Using an explicit computable expression of ordinary multinomials, we

establish three remarkable connections, with the q-generalized Fibonacci sequence,

the exponential partial Bell partition polynomials and the density of

convolution powers of the discrete uniform distribution. Identities and various

combinatorial relations are derived

Belbachir, Hacène

Bouroubi, Sadek

Khelladi, Abdelkader

EKE Repository of Publications

Annales Mathematicae et Informaticae35 (2008) pp. 21–30http://www.ektf.hu/amiConnection between ordinary multinomials,Fibonacci numbers, Bell polynomials anddiscrete uniform distribution∗Hacène Belbachir, Sadek Bouroubi, Abdelkader KhelladiFaculty of Mathematics, University of Sciences and Technology Houari Boumediene(U.S.T.H.B), Algiers, Algeria.Submitted 8 July 2008; Accepted 16 September 2008AbstractUsing an explicit computable expression of ordinary multinomials, weestablish three remarkable connections, with the q-generalized Fibonacci se-quence, the exponential partial Bell partition polynomials and the density ofconvolution powers of the discrete uniform distribution. Identities and vari-ous combinatorial relations are derived.Keywords: Ordinary multinomials, Exponential partial Bell partition poly-nomials, Generalized Fibonacci sequence, Convolution powers of discrete uni-form distribution.MSC: 05A10, 11B39, 11B65, 60C051. IntroductionOrdinary multinomials are a natural extension of binomial coefficients, for anappropriate introduction of these numbers see Smith and Hogatt [18], Bollinger [6]and Andrews and Baxter [2]. These coefficients are defined as follows: Let q > 1and L > 0 be integers. For an integer a = 0, 1, . . . , qL, the ordinary multinomial(La)qis the coefficient of the a-th term of the following multinomial expansion(1 + x+ x2 + · · ·+ xq)L=∑a>0(La)qxa, (1.1)with(La)1=(La)(being the usual binomial coefficient) and(La)q= 0 for a > qL.∗Research supported partially by LAID3 Laboratory of USTHB University.2122 H. Belbachir, S. Bouroubi, A. KhelladiUsing the classical binomial coefficient, one has(La)q=∑j1+j2+···+jq=a(Lj1)(j1j2)· · ·(jq−1jq). (1.2)Some readily well known established properties arethe symmetry relation (La)q=(LqL− a)q, (1.3)the longitudinal recurrence relation(La)q=q∑m=0(L− 1a−m)q, (1.4)and the diagonal recurrence relation(La)q=L∑m=0(Lm)(ma−m)q−1. (1.5)These coefficients, as for usual binomial coefficients, are built trough the Pascaltriangle, known as “Generalized Pascal Triangle”, see tables: 1, 2 and 3. One canfind the first values of the generalized triangle in SLOANE [17] as A027907 for q = 2,A008287 for q = 3 and A035343 for q = 4.As an illustration of recurrence relation, we give the triangles of trinomial,quadrinomial and pentanomial coefficients:Table 1: Triangle of trinomial coefficients:(La)2L\a 0 1 2 3 4 5 6 7 8 9 100 11 1 1 12 1 2 3 2 13 1 3 6 7 6 3 14 1 4 10 16 19 16 10 4 15 1 5 15 30 45 51 45 30 15 5 1Table 2: Triangle of quadrinomial coefficients:(La)3L\a 0 1 2 3 4 5 6 7 8 9 10 11 120 11 1 1 1 12 1 2 3 4 3 2 13 1 3 6 10 12 12 10 6 3 14 1 4 10 20 31 40 44 40 31 20 10 4 1Connection between ordinary multinomials, Fibonacci numbers,. . . 23Table 3: Triangle of pentanomial coefficients:(La)4L\a 0 1 2 3 4 5 6 7 8 9 10 11 12 130 11 1 1 1 1 12 1 2 3 4 5 4 3 2 13 1 3 6 10 15 18 19 18 15 10 6 3 14 1 4 10 20 35 52 68 80 85 80 68 52 35 20 · · ·Several extensions and commentaries about these numbers have been investi-gated in the literature, for example Brondarenko [7] gives a combinatorial interpre-tation of ordinary multinomials(La)qas the number of different ways of distributing“a” balls among “L” cells where each cell contains at most “q” balls.Using this combinatorial argument, one can easily establish the following rela-tion (La)q=∑L1+2L2+···+qLq=a(LL1)(L− L1L2)· · ·(L− L1 − · · · − Lq−1Lq)=∑L1+2L2+···+qLq=a(LL1, L2 · · · , Lq). (1.6)For a computational view of the relation (1.6) see Bollinger [6]. Andrews andBaxter [2] have considered the q-analog generalization of ordinary multinomials(see also [19] for an exhaustive bibliography). They have defined the q-multinomialcoefficients as follows[La](p)q=∑j1+j2+···+jq=aq∑ q−1l=1(L−jl)jl+1−∑q−1l=q−pjl+1[Lj1] [j1j2]· · ·[jq−1jq]where [La]=[La]q={(q)L / (q)a (q)L−a if 0 6 a 6 L0 