Some Relations Involving Hermite-Based Apostol-Genocchi

12 Pages·2012·0.15 MB·English

Applied Mathematical Sciences, Vol. 6, 2012, no. 82, 4091 - 4102
Some Relations Involving Hermite-Based
Apostol-Genocchi Polynomials
S. Gaboury
Department of Mathematics and Computer Sciences
Universit´e du Qu´ebec `a Chicoutimi
Chicoutimi, Qc. Canada G7H 2B1
[email protected]
B. Kurt
Department of Mathematics, Faculty of Science
University of Akdeniz, TR-07058 Antalya, Turkey
[email protected]
Abstract. Inrecentyears,thesubjectofApostol-Bernoullipolynomials,Apostol-Euler
polynomials and Apostol-Genocchi polynomials have been studied by a large number of
mathematicians. Many properties and interesting relationships for these polynomials
have been investigated. In this work, we present some further relations for the gener-
alized Apostol-Genocchi polynomials with parameters a,b and c introduced recently by
Srivastava et al. We also define the Hermite-based Genocchi polynomials with parameters
a, b, c. Some recurrences relations, addition theorem and relevant connections with the
generalized Hurwitz-Lerch zeta function ang the gaussian hypergeometric function are
given.
Mathematics Subject Classification: 05A10, 11B65, 28B99, 11B68
Keywords: Bernoulli numbers and polynomials, Euler numbers and polynomials,
Genocchi numbers and polynomials, Apostol-Bernoulli polynomials, Apostol-Euler poly-
nomials, Apostol-Genocchi polynomials, Hermite-based Apostol-Genocchi polynomials,
Hurwitz lerch zeta function
1. Introduction, Definitions and Notations
The generalized Bernoulli polynomials Bn(α)(x) of order α ∈ C, the generalized Euler
polynomials En(α)(x) of order α ∈ C and the generalized Genocchi polynomials Gn(α)(x) of
order α ∈ C, each of degree n as well as in α, are defined respectively by the following
4092 S. Gaboury and B. Kurt
generating functions (see,[6, vol.3, p.253 et seq.], [13, Section 2.8] and [15]):
(cid:2) (cid:3)
t α (cid:4)∞ tk
(1.1) ·ext = B(α)(x) (|t| < 2π;1α := 1),
et −1 k k!
k=0
(cid:2) (cid:3)
2 α (cid:4)∞ tk
(1.2) ·ext = E(α)(x) (|t| < π;1α := 1)
et +1 k k!
k=0
and
(cid:2) (cid:3)
2t α (cid:4)∞ tk
(1.3) ·ext = G(α)(x) (|t| < π;1α := 1).
et +1 k k!
k=0
The literature contains a large number of interesting properties and relationships in-
volving these polynomials [1, 3, 4, 6, 8, 18]. These appear in many applications in com-
binatorics, number theory and numerical analysis. Q.-M. Luo and Srivastava ([16, 17])
introduced the generalized Apostol-Bernoulli polynomials Bn(α)(x) of order α, Q.-M. Luo
[14] investigated the generalized Apostol-Euler polynomials En(α)(x) of order α and the
generalized Apostol-Genocchi polynomials G(nα)(x) of order α.
The generalized Apostol-Bernoulli polynomials Bn(α)(x;λ) of order α ∈ C, the gen-
eralized Apostol-Euler polynomials E(nα)(x;λ) of order α ∈ C, the generalized Apostol-
Genocchi polynomials G(nα)(x;λ) of order α ∈ C are defined respectively by the following
generating functions
(cid:2) (cid:3)
t α (cid:4)∞ tk
(1.4) ·ext = B(α)(x;λ) (|t+lnλ| < 2π;1α := 1)
λet −1 k k!
k=0
(cid:2) (cid:3)
2 α (cid:4)∞ tk
(1.5) ·ext = E(α)(x;λ) (|t+lnλ| < π;1α := 1)
λet +1 k k!
k=0
and
(cid:2) (cid:3)
2t α (cid:4)∞ tk
(1.6) ·ext = G(α)(x;λ) (|t+lnλ| < π;1α := 1).
