A probabilistic interpretation of the Volkenborn integral

0.17 MB·English

by  A. Bhandari

A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL
A.BHANDARI(1) ANDC.VIGNAT(2)
Abstract. In thispaper, weprovide aprobabilistic interpretation of the Volkenborn integral;
thisallowsustoextendresultsbyT.KimetalaboutsumsofEulernumberstosumsofBernoulli
numbers. We also obtain a probabilistic representation of the multidimensional Volkenborn
integral which allows us to derive a multivariate version of Raabe’s multiplication theorem for
thehigher-order BernoulliandEulerpolynomials.
2
1
0
1. Introduction
2

n The Volkenborn integral was introduced in 1971 by A. Volkenborn in his PhD dissertation and
a subsequently in the set of twin papers [4]; a more recent treatment of the subject can be found in
J [5]. The Volkenborn integral, or fermionic p−adic q−integral on Z , of a function f is defined as
p
8
1 pN−1
1+q
f(y)dµ (y)= lim f(x)(−q)x
] ˆZ −q N→+∞1+qpN
T p Xx=0
N In particular, the q =1-Volkenbornintegral satisfies [1, eq. (1.6)]
. 2 +∞ tn
h e(x+y)tdµ (y)= ext = E (x) .
at ˆZp −1 et+1 n=0 n n!
X
m
where E (x) is the Euler polynomial of degree n.
n
[ Another interesting case is the q =0-Volkenborn integral which satisfies [5, p. 271]

1 text +∞ tn
v e(x+y)tdµ (y)= = B (x)
1 ˆZ 0 et−1 n n!
p n=0
0 X
where B (x) is the Bernoulli polynomial of degree n. From this result we deduce the following
7 n
3
Theorem. If f(x) is analytic in a neighborhood of 0 then its q = 0-Volkenborn integral can be
.
1 computed as the expectation 1
0
2 (1.1) f(x)dµ (x)=Ef x+ıL − 1
1 ˆZ 0 B 2
: p (cid:18) (cid:19)
v where the random variable L follows the logistic distribution with density
B
i
X (1.2) πsech2(πx), x∈R.
r 2
a Moreover, its q =1-Volkenborn integral coincides with the expectation
1
f(x)dµ (x)=Ef x+ıL −
ˆZ 1 E 2
p (cid:18) (cid:19)
where the random variable L follows the hyperbolic secant distribution with density
E
(1.3) sech(πx), x∈R.
Proof. These results can be proved by computing the characteristic functions associated to the
logistic distribution
t t
Eexp(ıtL )= csch
B
2 2
(cid:18) (cid:19)
1intherestofthispaper,weusethenotationEXf(X)fortheprobabilisticexpectation f(x)pX(x)dxwhere
pX is the probability density function of the random variable X. When no ambiguity o´ccurs, we also denote
Ef(X1,X2,...)theexpectation overallrandom variablesthatappearasarguments ofthefunctionf.
1
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 2
and to the hyperbolic secant distribution
t
Eexp(ıtL )=sech .
E
2
(cid:18) (cid:19)
(cid:3)
The special case f(x) =xn yields the following moment representation for the Bernoulli poly-
nomial of degree n
n
1
(1.4) B (x)=E x+ıL −
n B
2
(cid:18) (cid:19)
and
n
1
E (x)=E x+ıL −
n E
2
(cid:18) (cid:19)
for the Euler polynomial of degree n. Moreover, choosing x = 0 yields the following moment
representationfor the n−th Bernoulli number
n
1
B =B (0)=E ıL −
n n B
2
(cid:18) (cid:19)
and
n
1
(1.5) E =E (0)=E ıL −
n n E
2
(cid:18) (cid:19)
for the n−th Euler number 2 where L and L follow respectively the logistic distribution (1.2)
B E
and the hyperbolic secant distribution (1.3).
An important feature of the logistic and hyperbolic secant random variables is the following
cancellation property:
Lemma1. IfU isacontinuousrandomvariable uniformlydistributedover[0,1]andindependent
B
of L then
B
n
1
E x+ıL − +U =xn.
