Modified Apostol-Euler numbers and polynomials of higher order Maged G. Bin-Saad1 and Ali Z. Bin-Alhag2 Abstract The purpose of this work is to give modified definitions of Apostol-Euler polynomials and numbers of higher order. We establish their elementary properties, sums, explicit relations, integrals and differential relations. A number of new results which introduced are generalization of known results and their special cases lead to the corresponding formulas of the classical Euler numbers and polynomials.
2010 MSC: Primary 11B68, Secondary 33C05, 11B73. Key Words. Euler polynomials; Euler polynomials of higher order, Apostol-Euler polynomials; Apostol-Euler polynomials of higher order. 1. Introduction and Definitions (µ)
The classical Euler polynomials En (a), Euler polynomials of higher order En (a) and Apostol-Euler polynomials En (a; x) are defined by the following generating function(see [13, p. 63-66, Eq (39),(40),(64)]), also (see [14]): ∞ X 2eaz zn = E (a) , n ez + 1 n=0 n!
2 z e +1
µ
eaz =
∞ X
En(µ) (a)
n=0
∞ X 2eaz zn = E (a; x) , n xez + 1 n=0 n!
(|z| < π), zn , n!
(1.1)
(|z| < π),
(1.2)
(|z + log x| < π),
(1.3)
respectively. In [7] Luo introduced analogous definitions for the classical Euler numbers and polynomials of higher order by using Apostol’s idea as follows (see [2, p. 165 (3.1)]): (µ)
Definition 1.1. (Luo [7]). Apostol-Euler polynomials of higher order En (a; x) are defined by means of the generating function:
2 z xe + 1
µ
eaz =
∞ X
En(µ)(a; x)
n=0
zn , n!
(|z + log x| < π),
(1.4)
with, of course, En(µ) (a) = En(µ) (a; 1), En (a) = En(1)(a; 1) and En (a; x) := En(1)(a; x), 1 Department
of Mathematics, Aden University, Kohrmakssar P.O.Box 6014,Yemen E-mail:
[email protected] 2 Department of Mathematics, Aden University, Kohrmakssar P.O.Box 6014,Yemen E-mail:
[email protected]
(1.5)
2
(µ)
Definition 1.2. Apostol-Euler numbers of higher order En (x) are defined by means of the generating function: µ X ∞ 2ez zn = En(µ) (x) , (|2z + log x| < π). (1.6) 2z xe + 1 n! n=0 with, of course, En(µ) = En(µ) (1), En = En(1)(1) and En (x) := En(1)(x),
(1.7)
(µ)
where En , En and En (x) denote the so-called Euler numbers of higher order, classical Euler numbers and Apostol-Euler numbers respectively. Certain properties of the Apostol-Euler polynomials such as asymptotic estimates, fourier expansions, multiplication formulas etc. are studied by several researchers, see for example [3, 5, 7, 8, 9, 11]. In this work we similarly give the analogous and modified definitions for the classical Euler numbers and polynomials of higher order by using Apostol’s idea as follows (µ;λ)
Definition 1.3. A Modified Apostol-Euler polynomials of higher order En defined by means of the generating function: µ ∞ X zn 2 az (µ;λ) e = E (a; x) , (λ 6= 0; |z + log x| < π), n xeλz + 1 n! n=0
(a; x) are
(1.8)
It easily follows from (1.8) that En(µ;λ) (a; x) = 2µ
∞ X
(µ)k (a + λk)n
k=0
(−x)k k!
(1.9)
In its special cases when λ = 1, (1.8) reduces to (1.4). Whereas, for µ = λ = 1, (1.8) reduces to (1.3). For x = µ = λ = 1, (1.8) reduces to (1.1). For x = λ = 1, (1.8) reduces to (1.2). (µ;λ)
Definition 1.4. A Modified Apostol-Euler numbers of higher order En defined by means of the generating function: µ X ∞ 2ez zn = En(µ;λ) (x) , (λ 6= 0, |2z + log x| < π). 2λz xe +1 n! n=0
(x) are
(1.10)
It easily follows from (1.10) that En(µ;λ) (x) = 2µ+n
∞ X
k=0
(µ)k
µ 2
+ λk
n (−x)k k!
