Download An Explicit Formula for the Euler Polynomials of Higher Order 1

Applied Mathematics & Information Sciences
– An International Journal
c
°2009
Dixie W Publishing Corporation, U. S. A.
3(1) (2009), 53–58
An Explicit Formula for the Euler Polynomials of Higher Order
Qiu-Ming Luo1,2
1
Department of Mathematics, East China Normal University
Dongchuan Road 500, Shanghai 200241, China
2
Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, China
Email Address: [email protected]
Received March 23, 2008; Revised June 18, 2008
We obtain an explicit formula for the Euler polynomials of higher order in terms of
the Gaussian hypergeometric function. The corresponding new formulas of the Euler
polynimials and numbers are also deduced. Finally, we also remark some possible
applications of these results in the number theory and information science fields.
Keywords: Euler numbers of higher order, Euler polynomials of higher order, Gaussian hypergeometric function, Stirling numbers of the second kind, difference operator,
coding theory and cryptography.
1 Introduction
The classical Bernoulli polynomials Bn (x) and Euler polynomials En (x) together
(α)
(α)
with their familiar generalizations Bn (x) and En (x) of (real or complex) order α, are
usually defined by means of the following generating functions (see, for details, [1, 3, 8]):
µ
and
z
ez − 1
µ
¶α
2
ez + 1
exz =
∞
X
Bn(α) (x)
n=0
¶α
exz =
∞
X
En(α) (x)
n=0
zn
n!
(|z| < 2π; 1α := 1) ,
(1.1)
zn
n!
(|z| < π; 1α := 1) ,
(1.2)
respectively. The classical Bernoulli and Euler polynomials are respectively found from
(1.1) and (1.2) as follows:
Bn (x) := Bn(1) (x)
and
En (x) := En(1) (x)
(n ∈ N0 ) .
(1.3)
54
Qiu-Ming Luo
Further the classical Bernoulli numbers Bn and Euler numbers En with their forms of
higher order are
µ ¶
1
Bn := Bn (0)
and En := 2n En
,
2
(1.4)
³α´
,
Bn(α) := Bn(α) (0)
and En(α) := 2n En(α)
2
respectively.
Recently, Luo et al. [6] gave certain new recursive formulas for the Euler numbers and
polynomials of higher order; Luo [7] further obtained several formulas involving the Stirling numbers of the second kind. Srivastava and Todorov [9] gave some elegant formulas
for the Bernoulli numbers and polynomials of higher order as follows
Bn(α)
=
n
X
µ
k
(−1)
k=0
¶µ
¶µ
¶−1
α+n α+k−1 n+k
S(n + k, k)
n−k
k
k
(1.5)
and
Bn(α) (x) =
¶
µ ¶
n µ ¶µ
k
X
n α + k − 1 k! X
k 2k
(−1)j
j (x + j)n−k
k
k
(2k)! j=0
j
k=0
h
j i
.
× 2 F1 k − n, k − α; 2k + 1;
x+j
(1.6)
Here 2 F1 [a, b; c; z] denotes the Gaussian hypergeometric function defined by (see [1,
15.1.1])
2 F1 [a, b; c; z] =
∞
X
(a)n (b)n z n
,
(c)n n!
n=0
(1.7)
where (λ)0 = 1, (λ)n = λ(λ + 1) · · · (λ + n − 1) = Γ(λ + n)/Γ(λ) (n ≥ 1).
2 An Explicit Formula of En(α) (x) in Terms of the Gaussian Hypergeometric Function
In the present section, we will prove an explicit formula for the Euler polynomials of
higher order which is an analogue of the formula (1.6). We need the following lemmas
before proving main result.
Lemma 2.1. For n = 0, 1, . . . , x ∈ R, α ∈ C, the following formulas are true
En(α) (x) =
n µ ¶ (α) ³
X
n E
k=0
k
k
2k
x−
α ´n−k
,
2
An Explicit Formula for the Euler Polynomials of Higher Order
55
En(α) (x) = (−1)n En(α) (α − x),
n µ ¶
³ ´ X
n (−x)n−k (α)
(α) x
En
=
Ek (x).
k
2
2n−k
k=0
Proof. From (1.2), we can readily obtain above formulas.
Clearly, by Lemma 2.1 we have
En(α) (α) = (−1)n En(α) (0),
n µ ¶
³ ´ X
n (−α)n−k (α)
(α) α
En
Ek (α).
=
2
2n−k
k
(2.1)
k=0
Theorem 2.1. For n = 0, 1, . . . , x ∈ R, α ∈ C, the following formula in terms of the
Gaussian hypergeometric function is true
µ ¶µ
¶ k
µ ¶
n
X
1 n α+k−1 X
j k
En(α) (x) =
(−1)
j k (x + j)n−k
2k k
k
j
j=0
k=0
(2.2)
h
j i
× 2 F1 k − n, k; k + 1;
.
x+j
Proof. By Taylor’s expansion and Leibniz’s rule, the generating relation (1.2) yields
n µ ¶
X
n n−s s n³ 2 ´α o¯¯
d
(α)
En (x) =
x
Dz
.
(2.3)
¯ , Dz =
z
s
e +1
dz
z=0
s=0
Since
(1 + w)−α =
¶
∞ µ
X
α+k−1
(−w)k
k
(|w| < 1),
k=0
setting 1 + w = (ez + 1)/2 and applying the binomial theorem, we find from (2.3) that
µ
¶
n µ ¶
s
X
n n−s X (−1)k α + k − 1
(α)
x
En (x) =
Dzs {(ez − 1)k }|z=0 .
(2.4)
k
s
2
k
s=0
k=0
Now we make use of the well-know formula (see [1, 24.1.4])
(ez − 1)k =
∞
X
zr
r=k
r!