B B
2
(cid:18) (cid:19)
Accordingly, if U is aRademacher distributedrandom variable (Pr{U =0}=Pr{U =1}= 1)
E E E 2
then
n
1
E x+ıL − +U =xn.
E E
2
(cid:18) (cid:19)
Both results canbe shownconsidering the characteristicfunctions of the involvedvariables; for
example, in the Bernoulli case,
1
Eexp t ıL − exp(tU )=1
B B
2
(cid:18) (cid:18) (cid:19)(cid:19)
so that all integer nonzero moments of the random variable ıL − 1 +U vanish.
B 2 B
The Volkenborn integrals were used by Kim et al [1] to obtain non-trivial identities on Euler
numbers E using integrals of the Bernstein polynomials defined as
n
n
B (x)= xk(1−x)n−k, 0≤x≤1.
k,n
k
(cid:18) (cid:19)
In this paper, we show that the probabilistic representation (1.1) of the Volkenborn integral
makes its computation very easy to handle. We illustrate this fact by extending the non-trivial
identities of[1]to the case ofBernoullinumbers. In the secondsection, wederive the probabilistic
equivalentofthe multidimensionalVolkenbornintegralsasintroducedin[2]andweuse itto prove
a multivariate version of Raabe’s multiplication theorem for Bernoulli and Euler polynomials.
2notethatitdiffersfromthen−thEulernumberofthefirstkinddefinedbyE˜n=2nEn(cid:0)1(cid:1).
2
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 3
2. Identities for Bernoulli numbers and polynomials
2.1. first-order identity. In order to obtain non-trivial identities on Bernoulli numbers, we re-
place the Bernstein polynomials used by Kim et al by the Beta polynomials
B (x)=xk(1+x)n−k, 0≤k ≤n
k,n
and, with X =ıL − 1, compute the expectation EB (X) in two different ways:
B 2 k,n
- the first way is by applying the binomial formula
n−k n−k
n−k n−k
(2.1) EB (X)=EXk Xj = B
k,n j j j+k
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
- the second way is by expressing X =(X +1)−1 so that
k k
k k
(2.2) EB (X)=E (−1)j(1+X)(n−k)+(k−j) = (−1)j{B +δ };
n,k j j n−j n−j−1
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
since j ≤k ≤n, the Kronecker adds a term n−1 (−1)n−1 if k =n−1 or n (−1)n−1 if k =n
n−1 n−1
and no term otherwise. We conclude the following
(cid:0) (cid:1) (cid:0) (cid:1)
Theorem 2. The Bernoulli numbers satisfy
k k (−1)jB if 0≤k ≤n−2, n≥2
n−k n−k j=0 j n−j
B = k k (−1)jB +(−1)n−1 if k =n−1, n≥1
j j+k Pj=0(cid:0)j(cid:1) n−j
Xj=0(cid:18) (cid:19) Pkj=0(cid:0)kj(cid:1)(−1)jBn−j +n(−1)n−1 if k =n, n≥0
We remark that the case k=Pn re(cid:0)ad(cid:1)s
n
n
B = (−1)jB +n(−1)n−1
n j n−j
j=0(cid:18) (cid:19)
X
2.2. polynomial identities. From (1.4), we deduce that the above results can be extended to
the case of Bernoulli polynomials by choosing X =x+ıL − 1. We deduce the following
B 2
Theorem 3. The Bernoulli polynomials satisfy, for all 0≤k ≤n,
n−k k
n−k k
(2.3) B (x)= (−1)jB (x)+(nx−(n−k))xn−k−1(x−1)k−1
j j+k j n−j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
Proof. The left-hand side is a direct consequence of that of (2.1); the left-hand side of (2.2) with
X =x+ıL − 1 yields
B 2
k
k
(−1)jB (x+1)
j n−j
j=0(cid:18) (cid:19)
X
and since B (x+U) = xn−j, we deduce B (x+1)−B (x) = (n−j)xn−j−1. replacing
n−j n−j n−j
in the above sum yields the result. (cid:3)
We note that the case k =0 reads
n
n
B (x)=B (x)+nxn−1,
j n
j
j=0(cid:18) (cid:19)
X
which can be restated as
B (x+1)−B (x)=nxn−1
n n
and is nothing but the expression of the cancellation principle
EB (x+U )=xn−1.
