Vol. 5, No. 2, 2012, 97-107
ISSN 1307-5543 – www.ejpam.com
Probabilistic Proofs of Some Relationships Between the Bernoulli
and Euler Polynomials
H. M. Srivastava
1,∗, Christophe Vignat
21Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W
3R4, Canada
2Information Theory Laboratory, École Polytechnique Fédérale de Lausanne,
Station 14, 1015 Lausanne, Switzerland
Abstract. The main purpose of this article is to provide probabilistic proofs of the relationships be-tween the generalized Bernoulli (or Nörlund) polynomials B(α)
n (x)and the generalized Euler
polyno-mialsE(α)
n (x)of (real or complex) orderαand degreeninx, which were proved recently by Srivastava
and Pintér[11]. Some other approaches to these relationships and their seemingly interesting gener-alizations are also investigated.
2010 Mathematics Subject Classifications: 11B68, 60E07, 11B83, 62E15.
Key Words and Phrases: Bernoulli polynomials; Euler polynomials; Generating functions; Probabilis-tic proofs; Umbral calculus.
1. Introduction, Definitions and Preliminaries
Throughout this paper, we use the following standard notations: N:={1, 2, 3,· · · }, N0:={0, 1, 2, 3,· · · }=N∪ {0} and
Z−:={−1,−2,−3,· · · }=Z−0 \ {0}.
Also, as usual,Zdenotes the set of integers,Rdenotes the set of real numbers andCdenotes the set of complex numbers.
The classical Bernoulli polynomials Bn(x)and the classical Euler polynomials En(x), to-gether with their familiar generalizationsB(α)n (x)and E(α)n (x)of (real or complex) orderα, are usually defined by means of the following generating functions (see, for details, [2, Vol. ∗Corresponding author.
Email addresses:harimsrimath.uvi.a(H. M. Srivastava),hristophe.vignatepfl.h(C. Vignat)
III, p. 253et seq.],[4, Section 2.8]and[9, p. 61et seq.]; see also[1],[2, Vol. I, p. 35et seq.], [5],[10, p. 81et seq.]and[8], and the references cited therein):
t
et−1
α
·ex t= ∞ X
n=0
B(α)n (x)t
n
n! (|t|<2π; 1
α:=1) (1)
and
2
et+1
α
·ex t = ∞ X
n=0
E(α)n (x)t
n
n! (|t|< π; 1
α:=1), (2)
so that, obviously, the classical Bernoulli polynomialsBn(x)and the classical Euler polynomi-alsEn(x)are given, respectively, by
Bn(x):=Bn(1)(x) andEn(x):=En(1)(x) n∈N0
. (3)
For the classical Bernoulli numbersBnand the classical Euler numbersEn, we have
Bn:=Bn(0) =Bn(1)(0) andEn:=En(0) =En(1)(0) n∈N0
, (4)
respectively.
Recently, for the generalized Bernoulli (or Nörlund) polynomialsB(α)n (x)and the general-ized Euler polynomials of orderαand degree nin x, Srivastava and Pintér[11]proved the following two theorems.
Theorem 1(see Srivastava and Pintér[11, p. 379, Theorem 1]). The following identity holds true:
B(α)n x+y =
n
X
k=0
n
k B
(α)
k y
+k
2B
(α−1)
k−1 y
En−k(x) (5)
(α∈C; n∈N0).
Theorem 2(see Srivastava and Pintér[11, p. 380, Theorem 2]). The following identity holds true:
En(α) x+y=
n
X
k=0 2
k+1
h
Ek(α+−11) y−E(α)k+1 y
i
Bn−k(x) (6)
(α∈C; n∈N0).
2. A Set of Useful Probabilistic Tools
In this section, we recall several probabilistic tools which will be needed in Section 3 for the probabilistic proofs of Theorems 1 and 2. First of all, Sun [12] gave the following probabilistic representation of the Bernoulli polynomialsBn(x)and the generalized Bernoulli (or Norlünd) polynomialsB(α)n (x)of orderα∈N0. Throughout this paper, we follow the usual convention andtacitlyassume that an empty sum and an empty product are interpreted to be 0 and 1, respectively.