∆k 0r ,
S(r, k) =
1 k r
∆ 0 ,
k!
(2.5)
where S(r, k) denotes the Stirling number of the second kind defined by
r µ ¶
X
x
xr =
k!S(r, k).
k
k=0
For convenience we write
∆k ar = ∆k xr |x=a =
µ ¶
k
X
k
(−1)k−j
(a + j)r ,
j
j=0
(2.6)
56
Qiu-Ming Luo
where ∆ is the difference operator defined by (see [1, pp. 822, III])
∆f (x) = f (x + 1) − f (x).
So, in general (see [1, pp. 823, 24.1.1]),
k
∆ f (x) =
k
X
k−j
(−1)
j=0
µ ¶
k
f (x + j).
j
From (2.5) we find that
Dzs {(ez − 1)k }|z=0 = ∆k 0s = k!S(s, k).
Substituting this value into (2.4) yields
µ
¶
n µ ¶
s
X
n n−s X (−1)k k! α + k − 1
(α)
En (x) =
x
S(s, k),
s
2k
k
s=0
(2.7)
(2.8)
k=0
or
En(α) (x) =
µ
¶
n µ ¶
s
X
n n−s X (−1)k α + k − 1
x
∆ k 0s .
k
s
2
k
s=0
(2.9)
k=0
If we rearrange the resulting duple series (2.9), we have
¶
µ
¶ n−k µ
n
X
n
(−1)k α + k − 1 X
xn−s−k ∆k 0s+k .
En(α) (x) =
s
+
k
2k
k
s=0
(2.10)
k=0
Further substituting for ∆k 0s+k from the definition (2.6) with a = 0 into (2.10), we get
that
µ ¶µ
¶
µ ¶
n
k
X
1 n α + k − 1 n−k X
j k
En(α) (x) =
x
(−1)
jk
2k k
k
j
j=0
k=0
(2.11)
h
i
−j
× 2 F1 k − n, 1; k + 1;
.
x
Finally, we apply the known transformation [1, 15.3.4]
h
z i
−a
2 F1 [a, b; c; z] = (1 − z)
2 F1 a, c − b; c;
z−1
and (2.11) leads immediately to the desired (2.2).
3
Further Remarks
Remark 3.1. Taking α = 1 in (2.2), we obtain the following new formula for the Euler
polynomials
µ ¶ k
µ ¶
n
h
X
1 n X
j i
j k
k
n−k
.
En (x) =
(−1)
j
(x
+
j)
F
k
−
n,
k;
k
+
1;
2
1
2k k j=0
x+j
j
k=0
An Explicit Formula for the Euler Polynomials of Higher Order
57
Remark 3.2. By [1, 15.1.20], we have
2 F1 [a, b; c; 1]
=
Γ(c)Γ(c − a − b)
Γ(c − a)Γ(c − b)
(c 6= 0, −1, −2, . . . , <(c − a − b) > 0),
which (for a = k − n, b = k, and c = k + 1) readily yields
2 F1 [k − n, k; k + 1; 1] =
µ ¶−1
n
,
k
(0 ≤ k ≤ n).
(3.1)
In view of (3.1) and the special case when x = 0 in formula (2.2), we get that
En(α) (0)
µ
¶ k
µ ¶
n
X
1 α+k−1 X
j k
(−1)
=
jn,
2k
k
j
j=0
(3.2)
µ
¶
n
X
(−1)k k! α + k − 1
S(n, k).
=
2k
k
(3.3)
k=0
or equivalently,
En(α) (0)
k=0
(α)
Moreover, we note that, with En
order can be written as
En(α) =
n µ ¶
X
n
k=0
k
(α)
:= 2n En (α/2) and (2.1), the Euler numbers of higher
2k αn−k
µ
¶
k
X
(−1)j j! α + j − 1
S(k, j).
2j
j
j=0
(3.4)
Further, setting α = 1 in formula (3.4), we deduce easily the following formula for the
Euler numbers
n µ ¶
k
X
n k X (−1)j j!
En =
S(k, j).
(3.5)
2
2j
k
j=0
k=0
Acknowledgements
This research was supported, in part, by the Henan Innovation Project For University
Prominent Research Talents, China under Grant 2007KYCX0021; Natural Science Foundation of Henan Province, China under Grant 0511015500; and Natural Science for Basic
Research of Education Office of Henan Province, China under Grant 2007110022.
References
[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied
Mathematics Series 55, 4th printing, Washington, 1965.
58
Qiu-Ming Luo
[2] C. Berge, Principles of Combinatorics, Academic Press, 1971.
[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental
Functions, Volume I, McGraw-Hill, New York, 1953.
[4] P. Garrett, The Mathematics of Coding Theory, Prentice Hall, 2003.
[5] K. Knopp, Infinite Sequences and Series, Dover, 1956.
[6] Q.-M. Luo, Y. Zheng and F. Qi, Euler numbers of higher order and Euler polynomials
of higher order, Henan Science, 21 (2003), 1–6 (in Chinese).
[7] Q.-M. Luo. The Euler polynomials of higher order involving the Stirling numbers of
the second kind, Austral. Math. Soc. Gaz. 31 (2004), 194–196.
[8] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special
Functions of Mathematical Physics, Third Enlarged Edition, Springer-Verlag, New
York, 1966.
[9] H. M. Srivastava and P. G. Todorov, An explicit formula for the generalized Bernoulli
polynoaials, J. Math. Anal. Appl. 130 (1988), 509–513.
[10] Y. Wang, H. Li and C. Liang, Information and coding theory, Higher Education Publishing Company, Beijing, 2005 (in Chinese).