n−1 B
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 4
3. A polynomial extension to Kim’s identity
In[1],thefollowingidentityisderivedusingtheBernsteinpolynomialsB (x)= n xk(1−x)n−k
k,n k
n−k n−k k k (cid:0) (cid:1)
(−1)jE = (−1)k−jE +2δ .
j j+k j n−j k
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
We now provide the following polynomial extension of this identity
Theorem 4. The Euler polynomials satisfy
n−k k
n−k k
(−1)jE (x)=(−1)n+k+1 E (x)+2xk(1−x)n−k.
j j+k j n−j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
Proof. We start from the identity
n−k k
n−k k
(−1)jxj+k = (−1)k−j(1−x)n−j
j j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
obtained by expanding either (left-hand side) the (1−x)n−k term of the (right-hand side) xk =
(x−1+1)k in the expression of the Bernstein polynomial. Replacing the variable x by x =
X+ıL − 1 and remarking that
E 2
1 n−j
E(1−x)n−j =E 1−X− ıL − =(−1)n−jE (X −1)
E n−j
2
(cid:18) (cid:18) (cid:19)(cid:19)
with, by the cancellation principle,
E (X−1)+E (X)=2(X −1)n−j,
n−j n−j
we deduce
k k
k k
(−1)k−j(1−x)n−j = (−1)k−j(−1)n−j −E (X)+2(X −1)n−j
j j n−j
Xj=0(cid:18) (cid:19) Xj=0(cid:18) (cid:19) n o
k k
k k
= (−1)n+k+1 E (X)+2 (−1)k−j(1−X)n−j
j n−j j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
this last sum being equal to 2Xk(1−X)n−k, hence the result. (cid:3)
We notice that the case X =0 is
n−k k
n−k k
(−1)jE =(−1)n+k+1 E +2δk.
j j+k j n−j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
It can be shown that
k k
k k
(−1)n+k+1 E = (−1)k−jE
j n−j j n−j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
as follows: using the moment representation (1.5) the right-hand side reads
n−k k n−k k
1 1 1 1
E ıL − 1− ıL − =E −ıL − 1− −ıL −
E E E E
2 2 2 2
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
by the symmetry of the hyperbolic secant distribution, and is thus equal to
1 n−k 3 k 1
(−1)n−kE ıL + ıL + =Ef ıL − +1
E E E
2 2 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18)(cid:18) (cid:19) (cid:19)
with f(x)=(−1)n−kxn−k(x+1)k. By the cancellation principle
1 1
Ef ıL − +1 +f ıL − =2f(0)=0
E E
2 2
(cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19)
so that the right-hand side if equal to −Ef ıL − 1 which coincides with the left-hand side;
E 2
hence we recover Kim’s identity.
(cid:0) (cid:1)
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 5
4. Multidimensional Volkenborn integral
4.1. Introduction. In [2], a multivariate version of the Volkenborn integral is defined as
f(x)dµ (x)= ... f(x ,...,x )dµ (x )...dµ (x ).
ˆ 0 ˆ ˆ 1 k 0 1 0 k
In particular, it satisfies, with y∈Rk and the notation |y|= k y ,
i=1 i
k
t P
e(x+|y|)tdµ (y)= ext.
ˆZkp 0 (cid:18)et−1(cid:19)
ThismultivariateversionoftheVolkenbornintegralcanagainbe expressedasanexpectationover
a simple random variable as shown now.
4.2. Momentrepresentationandelementaryproperties. TheBernoullipolynomialsB(k)(x|a)
n
of order k and degree n with x∈ R with parameter a ∈Rk, also called Nörlund polynomials, are
defined by the generating function [3, 1.13.1]
+∞ tn k a t
B(k)(x|a) =ext j
n n! eajt−1
n=0 j=1(cid:18) (cid:19)
X Y
and the corresponding Bernoulli numbers B(k)(a) by
n
k
a t
B(k)(a)=B(k)(0|a)= j .
n n eajt−1
j=1(cid:18) (cid:19)
Y
In particular, taking a =1 for all j ∈[1,k] and denoting
j
B(k)(x)=B(k)(x|1,1,...,1)
n n
we deduce
+∞ tn
e(x+|y|)tdµ (y)= B(k)(x) .