(1.11)
which, in the special case when λ = 1, (1.10) reduces to (1.6). The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the modified Apostol-Euler polynomials and numbers of higher order defined in (1.8) and (1.10). Some elementary properties, sums, explicit relations and expansions involving classical functions and
3
hypergeometric series for these functions are established. Some particular cases and consequences of our main results are also considered. 2. Some elementary properties In the present section we can readily proof of the following results in a straightforwad way by using the formulas (1.8)-(1.11). Theorem 2.1. For the Modified Apostol-Euler polynomials and numbers defined by (1.8) and (1.10) the following result holds µ En(µ;λ) (x) = 2n En(µ;λ) ;x . (2.1) 2
where µ 6= 0. P roof. If in formula (1.8), we replace a by µ2 , and z by 2z, then in view of definition (1.10) we get (2.1). In its special cases when λ = 1, (2.1) reduces to the formula (see [7, p. 918 (3)]): µ En(µ) (x) = 2n En(µ) ;x . (2.2) 2 It is easily seen that
En(0;λ)(a; x) = an .
(2.3)
Corollary 2.1. Z
(µ;λ)
(µ;λ)
β
En+1 (β; x) − En+1 (α; x) . n+1
En(µ;λ) (a; x) da = α
(2.4)
P roof. Integrating both the sides of equation (1.9) with respect to a, from α to β, we get (2.4). Theorem 2.2. (Additionformula). If n is positive integer, then En(µ+ν;λ) (a + b; x) =
n X n (µ;λ) (ν;λ) Ek (a; x)En−k (b; x). k
(2.5)
k=0
P roof. The generating relation (1.8) yields ∞ X
En(µ+ν;λ) (a
n=0
=
zn + b; x) = n!
µ
2 xeλz + 1
∞ X ∞ X
eaz .
2 xeλz + 1 2 xeλz + 1
µ+ν ν
e(a+b)z
ebz
z n+k n!k! n=0 k=0 " n # ∞ X X n (µ;λ) zn (ν;λ) = Ek (a; x)En−k (b; x) . k n! n=0 =
k=0
(µ;λ)
Ek
(a; x)En(ν;λ)(b; x)
(2.6)
4 n
We now compare the coefficients of zn! on both side of (2.6), and we obtain the formula (2.5) immediately. For λ = 1, (2.5) reduces to the formula (see [7, p. 919 (9)]): n X n (µ) (ν) En(µ+ν) (a + b; x) = Ek (a; x)En−k (b; x). (2.7) k k=0
Corollary 2.2. If n is a positive integer. Then n X n (µ;λ) (µ;λ) En (a + b; x) = bn−k Ek (a; x). k
(2.8)
k=0
For λ = 1, (2.10) reduces to the formula (see [14, p. 95 (38)]): n X n (µ) (µ) En (a + b; x) = bn−k Ek (a; x). k
(2.9)
k=0
Theorem 2.3. (Difference equation). Let a 6= −λ; λ 6= 0. Then xEn(µ;λ) (a + λ; x) + En(µ;λ) (a; x) = 2En(µ−1;λ)(a; x).
(2.10)
P roof. The generating relation (1.8) yields ∞ h X
xEn(µ;λ) (a
+ λ; x) +
i zn
En(µ;λ) (a; x)
n=0
= =2
2 xeλz
+1
n!
2
=x µ
xeλz
+1
µ−1
eaz =
n
2 xeλz + 1
µ
eaz xeλz + 1
∞ X
n=0
(a+λ)z
e
2En(µ−1;λ)(a; x)
+
zn . n!