Lemma 1 (see Sun[12]). Given a sequence
Ln n∈N of independent random variables, each with the Laplace distribution 1
2exp(−|x|) (x ∈R), define the random variableLBby LB=
∞ X
k=1
Lk
2πk. (7)
Then each of the following probabilistic representations holds true:
Bn(x) =E
ıLB+x−1
2
n
(n∈N0;x ∈R; ı2=−1) (8)
and
B(α)n (x) =E
x+
α X
i=1
ıLB(i)−1
2
!n
(n∈N0;α∈N0; x∈R), (9)
where the random variables
n
L(i)
B
o
1≤i≤α are independent and distributed asLB in(7).
Remark 1. The symbolEX denotes the expectation operator given by
EX
g(X) =
Z
fX(x)g(x)d x,
where fX is the probability density of the relevant random variable X . Moreover, in the absence
of ambiguity, we will use the simple notationE.
Remark 2. The random variableLB, defined by(7)as an infinite sum of independent random
variables, may seem to be difficult to use. We, therefore, propose the following characterization, which can be easily proved by looking at the characteristic function of the random variableLB as defined by(7).
Lemma 2. The random variableLB in(7)follows a logistic distribution with the density given by
fLB(x) = π
2 sech
Lemma 3(see Sun[12]). If the random variableLEis defined by
LE= ∞ X
k=1
Lk
(2k−1)π (11)
where
Lk k∈Nare independent Laplace random variables, then each of the following probabilistic
representations holds true:
En(x) =E
ıLE+x−1
2
n
(n∈N0; x∈R). (12)
More generally, forα∈C,
E(α)n (x) =E
x+
α X
i=1
ıLE(i)−1
2
!n
(n∈N0; α∈C; x ∈R) (13)
where the random variables
n
L(i)
E
o
1≤i≤α are independent and distributed asLE in(11).
Remark 3. As in the case of the Bernoulli polynomials, a more convenient characterization of the random variableLE is provided by the following lemma.
Lemma 4. The random variableLE follows the hyperbolic secant distribution
fLE(x) =sech(πx). (14) Lemma 5 below provides a fundamental property of each of the random variablesLBand LE.
Lemma 5. If UBis uniformly distributed over[0, 1]and independent ofLB,then, for any entire
functionϕ(x)and for all x∈C, E
ϕ
x+UB+ıLB−1
2
=ϕ(x). (15)
Furthermore, if UE is a Bernoulli random variable:
Pr
UE=0 =Pr
UE =1 =1
2
independent ofLE, then
E
ϕ
x+UE+ıLE−1
2
=ϕ(x) (16)
Proof. It suffices to check, with
L =LBorLE and U=UBorUE, that
E
x+U+ıL −1 2
n
=xn (n∈N0; x∈C). (17) The result (17) can be easily derived by using the moment generating functions of the cor-responding random variables. For example, in the case of the Bernoulli polynomials, we find that
E
exp zUB =
exp(z)−1
z
and
E
exp
z
ıLB− 1
2
= z
exp(z)−1; hence
E
exp
z
UB+ıLB− 1 2
=1, which demonstrates the result asserted by Lemma (5).
Remark 4. Lemma(5)expresses the fact that the independent random variables UB and ıLB−1
2
or
UE and ıLE− 1 2
cancel each other in the sense that any non-zero moment of their sum equals0. We will also need a corollary of Lemma(5)in the following form.
Lemma 6. If, for all x∈C,
E
ϕ(x+Z) =E
ψ(x+Z)
with
Z=UB, UE, ıLB−1
2 or ıLE− 1 2,
then
ϕ(x) =ψ(x) (x ∈C).