ˆZkp 0 nX=0 n n!
We provide a multidimensional extension of the moment representation (1.4) as follows
Theorem 5. The Bernoulli polynomials B(k)(x|a) satisfy
n
n
k
1
(4.1) B(k)(x|a)=E x+ a ıL(j)−
n  j B 2 
j=1 (cid:18) (cid:19)
X
 
where the random variables L(j) are independent and follow the logistic distribution (1.2).
B
1≤j≤k
As a consequence, the Benrnouloli numbers B(k)(a) satisfy
n
n
k
1
(4.2) B(k)(a)=E a ıL(j)−
n  k B 2 
j=1 (cid:18) (cid:19)
X
and the multivariate Volkenborn integral, with x∈Rk, 
k
1
f(|x|)dµ (x)=Ef x +ıL(j)− .
ˆZkp 0 Xj=1(cid:18) j B 2(cid:19)
 
Proof. Let us compute the generating function
n
+∞Ex+ k aj ıL(Bj)− 12  tnn! =Eexptx+t k aj ıL(Bj)− 12 =etx k Eetaj(cid:16)ıLB(j)−21(cid:17)
n=0 j=1 (cid:18) (cid:19) j=1 (cid:18) (cid:19) j=1
X X X Y
with,for a logistic distributedrandom variable L , 
j
Eetaj(cid:16)iLB(j)−12(cid:17) = ajt
eajt−1
hence the result. (cid:3)
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 6
The moment representation (4.1) allows to recover easily some well-known results about the
higher-order Bernoulli polynomials.
Proposition 6. The higher-order Bernoulli polynomials satisfy the identities
x
(4.3) B(1)(x|a)=anB ,
n n a
(cid:16) (cid:17)
n
n
(4.4) xlB(k) (y|a)=B(k)(x+y|a)
l n−l n
l=0(cid:18) (cid:19)
X
and
a +···+a
(4.5) B(k) 1 k|a =0.
2n+1 2
(cid:18) (cid:19)
Proof. Identity (4.3) is a direct consequence of the moment representation (4.1); identity (4.4) is
obtained using a binomial expansion of (4.1) and identity (4.5) by computing
2n+1
k
a +···+a
B(k) 1 k|a =E ı a L(l)
2n+1 2 l B
(cid:18) (cid:19) l=1 !
X
and using the fact that the logistic density (1.2) is an even function. (cid:3)
Wealsodeducestraightforwardlyfromamultinomialexpansionoftherepresentations(4.1)and
(4.2) the following
Proposition 7. The higher-order Bernoulli polynomials satisfy
n
B(k)(x +···+x |a)= B (x |a )...B (x |a )
n 1 k i ,...,i i1 1 1 ik k k
i1+·X··+ik=n(cid:18) 1 k(cid:19)
and the higher-order Bernoulli numbers
n
B(k)(a)= B (a )...B (a )
n i ,...,i i1 1 ik k
i1+·X··+ik=n(cid:18) 1 k(cid:19)
These results extend Corollary 5 and Corollary 6 in [2] which correspond to the case a =
(1,...,1).
4.3. Kim’s identity for Nörlund polynomials. In order to highlight the efficiency of the mo-
mentrepresentation(4.1),wederivenowanextensionofKim’sidentity(2.3)tothecaseofNörlund
polynomials as follows.
Theorem 8. For p∈N and 0≤k ≤n,
n−k k
n−k k
B(p) (x)= (−1)j B(p) (x)+(n−j)B(p−1) (x) .
j j+k j n−j n−j−1
Xj=0(cid:18) (cid:19) Xj=0(cid:18) (cid:19) n o
Proof. We start from the identity
n−k k
n−k k
(4.6) Xj+k = (−1)j(1+X)n−j
j j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
and replace X by x+ p ıL(l)− 1 so that, from 4.1, the left-hand side reads
l=1 B 2
(cid:16) (cid:17)
P n−k
n−k
B(p) (x)
j j+k
j=0(cid:18) (cid:19)
X
while the right-hand side is
k
k
(−1)jB(p) (x+1).