2 xeλz + 1
µ
eaz
(2.11)
We now compare the coefficients of zn! on both side of (2.11), and we obtain the formula (2.10) immediately. For λ = 1, (2.10) reduces to the formula (see [7, p. 918 (5)]): xEn(µ)(a + 1; x) + En(µ) (a; x) = 2En(µ−1) (a; x). Next, by combining (2.8) and (2.10) ( with b = λ ), we find that " n # X n 1 (µ−1;λ) (µ;λ) n−k (µ;λ) En (a; x) = x λ Ek (a; x) + En (a; x) . k 2
(2.12)
(2.13)
k=0
which, in the special cases when λ = 1, (2.13) reduces to the formula (see [14, p. 96 (42)]): " n # X n (µ) 1 (µ−1) (µ) En (a; x) = x Ek (a; x) + En (a; x) . (2.14) k 2 k=0
For µ = λ = 1, (2.13) reduces to the formula (see [14, p. 96 (43)]): " n # X n 1 n a = x Ek (a; x) + En (a; x) . k 2 k=0
(2.15)
5
Theorem 2.4. (T wo recursion formulas). Let a 6= −λ; λ 6= 0. Then (µ;λ)
En+1 (a; x) = aEn(µ;λ)(a; x) −
λµx (µ+1;λ) E (a + λ; x), 2 n
(2.16)
2 (µ;λ) 2(λµ − a) (µ+1;λ) E (a; x) + En (a; x). (2.17) λµ n+1 λµ P roof. Denote, for the convenience, the right-hand side of formula (2.16) by I. Then in view of formula (1.9), it is easily seen that: En(µ+1;λ) (a; x) =
I = a2µ
∞ X
∞
(µ)k (a + λk)n
k=0
k+1 X (−x)k n (−x) + λ2µ (µ)k+1 (a + λ(k + 1)) . k! k!
(2.18)
k=0
Now, let k → k − 1 in the second summation of (2.18) we get I = 2µ
∞ X
(µ)k (a + λk)n+1
k=0
(−x)k . k!
Then in view of formula (1.9), we are finally led to the left-hand side of formula (2.16). Similarly, one can prove the formula (2.17). For λ = 1, (2.16) and (2.17) reduce to the formulas (see [7, p. 919 (12), (13)]): µx (µ+1) (µ) E (a + 1; x), (2.19) En+1 (a; x) = aε(µ) n (a; x) − 2 n En(µ+1) (a; x) =
2 (µ) 2(µ − a) (µ+1) E (a; x) + En (a; x), µ n+1 µ
(2.20)
respectively. Theorem 2.5. Let a 6= λµ. Then En(µ;λ) (λµ − a; x) =
(−1)n (µ;λ) E (a; x−1). xµ n
(2.21)
P roof. The generating relation (1.8) yields ∞ X
En(µ;λ) (λµ − a; x)
n=0
=
zn = n!
2eλz xeλz + 1
2 xeλz + 1
µ
µ
e(λµ−a)z
e−az
µ 1 2 = µ e−az x x−1 e−λz + 1 ∞ n X (−1)n (µ;λ) −1 z = E (a; x ) . n xµ n! n=0
(2.22)
n
We now compare the coefficients of zn! on both side of (2.22), and we obtain the formula (2.21) immediately. For λ = 1, (2.21) reduces to the formula (see [7, p. 919 (10)]): En(µ) (µ − a; x) =
(−1)n (µ) E (a; x−1 ). xµ n
(2.23)
6
3. Explicit Representations In this section, we geve some explicit formulas for the modified Apostol-Euler polynomials and numbers of higher order. Theorem 3.1. If n, is a positive integer and µ and x are arbitrary real or complex parameters, then we have (µ;λ) n X Ek (x) µ n−k n En(µ;λ) (a; x) = a − . (3.1) k 2k 2 k=0
P roof. Denote, for the convenience, the right-hand side of formula (3.1) by I. Then in view of Theorem 2.1 and formula (1.9), it is easily seen that: n ∞ k X µ n−k µ (−x)m X n + λm a− I= 2µ (µ)m k m! 2 2 m=0 k=0
=
∞ X
2µ (µ)m (a + λm)n
m=0
(−x)m . m!