Proof. If, for example, E
ϕ x+UB =E
ψ x+UB (x∈C), then the result asserted by Lemma 6 follows upon setting
x7→x+ıLB−1
3. Probabilistic Proofs of Theorems 1 and 2
We now use the tools presented in the preceding section in order to prove the Srivastava-Pintér identities (5) and (6).
3.1. Proof of Theorem 1.
Assuming first thatα∈N, let us replace the variablesx and y in (5) by
x+UE and y+ α X
i=1
UB(i),
respectively. The left-hand side of the Srivastava-Pintér identity (5) reads as follows:
E
B(α)n x+UE+y+ α X
i=1
UB(i)
!
=E
x+y+UE
n
=1
2 x+ y+1
n +1
2 x+y
n
. (18)
The same operation in the right-hand side of the Srivastava-Pintér identity (5) yields
E
n
X
k=0
n
k
B(α)k y+ α X
i=1
UB(i)
!
En−k x+UE
=E n X
k=0
n k y+ α X
i=1
ıLB(i)−1
2
+
α X
i=1
UB(i)
!k
xn−k
= n X
k=0
n k
ykxn−k
= x+ yn
(19) for the first term, and
1 2 d d y n X
k=0
n k E
y+
α X
i=0
UB(i)+ α−1
X
i=1
ıLB(i)−1
2
!k
xn−k
= 1 2 d d y § E
x+y+UB(α)
nª
= 1
2
x+y+1n− x+yn (20)
3.2. Proof of Theorem 2.
In (6) we replace the variables x and y by
x+UB and y+
α X
i=1
UE(i),
respectively. We thus obtain
E
E(α)n x+UB+ y+ α X
i=1
UE(i)
!
=E
x+y+UB
n
= 1
n+1
x+ y+1n+1
− x+yn+1
(21) for the left-hand side. For the right-hand side, we similarly obtain
E
E
(α−1)
k+1 y+
α X
i=1
UE(i)
!
−Ek(α)+1 y+ α X
i=1
UE(i)
!!
Bn−k x+UB
=
1
2 y+1
k+1 +1
2y k+1
−yk+1
xn−k
= 1
2 y+1
k+1
−1 2y
k+1
xn−k. (22)
The derivative of the right-hand side sum in (22) is given by n
X
k=0
n k
y+1k
− yk
xn−k= x+y+1n
− x+yn
,
which obviously coincides with the derivative of the left-hand side sum in (21). Applying the assertion of Lemma 6 once again, we are led to the Srivastava-Pintér identity (6).
4. Further Remarks and Observations
In this concluding section, we begin by presenting several further remarks and observa-tions concerning (for example) the scope and prospects of our probabilistic and other ap-proaches to the Srivastava-Pintér identities (5) and (6) asserted by Theorems 1 and 2, respec-tively.
Remark 5. Although the probabilistic proofs of Theorems 1 and 2 were given in the preceding section only in the case whenα ∈N0, yet they can be extended appropriately to any complex-valued parameter α, since the function α 7→ B(α)n (x) is a polynomial of degree n in α. For example, we have
B(α)0 (x) =1, B(α)1 (x) =x−α
B2(α)(x) =x2−xα+α
6 +
α(α−1)
4 ,
and so on. Hence any identity that holds true for allα∈N0 extends also to the whole complex
α-plane. Furthermore, the Bernoulli (or Nörlund) polynomials Bn(−α)(x) (α∈N0)generated by
∞ X
n=0
B(n−α)(x)t
n
n! =
et−1
t
α
·ex t (|t|<2π; α∈N0) (23)
can be expressed as follows as moments:
B(n−α)(x) =E
x+
α X
i=1
UB(i)
!n
(α∈N0). (24)
Similarly, for the Euler polynomials E(n−α)(x) (α∈N0)generated by
∞ X
n=0
E(n−α)(x)t
n
n! =
et+1 2
α
·ex t (|t|< π;α∈N0), (25)
we have
E(n−α)(x) =E
x+
α X
i=1
UE(i)
!n
(α∈N0). (26)
Remark 6. Another approach to the identities(5) and(6)forα ∈N0 consists in proving first their cases whenα=0, namely
Bn(0) x+ y=
n
X
k=0
n
k B
(0)
k y
+ k
2B
(−1)
k−1 y
En−k(x) (27)
for the identity(5).