λet +1 k k!
k=0
Recently, Srivastava et al. in [20, 21] 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. Let a, b, c ∈ R+, (a (cid:3)= b) and n ∈ N . The generalized Apostol-Bernoulli
0
polynomials B(nα)(x;λ;a,b,c) oforder α, thegeneralizedApostol-Euler polynomialsEn(α)(x;λ;a,b,c)
of order α and the generalized Apostol-Genocchi polynomials Gn(α)(x;λ;a,b,c) of order α
are defined respectively by the following generating functions
(cid:2) t (cid:3)α (cid:4)∞ tn (cid:5)(cid:6)(cid:6) (cid:5)a(cid:7) (cid:6)(cid:6) (cid:7)
(1.7) cxt = B(α)(x;λ;a,b,c) , (cid:6)tln +lnλ(cid:6) < 2π;1α := 1 ,
λbt −at n n! b
n=0
Some relations involving Hermite-based Apostol-Genocchi polynomials 4093
(cid:2) (cid:3)
2 α (cid:4)∞ tn (cid:5)(cid:6)(cid:6) (cid:5)a(cid:7) (cid:6)(cid:6) (cid:7)
(1.8) cxt = E(α)(x;λ;a,b,c) , (cid:6)tln +lnλ(cid:6) < π;1α := 1
λbt +at n n! b
n=0
and
(cid:2) 2t (cid:3)α (cid:4)∞ tn (cid:5)(cid:6)(cid:6) (cid:5)a(cid:7) (cid:6)(cid:6) (cid:7)
(1.9) cxt = G(α)(x;λ;a,b,c) , (cid:6)tln +lnλ(cid:6) < π;1α := 1 .
λbt +at n n! b
n=0
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 B (x), classical Euler polynomials E (x)
n n
and classical Genocchi polynomials G (x).
n
Recently, LinandSrivastava[11]introducedthefollowinggeneralizationoftheHurwitz-
Lerch zeta function Φ(z,s,a)
(cid:4)∞ (μ) zn
(1.10) Φ(ρ,σ)(z,s,a) := ρn
μ,ν (ν) (n+a)s
σn
n=0
(cid:8)
μ ∈ C; a,ν ∈ Z−; ρ,σ ∈ R+; ρ < σ when s,z ∈ C;
0
(cid:9)
ρ = σ and s ∈ C when |z| < 1; ρ = σ and Re(s−μ+ν) when |z| = 1 ,
where (λ) denotes the Pochhammer symbol defined by
ν
Γ(λ+ν)
(λ) := ; (λ) = 1.
ν Γ(λ) 0
It is obvious that
(1.11) Φ(σ,σ)(z,s,a) = Φ(0,0)(z,s,a) = Φ(z,s,a)
ν,ν μ,ν
and
(cid:4)∞ (μ) zn
(1.12) Φ(1,1)(z,s,a) = Φ∗ (z,s,a) := n .
μ,1 μ n! (n+a)s
n=0
The reader should see the works of Garg et al. [7] and Lin et al. [12] for further results
involving the generalized Hurwitz-Lerch zeta function. Making use of this generalization,
Srivastava et al. in[20, 21] gave relationships between thesepolynomialsandthe Hurwitz-
Lerch zeta function. They also obtained other relationships, in particular, with the Gauss
hypergeometric function and the Stirling numbers of the second kind.
Finally, Brychkov in [3] by considering the well-known formula for the product of power
(cid:4)∞
series, that is, if f (z) = a(i)zk, then
i k
k=0
(cid:10)m (cid:4)∞
(1.13) f (z) = b zk,
i k
i=1 k=0
4094 S. Gaboury and B. Kurt
where
(cid:4) (cid:10)m a(i)
(1.14) b = k zk,
k k !
i
k1+···+km=k i=1
obtained the following relation for the generalized (classical) Bernoulli polynomials:
(cid:4) (cid:10)m
1 1
(1.15) B(αi)(z ) = B(α)(z)
k ! ki i n! n
i
k1+···+km=ni=1
with α = α +···+α and z = z +···+z .
1 m 1 m
In this paper, we give further results involving the generalized Apostol-Genocchi poly-
nomials G(nα)(x;λ;a,b,c). A multiplication formula analogue of that of Brychkov is
given. We also define the generalized Hermite based Apostol-Genocchi polynomials
HG(nα)(x,y;λ;a,b,c) and we present several properties for that new class of polynomi-
als. In particular, we obtain some relationships involving the generalized Hurwitz-Lerch
zeta function Φ∗ (z,s,a) and the gaussian hypergeometric function.
μ
2. Some properties and multiplication formula for the generalized
Apostol-Genocchi polynomials
In this section, we present some further properties (Theorem 2 and Theorem 3) for the
generalized Apostol-Genocchi polynomials Gn(α)(x;λ;a,b,c). We also give, in Theorem 4,
a multiplication formula based on the result obtained recently by Brychkov [3].
