0, ν > 0, α ∈ N, n ≥ 0, for a, b, c ∈ R+
(a 6= b) and for λ ∈ C, we have the following identity:
!
(µ − 1)ν ln a
n
(α)
; −λ; a, b, c µn−k ν k+1
Gn−k νx +
k
µ ln c
k=0
!
k−i
k
X k
b
(α−1)
(µy; −λ; a, b, c) ln
×
Sk−i (µ − 1, λ) Gi
i
a
i=0
!
n
X
(ν − 1)µ ln a
n
(α)
=
Gn−k µx +
; −λ; a, b, c ν n−k µk+1
ν ln c
k
k=0
!
k−i
k
X
k
b
(α−1)
×
Sk−i (ν − 1, λ) Gi
(νy; −λ; a, b, c) ln
.
i
a
i=0
n
X
(3.23)
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B.-Jean Fugère
Department of Mathematics and Computer Science,
Royal Military College, Kingston, Ontario, Canada, K7K 5L0
E-mail address: fugerej@rmc.ca
Sebastien Gaboury
Department of Mathematics and Computer Science,
University of Quebec at Chicoutimi, Quebec, Canada, G7H 2B1
E-mail address: s1gabour@uqac.ca
Richard Tremblay
Department of Mathematics and Computer Science,
University of Quebec at Chicoutimi, Quebec, Canada, G7H 2B1
E-mail address: rtrembla@uqac.ca