Hence, the left-hand side of formula (3.1) follows. For λ = 1, (3.1) reduces to the formula (see [7, p. 918 (3)]): (µ) n X Ek (x) µ n−k n a − . En(µ) (a; x) = k 2k 2
(3.2)
k=0
Theorem 3.2. F or {µ, n} ∈ N, we have 1
(n;λ)
En−1 (a; x) =
2(µ−1)n
(µ−1)n
X
k=0
(µ − 1)n k
(µn;λ)
xk En−1 (a + λk; x).
(3.3)
P roof. Denote, for the convenience, the right-hand side of formula (3.3) by I. Then in view of formula (1.9), it is easily seen that: ∞ (µ−1)n X X (−x)m+k (µ − 1)n k I= (−1) 2n (µn)m (a + λ(k + m))n−1 . k m! m=0
k=0
Now, letting m → m − k in (3.4) using the formulas µ (−µ) (−1)k (µ) = (−1)kkk! ; (µ)m−k = (1−µ−m)mk and (−m)k = k I=
∞ X
2n (µn)m (a + λm)n−1
m=0
(−1)k m! (m−k)!
we get
(µ−1)n (−x)m X (n − µn)k (−m)k 1m . m! (1 − µn − m)k m! k=0
Now, using the formulas n X (a)k (b)k 1k Γ(c)Γ(c − a − b) = ; (c)k k! Γ(c − a)Γ(c − b) k=0
Γ(a − k) = (a)−k Γ(a)
(3.4)
7 k
(−1) and (a)−k = (1−a) , gives us the left-hand side of formula (3.3). k For s = λ = 1, (3.3) reduces to the formula (see [6, p. 8 (2.9)]):
1
(n)
En−1 (a; x) =
2(µ−1)n
(µ−1)n
X
k=0
(µ − 1)n k
(µn)
xk En−1 (a + k; x).
(3.5)
Theorem 3.3. If n is a positive integer and µ and x are arbitrary real or complex parameters, then we have En(µ;λ)
n X a n−k (µ;λ) n ;x = − Ek (a; x). k 2 2
a
(3.6)
k=0
P roof. The generating relation (1.8) yields ∞ X
z a En(µ;λ) ( ; x) 2
n=0
n
n!
2
=
2 xeλz
µ
+1
µ
a
e2z
a
eaz .e− 2 z xeλz + 1 ∞ ∞ X z n+k a n X (µ;λ) = − Ek (a; x) . 2 n!k! n=0 k=0 " n # ∞ X X n a n−k (µ;λ) zn (a; x) . = − Ek k 2 n! n=0 =
(3.7)
k=0
n
We now compare the coefficients of zn! on both side of (3.7), and we obtain the formula (3.6) immediately. For λ = 1, (3.6) reduces to the formula (see [7, p. 922 (27)]): En(µ)
n X a n−k (µ) n ;x = − Ek (a; x). k 2 2
a
(3.8)
k=0
Theorem 3.4. F or k ∈ N0 , we have ∞ X
(µ;λ)
En+k (a; x)
k=0
wk = eaw En(µ;λ) (a; xeλw ). k!
(3.9)
P roof. Denote, for the convenience, the right-hand side of formula (3.9) by I. Then, in view of formula (1.9), it is easily seen that: I=
∞ X
m=0
=
∞ X ∞ X
k=0 m=0
2µ (µ)m (a + λm)n
(−x)m (a+λm)w e m!
2µ (µ)m (a + λm)n+k
(−x)m w k . m! k!
(3.10)
8
Now, with the help of definition (1.9), equation (3.10) gives us the left-hand side of formula (3.9). In its special cases when λ = 1, (3.9) reduces to the formula ∞ X
(µ)
En+k (a; x)
k=0
wk = eaw En(µ)(a; xew ). k!