Theorem 1. Let a, b, c ∈ R+, α and β an arbitrary complex number and n ∈ N .
0
Then the generalized Apostol-Genocchi polynomials Gn(α)(x;λ;a,b,c) satisfy the following
addition theorem
(cid:2) (cid:3)
(cid:4)n n
(2.1) G(α+β)(x+y;λ;a,b,c) = G(α)(x;λ;a,b,c)G(β) (y;λ;a,b,c).
n k k n−k
k=0
The proof follows easily from (1.9).
Theorem 2. Let a, b, c ∈ R+, α an arbitrary complex number and n,r ∈ N . Then the
0
generalized Apostol-Genocchi polynomials Gn(α)(x;λ;a,b,c) satisfy he following relation
(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)
n (cid:4)∞ m+r −1 b n−r
(2.2) G(r)(x;λ;a,b,c) = 2r r! mln +xlnc−rlna .
n r m a
m=0
Proof. From (1.9), we have
(cid:4)∞ tn
G(r)(x;λ;a,b,c)
n n!
n=0
Some relations involving Hermite-based Apostol-Genocchi polynomials 4095
(cid:2) (cid:3)
(cid:5) (cid:7)
2t r −r
= cxt = (2t)re−trlna 1+λetlnab extlnc
λbt +at
(cid:11) (cid:12)
(cid:2) (cid:3) (cid:2) (cid:3)
(cid:4)∞ (cid:4)∞ m+r −1 b n tn
= 2rtr (−λ)m mln +xlnc−rlna
m a n!
(cid:11)n=0 m=0 (cid:12)
(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)
(cid:4)∞ n+r (cid:4)∞ m+r−1 b n tn+r
= 2r r! mln +xlnc−rlna .
r m a (n+r)!
n=0 m=0
Now, upon replacing n by n−r (n ≥ r) and comparing the coefficient of tn, we arrive
at (2.2).
If we put a = 1, b = c = e in (2.2), we recover a result obtained by Ozden et al. in [19].
Theorem 3. Let a, b, c ∈ R+, (a (cid:3)= b), α an arbitrary complex number and for n ∈
N . The following relation holds between the generalized Apostol-Genocchi polynomials
0
G(nα)(x;λ;a,b,c), the generalized Apostol-Euler polynomials En(α)(x;λ;a,b,c) and the gen-
eralized Apostol-Bernoulli polynomials Bn(α)(x;λ;a,b,c)
(cid:2) (cid:3)
(cid:4)n n
(2.3) G(α)(x;λ2;a,b,c) = 2α−n E(α) (2x;λ;a,b,c)B(α)(0;λ;a,b,c).
n k n−k k
k=0
Proof. From (1.7), (1.8) and (1.9), we write
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
(cid:4)∞ tn 2t α 2 α t/2 α
G(α)(x;λ2;a,b,c) = cxt = cxt·2α
n n! λ2bt+at λbt/2+at/2 λbt/2−at/2
n=0
(cid:4)∞ tn (cid:4)∞ tk
= 2α E(α)(2x;λ;a,b,c) · B(α)(0;λ;a,b,c)
n 2nn! n 2kk!
n(cid:13)=0 k=0 (cid:14)
(cid:2) (cid:3)
(cid:4)∞ (cid:4)n n tn
(2.4) = 2α−n E(α) (2x;λ;a,b,c)B(α)(0;λ;a,b,c) .
k n−k k n!
n=0 k=0
tn
Comparing the coefficients of , we obtain (2.3).
n!
In view of the result given by Brychkov [3], the next theorem holds for the generalized
Apostol-Genocchi polynomials G(nα)(x;λ;a,b,c).
Theorem 4. The generalizedApostol- Genocchipolynomials Gn(α)(x;λ;a,b,c) satisfies the
following multiplication relation
(cid:4) (cid:10)m
1 1
(2.5) G(αi)(x;λ;a,b,c) = G(α)(x;λ;a,b,c).
k ! i n! k
i
k1+···+km=ni=1
4096 S. Gaboury and B. Kurt
Proof. From (1.9) and (1.13)-(1.15),
(cid:2) (cid:3) (cid:2) (cid:3)
(cid:10)m (cid:4)∞ tk (cid:10)m 2t αi 2t α1+···+αm
G(αi)(x ;λ;a,b,c) = cxit = c(x1+···+xm)t
ki i k! λbt +at λbt +at
i=1k =0 i=1
i (cid:2) (cid:3)
2t α (cid:4)∞ tn
= cxt = G(α)(x;λ;a,b,c) ,
λbt +at n n!
n=0
where α +···+α = α, k +···+k = n. Comparing the coefficients, we have (2.5).