j n−j
j=0(cid:18) (cid:19)
X
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 7
Since EB(p) (x+U) = B(p−1) (x) with U uniform on [0,1], we deduce by the cancellation
n−j−1 n−j−1
principle
B(p) (x+1)−B(p) (x)
n−j n−j =B(p−1) (x)
n−j n−j−1
which yields the final result. (cid:3)
4.4. Kim’s identity extended to multidimensional Euler polynomials. We now provide a
multidimensional version of the polynomial Kim identity derived in Theorem 4 as follows:
Theorem 9. The multidimensional Euler polynomials satisfy the identity
n−k k
n−k k
(−1)jE(p) (x) = (−1)n+k+1 E(p) (x)
j j+k j n−j
j=0(cid:18) (cid:19) j=0(cid:18) (cid:19)
X X
k
k
+2(−1)n+k E(p−1)(x−1)
j n−j
j=0(cid:18) (cid:19)
X
Proof. Starting again from identity (4.6), we take x=X+ p ıL(l)− 1 and copute
l=1 E 2
(cid:16) (cid:17)
p n−j P
1
(1−x)n−j = 1−X− ıL(l)− =(−1)n−jE(p) (X −1).
E 2 n−j
l=1(cid:18) (cid:19)!
X
However,by the cancellation rule
E(p) (X −1)+E(p) (X)=2E(p−1)(X −1)
n−j n−j n−j
and the result follows. (cid:3)
4.5. Raabe’s and Nielsen’s multiplication theorem for Nörlund polynomials. Raabe’s
usual multiplication theorem
m−1
l
m1−nB (mx)= B x+
n n
m
l=0 (cid:18) (cid:19)
X
and
m−1
l
m−nE (mx)= (−1)lE x+ , m odd
n n
m
l=0 (cid:18) (cid:19)
X
and Nielsen’s multiplication theorem
m−1
2 l
m−nE (mx)=− (−1)lB x+ , m even
n n+1 n+1 m
l=0 (cid:18) (cid:19)
X
are an interesting feature of the Bernoulli polynomials since, as noted by Nielsen, [8, p. 54]
It is very curious, it seems to me, that there exist polynomials, with arbitrary
degree,thatsatisfyequationsoftheaboveform. However,itiseasytoprovethat,
uptoanarbitraryconstantfactor,theB (x)andE (x)aretheonlypolynomials
n n
that satisfy the mentioned property.
Usingthe momentrepresentationandbasicresultsfromprobabilitytheory,weproposethe follow-
ing extension of Raabe’s celebrated multiplication theorem to the multivariate case.
Theorem 10. If m∈N,
m−1 k
1
(4.7) mk−nB(k)(mx|a)= B(k) x+ a l |a
n n m i i
!
l1,.X..,lk=0 Xi=1
and if moreover m is odd,
m−1 k
1
(4.8) m−nE(k)(mx|a)= (−1)l1+···+lkE(k) x+ a l |a .
n n m i i
!
l1,.X..,lk=0 Xi=1
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 8
Proof. Letus denote U˜(i) a setof k discrete randomvariablesindependent anduniformly
1≤i≤k
distributed in the setn{0,.o..,m−1} and U(i) a set of k continuous random variables
B
1≤i≤k
independent and uniformly distributed on tnhe intoerval [0,1]. For the Bernoulli case, we have
n
m−1 k k k
1 1 1 1
B(k) x+ a l |a =E x+ a ıL(i)− + a U˜(i)
mk n m i i i B 2 m i
l1,.X..,lk=0 Xi=1 ! Xi=1 (cid:18) (cid:19) Xi=1 !
n
k k
1 1
= E mx+m a ıL(i)− + a U˜(i)
mn i B 2 i
i=1 (cid:18) (cid:19) i=1 !
X X
n
k k k k
1 1 1
= E mx+m a ıL(i)− + a U˜(i)+ a ıL˜(i)− + a U(i)
mn i B 2 i i B 2 i B
i=1 (cid:18) (cid:19) i=1 i=1 (cid:18) (cid:19) i=1 !
X X X X
Now we use the fact that U˜(i)+U(i) has the same distribution as mU(i) so that we obtain
B B
n
k k k
1 1 1
E mx+m a ıL(i)− + a ıL˜(i)− +m a U(i)
mn i B 2 i B 2 i B
i=1 (cid:18) (cid:19) i=1 (cid:18) (cid:19) i=1 !
