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) References [1] T. Kim, J. Choi, Y.-H. Kim and C. S. Ryoo, On the Fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, Journal of Inequalities and Applications,Volume2010(2010), ArticleID864247 [2] M.-S. Kim and J.-W. 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Raabe, Zurückführung einiger Summen und bestimmten Integrale auf die Jacob Bernoullische Function,Journal fürdiereineundangewandte Mathematik, 1851,42,348-376 [7] L.Carlitz,TheMultiplicationFormulasfortheBernoulliandEulerPolynomials,Mathematics Maga- zine,27-2,59-64,1953 [8] N.Nielsen,Traitéélémentaire desnombresdeBernoulli,Gauthier-Villars,1923 (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]