1 m 1 m
Similiary, for the generalized Apostol-Bernoulli polynomials Bn(α)(x;λ;a,b,c) and the
generalized Apostol-Euler polynomials En(α)(x;λ;a,b,c)
(cid:4) (cid:10)m
1 1
(2.6) B(αi)(x;λ;a,b,c) = B(α)(x;λ;a,b,c)
k ! i n! n
i
k1+···+km=ni=1
and
(cid:4) (cid:10)m
1 1
(2.7) E(αi)(x;λ;a,b,c) = E(α)(x;λ;a,b,c).
k ! i n! n
i
k1+···+km=ni=1
3. Hermite-based Apostol-Genocchi polynomials
In this section, we recall the definition of the two variable Hermite-Kamp´e de F´eriet
polynomials (2VHKdFP) H2(x,y). Next, we define and give some properties of the gen-
n
eralized Hermite-based Apostol-Genocchi polynomials HGn(α)(x,y;λ;a,b,c). Finally, we
establish some relations between this new class of polynomials and the Hurwitz-Lerch
zeta function and the familiar gaussian hypergeometric function.
The two variable Hermite-Kamp´e de F´eriet polynomials (2VHKdFP) H2(x,y) (see,
n
[2, 5, 23]) is defined by
[n]
(cid:4)2 yrxn−2r
(3.1) H(2)(x,y) = n!
n r!(n−2r)!
r=0
with the following generating function
(cid:4)∞ tn
(3.2) H(2)(x,y) = ext+yt2.
n n!
n=0
WedefinethegeneralizedHermite-basedApostol-GenocchipolynomialsHGn(α)(x,y;λ;a,b,c)
by means of the following generating function.
Definition 2. Let a, b, c ∈ R+, (a (cid:3)= b), α an arbitrary complex number and n ∈ N . The
0
generalized Hermite-based Apostol-Genocchi polynomials HGn(α)(x,y;λ;a,b,c) are defined
Some relations involving Hermite-based Apostol-Genocchi polynomials 4097
by
(3.3)
(cid:2) (cid:3)
(cid:4)∞ tn 2t α (cid:5)(cid:6)(cid:6) (cid:5)a(cid:7) (cid:6)(cid:6) (cid:7)
G(α)(x,y;λ;a,b,c) = cxt+yt2 (cid:6)tln +lnλ(cid:6) < π;1α := 1 .
H n n! λbt +at b
n=0
Obviously, if we set a = 1, b = c = e in (3.3), we recover the classical Hermite-based
Apostol-Genocchi polynomials investigated by Khan et al. in [10].
From (3.3) and with the help of the following elementary serie identity [22, p. 100]
[n]
(cid:4)∞ (cid:4)∞ (cid:4)∞ (cid:4)2
(3.4) A(k,n) = A(k,n−2k)
n=0 k=0 n=0 k=0
we find
[n]
(cid:4)2 G(α) (x,λ;a,b,c)(y ·lnc)l
(3.5) G(α)(x,y;λ;a,b,c) = n! n−2l .
H n l!(n−2l)!
l=0
Theorem 5. Let a, b, c ∈ R+, α and β arbitrary complex numbers n ∈ N . The general-
0
izedHermite-based Apostol-GenocchipolynomialsHGn(α)(x,y;λ;a,b,c)satisfy thefollowing
relation
(3.6)
(cid:2) (cid:3)
(cid:4)n n
G(α+β)(x +x ,y +y ;λ;a,b,c) = G(α)(x ,y ;λ;a,b,c) G(β)(x ,y ;λ;a,b,c).
H n 1 2 1 2 k H n 1 1 H n 2 2
k=0
Proof. From (3.3), the result follows easily.
Theorem 6. For a,b,c ∈ R+, a (cid:3)= b, l ∈ N , the generalized Hermite-based Apostol-
0
Genocchi polynomials
HGn(l)(x,y;λ;a,b,c) and the generalized Hurwitz-Lerch zeta function Φ∗μ(z,s,a) are related
by
(3.7)
[n] (cid:8) (cid:9) (cid:2) (cid:2) (cid:3)(cid:3)
(cid:4)2 Φ∗ −λ,l−n+2j, xlnc−llna b n−l−2j
G(l)(x,y;λ;a,b,c) = 2ln! l lnb−lna · ln (ylnc)j.