(3.11)
Theorem 3.5. If n is a positive integer and µ and x are arbitrary real or complex parameters, then we have n X n µ+l−1 (µ;λ) µ En (a; x) = 2 (λx)l (x + 1)−µ−l l l l=0
×
l X
(−1)k
k=0
l k
k l (a + λk)n−l 2 F1 l − n, l; l + 1;
λk , (a + λk)
(3.12)
where 2 F1 denotes the Gaussian hypergeometric funtion [13, p.44 (4)]. P roof. we differentiate both side of the generating relation (1.8) with respect to the variable z. By using Leibniz’s rule, we get µ 2 d az En(µ;λ) (a; x) = Dzn e . , Dz = λz xe + 1 dz z=0 = 2µ
n n X −µ o n an−s Dzs (x + 1) + x(eλz − 1) . s z=0
(3.13)
s=0
Applying the series expansion: ∞ X µ+l−1 −µ (x + ω) = x−µ−l (−ω)l , l
(|ω| < |x|),
(3.14)
l=0
and the well-known formula (see [13, p.58 (15)]) (eλz − 1)l = l!
∞ X r=l
S(r, l)
(λz)r , r!
(3.15)
we have En(µ;λ) (a; x) = 2µ
n s X X n µ+l−1 an−s (x + 1)−µ−l (−x)l λs l!S(s, l). s l s=0
l=0
(3.16) We now change the order of summation in the above double series, and make use of the following formula:(see [13, p.58 (20)]) S(s, l) =
l 1X l (−1)l−k ks, k l! k=0
so that En(µ;λ) (a; x)
n X n µ+l−1 =2 (λx)l (x + 1)−µ−l an−l l l µ
l=0
(3.17)
9
×
l X
(−1)k
k=0
l k
−λk , k l 2 F1 l − n, 1; l + 1; a
(3.18)
in terms of the Gaussian hypergeometric funtion. Finally, if we apply the known transformation [1, p.284 (23)]: z −a F [a, b; c; z] = (1 − z) F a, c − b; c; 2 1 2 1 (z − 1) in (3.18), we are led immediately to the explicit formula (3.12) asserted by Theorem 3.5. For λ = 1, (3.12) reduces to the formula (see [7, p. 920 (16)]): n X n µ+l−1 En(µ) (a; x) = 2µ xl (x + 1)−µ−l l l l=0
×
l X
(−1)k
k=0
l k
k l (a + k)n−l 2 F1 l − n, l; l + 1;
k . (a + k)
(3.19)
Corollary 3.1. If n is a positive integer and µ and x are arbitrary real or complex parameters, then we have n X n µ+l−1 En(µ;λ) (x) = 2l+µ (λx)l (x + 1)−µ−l l l l=0
×
l X
(−1)k
k=0
l k
k l (µ + 2λk)n−l 2 F1 l − n, l; l + 1;
2λk . (µ + 2λk)
(3.20)
P roof. For a = µ2 in Theorem 3.5, then in view of Theorem 2.1, we obtain the formula (3.20) immediately. The following extended Hurwitz-Larch Zeta function introduced and studied earlier by Bin-Saad [4, p.271 (2.7)] Φ∗µ,λ (x, z, a) =
∞ X
m=0
(µ)n xm , z (a + λm) m!
µ ∈ C\Z− 0 ; a ∈ C\{(−λn)}; |x| < 1.
(3.21)
Below we give the following relationships between the modified Apostol-Euler polynomials defined by (1.8) and Bin-Saad-Hurwitz-Larch Zeta function (3.21). Theorem 3.6. Let n ∈ N; − 1 < x ≤ 1; µ ∈ C\Z− 0 ; λ 6= 0. Then En(µ;λ) (a; x) = 2µ Φ∗µ,λ (−x, −n, a). P roof. By (1.8) and the generalized binomial theorem, we have µ ∞ X zn 2 En(µ;λ) (a; x) = eaz λz + 1 n! xe n=0 = 2µ
∞ X (µ)k k=0
k!