The special identity(27)can be checked easily, since B(n0)(x) =xn and Bk(−−11) y
=E
y+UBk−1
so that the left-hand side of (27)reads
x+ yn
,
while the right-hand side of (27)is given by
E
n
X
k=0
n
k y
k+ k
2 y+UB
k−1
En−k(x) !
=En x+y + 1
2
En x+y+1
=E
En x+y+UE
= x+yn.
Thus, upon replacing y by
y+ α−1
X
i=1
ıLB(i)−1
2
,
we are led to the identity(5).
Remark 7. The approach indicated in Remark 6 suggests a generalization of the identity(5)to the Bernoulli polynomials B(α)n (x|a)of orderα∈N, degree n and parametera∈Rα defined by the following generating function (see[2]):
∞ X
n=0
B(α)n (x|a)t
n
n!=exp(x t)
α Y
k=1
akt
exp akt
−1
. (28)
It can be easily verified that
Bn(α)(x|a) =E
x+
α X
i=1
ai
ıLB(i)−1
2
!n
(29)
and that the casea= (1,· · ·, 1)corresponds to the Bernoulli (or Nörlund) polynomials B(α)n (x) (α∈N0). The Euler case reads analogously as follows:
∞ X
n=0
E(α)n (x|a)t
n
n! =exp(x t)
α Y
k=1
2ak
exp akt
+1
(30)
and
En(α)(x|a) =E
x+
α X
i=1
ai
ıLE(i)−1
2
!n
. (31)
Finally, we state and prove the following result.
Theorem 3. For n∈N0,α∈C,a∈Cα and any j(1≦ j≦α)such that aj6=0,
B(α)n x+y|a =
n
X
k=0
n
k B
(α)
k y|a
+aj
k
2B
(α−1)
k−1
y|a\aj
·En−kx|aj (32)
and
En(α) x+ y|a =
n
X
k=0 2
k+1
1
aj
Ek(α+−11)y|a\aj− 1 aj
Ek(α)+1 y|a
·B(n1−)kx|aj, (33)
where
a\aj:=a1,· · ·,aj−1,aj+1,· · ·,aα
Proof. Starting from the identity (5) withα=1, if we replacex and y by
x
aj and y aj,
respectively, we obtain
E
x aj +
y aj +
ıLB(j)−1
2
n =E
n X
k=0
n
k
·
y aj
+
ıLB(j)− 1
2
k + k
2
y aj
k−1
x aj
+ıLE−1
2
n−k
,
which, when multiplied byanj on both sides, yields
E
x+y+aj
ıLB(j)−1
2
n =E
n X
k=0
n
k
·
y+aj
ıLB(j)−1
2
k +k
2ajy k−1
x+aj
ıLE− 1 2
n−k
. (34)
Upon replacing y by
y+ α X
i=1(i6=j)
ai
ıLB(i)− 1
2
in (34), if we evaluate the resulting expectations, we get the first assertion (32) of Theorem 3. The second assertion (33) of Theorem 3 can indeed be proven similarly.
Remark 8. The underlying principle of the approach involved in Remarks 6 and 7 (and leading to Theorem 3 above) is that any Bernoulli or Euler polynomial can be represented as a moment of a shifted monomial as (for example) in (8) and (12). This can be related to the notion of polynomials of the binomial type which appears in the theory of operator calculus (see[6]).
Remark 9. Such other approaches as the umbral-calculus approach would allow an equally simple path to these proofs. In this connection, we refer the reader to the seminal paper by Rota and Taylor[7], where the notion of the cancellation properties exhibited by(15)and(16)
corresponds to the notion of the inverse umbras. The paper by Gessel [3], too, provides simple derivations of several identities for the Bernoulli polynomials by using umbral calculus.
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