H n (n−l−2j)!j! a
j=0
Proof. From (3.3), we write as
(cid:4)∞ tn
G(l)(x,y;λ;a,b,c)
H n n!
n=0
4098 S. Gaboury and B. Kurt
(cid:2) (cid:3)
(cid:5) (cid:7)
2t l −l
= cxt+yt2 = (2t)le−tllna 1+λetlnab extlnceyt2lnc
λbt +at
(cid:2) (cid:2) (cid:3)(cid:3)
(cid:4)∞ (l) (−λ)k (cid:4)∞ b n tn (cid:4)∞ t2j
= 2ltl k xlnc−llna+kln · (ylnc)j
k! a n! j!
k=0 n=0 j=0
(cid:2) (cid:2) (cid:3)(cid:3)
(cid:4)∞ (cid:4)∞ (cid:4)∞ (l) (−λ)k b n tn+l+2j
= 2l k xlnc−llna+kln ·(ylnc)j
k! a n!j!
n=0 j=0 k=0
(cid:2) (cid:2) (cid:3)(cid:3)
(cid:4)∞ (cid:4)∞ (cid:4)∞ (l) (−λ)k b n tn+l+2j
= 2l k xlnc−llna+kln ·(ylnc)j
k! a (n)!j!
n=0 j=0 k=0
(cid:11) (cid:8) (cid:9) (cid:2) (cid:2) (cid:3)(cid:3) (cid:12)
(cid:4)∞ (cid:4)∞ Φ∗ −λ,−n, xlnc−llna b n
= 2l l lnb−lna · ln ·(ylnc)j tn+l+2j.
(n)!j! a
n=0 j=0
Now, upon replacing n by n − l (n ≥ l), using equation (3.5) and comparing the
coefficient of tn, we arrive at Theorem 6.
Theorem 7. For n ∈ N , a,b,c ∈ R+, the generalized Hermite-based Apostol-Genocchi
0
polynomials HGn(l)(x,y;λ;a,b,c) and the gaussian hypergeometric function (see for details,
[22, p. 29])
(cid:4)∞ (α) (β) zn
(3.8) F (α,β;γ;z) = n n
2 1 (γ) n!
n
n=0
where (λ) denotes the Pochhammer symbol defined by
ν
Γ(λ+ν)
(λ) := ; (λ) = 1
ν Γ(λ) 0
are related by the following relation
HG(nl)(x,y;λ;a,b,c)
(cid:4)[n2] (y·lnc)i 2ln−2i! n−(cid:4)2i−l(cid:2)n−2i−l(cid:3)(cid:2)k+l−1(cid:3) λk (cid:15) (cid:2)b(cid:3)(cid:16)k
=n! · ln
i!(n−2i)! (n−2i−l)! k k (λ+1)k+l a
i=0 k=0
(3.9) (cid:2) (cid:3) (cid:13) (cid:8) (cid:9) (cid:14)
(cid:4)k k jln b
·(xlnc−llna)n−2i−k−l· (−1)j j jk 2F1 k+l−n−2i,1;k+1;−xlnc−allna .
j=0
Proof. Making use of equation (3.5) and the following result obtained by Srivastava et al.
in [21, p. 301, Eq. 74] for the Apostol-Genocchi polynomials
(3.10)
(cid:2) (cid:3)(cid:2) (cid:3) (cid:15) (cid:2) (cid:3)(cid:16)
2ln! (cid:4)n−l n−l k+l−1 λk b k
G(l)(x,λ;a,b,c) = ln (xlnc−llna)n−k−l
n (n−l)! k k (λ+1)k+l a
k=0
(cid:2) (cid:3) (cid:13) (cid:8) (cid:9) (cid:14)
(cid:4)k k jln b
· (−1)j j jk 2F1 k+l−n,1;k+1;−xlnc−allna
j=0
Some relations involving Hermite-based Apostol-Genocchi polynomials 4099
the result follows.
At this point, we must recall a basic identity which is needed for the sequel.