(−x)k e(a+λk)z
(3.22)
10
=
∞ X
n=0
=
∞ X
n=0
"
µ
"
µ
2
2
∞ X (µ)k
k!
k=0 ∞ X
k=0
Hence the result (3.22) follows.
k
n
(−x) (a + λk)
(−x)k (µ)k (a + λk)−n k!
#
#
zn n!
zn . n!
(3.23)
4. Differential Relations (µ;λ)
The modified Apostol-Euler polynomials of higher order En (a; x) satisfies some (µ;λ) differential recurrence relation. Fortunately these properties of En (a; x) can be developed directly from the formula (1.8). First, by recalling the familiar derivative formula from calculus in terms of the gamma function [10] Dxm xn =
Γ(n + 1) xn−m , Γ(n − m + 1)
n − m ≥ 0, Dx =
d , dx
(4.1)
where {m, n} ∈ N, we aim now to derive the following differential relation . Theorem 4.1. Let λ ∈ C\{0}. Then ∂ (µ;λ) nxµ (µ+1;λ) En (a; x) = − E (a + λ; x). ∂λ 2 n−1
(4.2)
P roof. The generating relation (1.8) yields ∞ X −µ ∂ (µ;λ) zn ∂ En (a; x) = 2µ eaz xeλz + 1 ∂λ n! ∂λ n=0
µ+1 zxµ 2 e(a+λ)z =− 2 xeλz + 1 ∞ xµ X (µ+1;λ) z n+1 =− En (a + λ; x) . 2 n=0 n!
(4.3)
Now, letting, n → n − 1 in the right-hand side of (4.3) and comparing the coefficients of zn n! on both side (4.3) we get connection formula (4.2) immediately. Corollary 4.1. Let k ∈ N. Then ∂ k (µ;λ) n! (µ;λ) En (a; x) = E (a; x). k ∂a (n − k)! n−k
(4.4)
For λ = 1, (4.4) reduces to the formula (see [7, p. 918 (6)]): ∂ k (µ) n! (µ) E (a; x) = E (a; x). ∂ak n (n − k)! n−k
(4.5)
Theorem 4.2. Let k ∈ N. Then Dxk En(µ;λ) (a; x)
=
−1 2
k
(µ)k En(µ+k;λ)(a + λk; x).
(4.6)
11
P roof. By starting from the left-hand side of (4.6) and in view of (1.9) and by using the relation (4.1), we get : Dxk En(µ;λ) (a; x) = 2µ
∞ X
(−1)m (µ)m (a + λm)n
m=0
xm−k . (m − k)!
(4.7)
Now, letting m → m + k in (4.7), using the formula (µ)m+k = (µ)k (µ + k)m and considering the formula (1.9), we get the right-hand side of formula (4.6). For λ = 1, (4.6) reduces to the formula k −1 k (µ) Dx En (a; x) = (µ)k En(µ+k)(a + k; x). (4.8) 2 Theorem 4.3. Let 0. Then h i Γ(ν) 2µ−ν xµ−1 En(ν;λ)(a; x). Dxν−µ xν−1 En(µ;λ)(a; x) = Γ(µ)
(4.9)
P roof. By starting from the left-hand side of formula (4.9) and in view of (1.9) and by using the relation (4.1), we get : Dxν−µ
h
i
xν−1 En(µ;λ) (a; x)
=
=2
µ
∞ X
(−1)m (µ)m (a + λm)n
m=0
Γ(ν + m) xm+µ−1 Γ(µ + m) m!
∞ Γ(ν) µ−ν µ−1 ν X (−x)m 2 x 2 (ν)m (a + λm)n Γ(µ) m! m=0
=
Γ(ν) µ−ν µ−1 (ν;λ) 2 x En (a; x). Γ(µ)
For λ = 1, (4.9) reduces to the formula h i Γ(ν) 2µ−ν xµ−1 En(ν)(a; x). Dxν−µ xν−1 En(µ)(a; x) = Γ(µ)
(4.10)
Closely associated with the derivative of the gamma function is the digamma function defined by [1,p.74(2.51)]: ψ(x) =
´ d Γ(x) ln Γ(x) = , dx Γ(x)
x 6= 0, −1, −2, .... .