X X X
and applying the cancellation principle, we deduce
n
k
1 1 1
E mx+ a ıL˜(i)− = B(k)(mx|a).
mn i B 2 mn n
i=1 (cid:18) (cid:19)!
X
For the Euler case, we need to use a signed measure (and then depart temporarily from the
probabilistic context) defining the set Uˆ(i) of k discrete variables independent such as
1≤i≤k
each Uˆ(i) takes values in {0,...,k,...,mn−1}owith a weight (−1)k. Then
n
m−1 k k k
1 1 1
(−1)l1+···+lkE x+ a l |a =E x+ a ıL(i)− + a Uˆ(i)
n m i i i E 2 m i
l1,.X..,lk=0 Xi=1 ! Xi=1 (cid:18) (cid:19) Xi=1 !
n
k k
1 1
= E mx+m a ıL(i)− + a Uˆ(i)
mn i E 2 i
i=1 (cid:18) (cid:19) i=1 !
X X
n
k k k k
1 1 1
= E mx+m a ıL(i)− + a Uˆ(i)+ a ıL˜(i)− + a U(i)
mn i E 2 i i E 2 i E
i=1 (cid:18) (cid:19) i=1 i=1 (cid:18) (cid:19) i=1 !
X X X X
wherenow U(i) areindependentRademacherrandomvariablesand,sincemisodd,eachU(i)+
E E
Uˆ(i) has thensamoe distribution as mU(i) so that we obtain
E
n
k k k
1 1 1
E mx+m a ıL(i)− + a ıL˜(i)− +m a U(i)
mn i E 2 i E 2 i E
i=1 (cid:18) (cid:19) i=1 (cid:18) (cid:19) i=1 !
X X X
and from the cancellation principle, we deduce the result. (cid:3)
Raabe’s identity (4.7) and (4.8) are in fact given without proof in [7, eq. (1.6) and (1.7)]. The
case for m even is not provided, so we prove now
Theorem 11. With n∈N and m even,
1 k n! k m−1 1 k
mk−n − a E(k) (mx)= (−1)l1+···+lkB(k) x+ a l |a .
2 (n−k)! i n−k n m i i
(cid:18) (cid:19) iY=1 ! l1,.X..,lk=0 Xi=1 !
A PROBABILISTIC INTERPRETATION OF THE VOLKENBORN INTEGRAL 9
Proof. Let us define a variable W ={0,...,m−1} with weights (−1)l. Then the right-hand side
reads, with B =ıL(i)− 1 and E =ıL(i)− 1,
i B 2 i E 2
n n
k k
W
E x+ a i +B = m−nE x+ a W +mB +E +U(i)
i m i i i i i E
Xi=1 (cid:18) (cid:19)! Xi=1 (cid:16) (cid:17)!
k
= m−nE x+ a W +mB +U(i)
n i i i E
!
Xi=1 (cid:16) (cid:17)
but it can be checkedthat each W +U(i) takes values 0 and m with respective weights 1 and −1
i E 2 2
so that we obtain
k k−1 k k−1
1
− m−n E x+m a B + W +U(i) +ma −E x+m a B + W +U(i)
2 n i i i E k n i i i E
! !!
Xi=1 Xi=1(cid:16) (cid:17) Xi=1 Xi=1(cid:16) (cid:17)
which coincides with
k k−1 k−1
1 1
− m−nE x+m a B + W +U(i) +ma U(k) =− m−n(ma )nE x+ a W +mB +U(i)
2 n i i i E k B 2 k n−1 i i i E
! !
Xi=1 Xi=1(cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
since ma B and ma U(k) cancel out. We are now back, up to a factor, to the same quantity as
k k k B
before except that n is replaced by n−1 and k by k−1, hence the result. (cid:3)
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(1) A.Bhandari was with the Biomedical Imaging Laboratory, E.P.F.L., Lausanne, Switzerland,
duringthecompletionofthiswork(2)InformationTheoryLaboratory,L.T.H.I.,E.P.F.L.,Lausanne,
Switzerland
E-mail address: [email protected], [email protected]