Lemma 1. (Generalized multinomial identity [4, p. 41]) If x , x ,..., x are commuting
1 2 r
elements of a ring, then for α ∈ C, we have
(cid:2) (cid:3)
(cid:4)
α
(3.11) (1+x +···+x )α = xν1···xνm
1 m ν ,··· ,ν 1 m
1 m
ν1,···,νm≥0
where summation takes place over all integers ν ≥ 0, with
i
(cid:2) (cid:3)
α α(α−1)(α−2)...(α−ν −ν −···−ν +1)
(3.12) = 1 2 m .
ν ,··· ,ν ν ! ν !...ν !
1 m 1 2 m
Theorem 8. For m ∈ N, n ∈ N , a,b,c ∈ R+, the following multiplication formula for
0
the Hermite-based Apostol-Genocchi polynomials of higher order holds true;
G(α)(mx,y;λ;a,b,c) =
H n
(3.13)
(cid:2) (cid:3) (cid:2) (cid:3)
mn−α (cid:4) ν1,···α,νm−1 (−λ)r HG(nα) x+ r(lnb−lnam)+lnαc(m−1)lna,my2;λm;a,b,c
ν1+···+νm−1≥0
where r = ν +2ν +···+(m−1)ν .
1 2 m−1
Proof. For m odd, we have
m(cid:17)−1(cid:8) (cid:9)
−λemt(lnb−lna) k
2t
(3.14) = 2te−lna k=0 .
λbt +at 1+(λet(lnb−lna))m
From the generalized multinomial identity (3.11), the last equation (3.14) and the gener-
ating function for the generalized Hermite-based Apostol-Genocchi polynomials (3.3), we
have
(cid:4)∞ tn
HGn(α)(mx,y;λ;a,b,c) n!
n=0
4100 S. Gaboury and B. Kurt
(cid:2) (cid:3)
= 2t αcmxt+yt2
λbt+at
(cid:13) (cid:14)
= (cid:2) 2te−lna (cid:3)α m(cid:4)−1(−λ)kekt(lnb−lna) αe(mxt+yt2)lnc
1+λmemt(lnb−lna)
k=0
(cid:2) (cid:3) (cid:2) (cid:3)
= (cid:4) m−α α (−λ)r 2tm αcmt x+r(lnb−lnam)+lnαc(m−1)lna +y(mm2t)2
ν1,··· ,νm−1 λmbmt+atm (cid:0) (cid:2)
ν1,···⎛,νm≥0
(cid:2) (cid:3)
(cid:4)∞ (cid:4)
α
= ⎝mn−α (−λ)r
ν ,··· ,ν
1 m−1
n=0 ν +···+ν ≥0
(cid:2) 1 m−1 (cid:3)(cid:3)
r(lnb−lna)+α(m−1)lna y tn
×HG(nα) x+ mlnc ,m2;λm;a,b,c n!.
tn
Comparing the coefficients of , the result follows.
n!
If we put λ = 1 and y = 0 in theorem 9, we obtain a result given by Jolany et al. in [9],
namely
G(α)(mx;a,b,c)
n
(3.15)
(cid:2) (cid:3) (cid:2) (cid:3)
(cid:4)
α r(lnb−lna)+α(m−1)lna
= mn−α (−1)r G(α) x+ ;a,b,c
ν ,··· ,ν n mlnc
1 m−1
ν +···+ν ≥0
1 m−1
where G(nα)(x;a,b,c) are the Genocchi polynomials of higher order.
Corollary 1. For a, b, p, q ∈ R+, n ∈ N and α ∈ C, the following equation holds true.
G(α)(px,qy;λ;a,b,c)
H m
(3.16)
[k]
(cid:4)n (cid:4)2 tn
= n! G(α) (x,y;λ;a,b,c)((p−1)xlnc)k−2j((q −1)ylnc)j .
H n−k (n−k)!(k −2j)!j!
k=0 j=0
Proof. Using definition (3.3), we have
(cid:2) (cid:3)
(cid:4)∞ tn 2t α
(3.17) G(α)(px,qy;λ;a,b,c) = cxt+yt2c(p−1)xtc(q−1)yt2.
H n n! λbt +at
n=0
Expanding the right hand side in power series and using series rearrangement techniques,
we obtain
(cid:4)∞ tn
G(α)(x,y;λ;a,b,c)
H n n!
n=0
(cid:13) (cid:14)(cid:13) (cid:14)(cid:13) (cid:14)
(cid:4)∞ tn (cid:4)∞ tk (cid:4)∞ t2j
= G(α)(x,y;λ;a,b,c) ((p−1)xlnc)k ((q−1)ylnc)j
H n n! k! j!
n=0 k=0 j=0