(4.11)
(µ)
Now, we wish to establish the derivative of the function En (a; x) with respect to the parameter µ. Theorem 4.4. Let µ ∈ C\Z− 0 . Then ∞ X (−x)m ∂ (µ;λ) En (a; x) = 2µ (µ)m (a + λm)n [log 2 + ψ(µ + m) − ψ(µ)]. ∂µ m! m=0
(4.12)
P roof. By starting from the left-hand side of formula (4.12), then according to the result: d µ d µ Γ(µ + m) µ d [2 (µ)m ] = (µ)m 2 + 2 = 2µ (µ)m [log 2 + ψ(µ + m) − ψ(µ)], dµ dµ dµ Γ(µ)
12
REFERENCES
we obtain the right-hand side of formula (4.12).
Theorem 4.5. Let n ∈ N. Then n X ∂ n (µ;λ) En−k (a; x)yk = ey ∂a En(µ;λ) (a; x). k
(4.13)
k=0
= En(µ;λ) (a + y; x).
(4.14)
P roof. Denote, for convenience, the right-hand side of equation (4.13) by I. Then ∂
I = ey ∂a En(µ;λ)(a; x) ∞ X yk ∂ (µ;λ) = E (a; x). k! ∂a n k=0
Upon using (4.4), we led finally to the left-hand side of the formula (4.13). Moreover, if we apply the formula (2.8) ( replace b by y and k by n − k), we get (4.14). Theorem 4.6. Let n ∈ N. Then −α n X ∂ n (µ;λ) (α)k En−k (a; x)yk = 1 − y En(µ;λ) (a; x), k ∂a
(4.15)
k=0
n X
(µ;λ)
(α)k (β)k En−k (a; x)
k=0
yk ∂ = 2 F1 α, β; −n; −y En(µ;λ)(a; x), k! ∂a
(4.16)
where 2 F1 is the Gaussian hypergeometric function (see [12] and [13]). P roof. We refer to the proof of Theorem 4.5.
References [1] Andrews, L.C., Special Functions for Engineers and Applied Mathematician, MacMillan, New York, 1985. [2] Apostol, T. M., On the Lerch Zeta function, Pacific J. Math., 1 (1951), 161-167. [3] Bayad A., Fourier expansions for Apostol-Bernoulli, Apostol-Euler and ApostolGenocchi polynomials, Math. Comp. 80 (2011) 2219-2221. [4] Bin-Saad Maged G., Hypergeometric Series Associated with the Hurwitz-Lerch Zeta Function, Acta Math. Univ. Comenianae, 2(2009), 269-286. [5] Chen S., Cai Y. and Luo Q. M.,An extension of generalized Apostol-Euler polynomials, Adv. Dierence Equ. (2013) 61. 10 pp. [6] Ghazala Yasmin, Some identities of the Apostol type polynomials arising from umbral calculus, Palestine Journal of Mathematics Vol. 7(1)(2018),35-52. [7] Luo Q.-M. , Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), 917-925.
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[8] Luo Q. M., Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 7(8)(2009) 2193-2208. [9] Luo Q. M., Srivastava H. M., Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308(1) (2005) 290-302. [10] Miller, K.S.and Rose, B., An introduction to The Fractional Calculus and Fractional Differential Equations, New York, 1993. [11] Srivastava H. M., Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Camb. Philos. Soc. 129 (2000) 77-84. [12] Srivastava H. M. and Manocha H. L.,A Treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984. [13] Srivastava H. M. and Choi J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London, 2001. [14] Srivastava H. M. and Choi J., Zeta and q-Zeta Functions and Associate Series and Integrals, Elsevier Science, Publishers, Amsterdam, London and New York, 2012.