otherwiseis the usual q-binomial coefficient, and where (q)k =∏∞m=1 (1− qm) /(1− qk+m),is called q-series. This definition is motivated by the relation (1.2).Another extension, the supernomials, has also been considered by Schilling andWarnaar [16]. These coefficients are defined to be the coefficients of xa in theexpression of∏Nj=1(1 + x+ · · ·+ xj)LjA refinement of the q-multinomial coefficient is also considered for the trinomialcase by Warnaar [20].Barry [3] gives a generalized Pascal triangle as(nk)a(n):=k∏j=1a (n− j + 1) /a (j) ,24 H. Belbachir, S. Bouroubi, A. Khelladiwhere a (n) is a suitably chosen sequence of integers.Kallas [11] and Noe [14] give a generalization of Pascal’s triangle by consideringthe coefficient of xa in the expression of (a0 + a1x+ · · ·+ aqxq)L.The main goal of this paper is to give some connections of the ordinary multino-mials with the generalized Fibonacci sequence, the exponential Bell polynomials,and the density of convolution powers of discrete uniform distribution. We willgive also some interesting combinatorial identities.2. A simple expression of ordinary multinomialsIf we denote xi the number of balls in a cell, the previous combinatorial inter-pretation given by Brondarenko is equivalent to evaluate the number of solutionsof the system {x1 + · · ·+ xL = a,0 6 x1, . . . , xL 6 q.(2.1)Now, let us consider the system (2.1). For t ∈ ]−1, 1[, we have (see also Comtet [8,Vol. 1, p. 92 (pb 16).])∑a>0(La)qta = (1 + t+ · · ·+ tq)L =∑06x1,...,xL6qtx1+···+xL ,and(1 + t+ · · ·+ tq)L=(1− tq+1)L(1− t)−L= L∑j=0(−1)j(Lj)tj(q+1)∑j>0(j + L− 1L− 1)tj .By identification, we obtain the following theorem.Theorem 2.1. The following identity holds(La)q=⌊a/(q+1)⌋∑j=0(−1)j(Lj)(a− j (q + 1) + L− 1L− 1). (2.2)This explicit relation seems to be important since in contrast to relations (1.2),(1.3) and (1.5), it allows to compute the ordinary multinomials with one summationsymbol.In 1711, de Moivre (see [13] or [12, 3rd ed. p. 39]) solves the system (2.1) as theright hand side of (2.2).Corollary 2.2. We have the following identity⌊n/2⌋∑j=0(nj)(n− jj)=⌊n/3⌋∑j=0(−1)j(nj)(2n− 3j − 1n− 1).Connection between ordinary multinomials, Fibonacci numbers,. . . 25Proof. It suffices to use relation (6) in Theorem 2.1 for q = 2 and a = L = n. The left hand side of the equality has the following combinatorial meaning. Itcomputes the number of ways to distribute n balls into n boxes with 2 balls atmost into each box. Put a ball into each box, then choose j boxes for removingthe boxes located in them into j boxes chosen from the remaining n− j boxes.3. Generalized Fibonacci sequencesNow, let us consider for q > 1, the “multibonacci” sequence (Φ(q)n )n>−q definedby Φ(q)−q = · · · = Φ(q)−2 = Φ(q)−1 = 0,Φ(q)0 = 1,Φ(q)n = Φ(q)n−1 +Φ(q)n−2 + · · ·+Φ(q)n−q−1 for n > 1.In [4], Belbachir and Bencherif proved thatΦ(q−1)n =∑k1+2k2+···+qkq=n(k1 + k2 + · · ·+ kqk1, k2, · · · , kq),and, for n > 1Φ(q−1)n =⌊n/(q+1)⌋∑k=0(−1)k n− k (q − 1)n− kq(n− kqk)2n−1−k(q+1),leading to∑k1+···+qkq=n(k1 + · · ·+ kqk1, · · · , kq)=⌊n/(q+1)⌋∑k=0(−1)k n− k (q − 1)n− kq(n− kqk)2n−1−k(q+1).This is an analogous situation in writing above a multiple summation with onesymbol of summation. On the other hand, we establish a connection between theordinary multinomials and the generalized Fibonacci sequence:Theorem 3.1. We have the following identityΦ(q)n =qm−r∑l=0(n− ll)q, (3.1)where m is given by the extended euclidean algorithm for division: n = m (q + 1)−r,0 6 r 6 q.26 H. Belbachir, S. Bouroubi, A. KhelladiProof. We haveΦ(q)n =∑k1+2k2+···+(q+1)kq+1=n(k1 + k2 + · · ·+ kq+1k1, k2, · · · , kq+1)=∑L>0∑k1+2k2+···+(q+1)kq+1=n(Lk1, k2, · · · , kq+1)=∑L>0∑k2+2k3+···+qkq+1=n−L(LL− k2 − · · · − kq+1, k2, · · · , kq+1)=∑L>0(Ln− L)q=n∑L> nq+1(Ln− L)q,using the fact that(La)q= 0 for a < 0 or a > qLNow consider the unique writing of n given by the extended euclidean algorithmfor division: n = m (q + 1)− r, 0 6 r < q + 1 then nq+1 = m−rq+1 , which givesΦ(q)n =qm−r∑k=0(m+ kqm− r − k)q=qm−r∑k=0(m+ k(q + 1) k + r)q=qm−r∑l=0(n− ll)q.As an immediate consequence of Theorem 3.1, we obtain the following identitiesΦ(q)(q+1)m =qm∑l=0((q + 1)m− ll)q=qm∑k=0(m+ k(q + 1) k)q,Φ(q)(q+1)m−1 =qm−1∑l=0((q + 1)m− l − 1l)q=qm∑k=0(m+ k(q + 1) k + 1)q,...Φ(q)(q+1)m−r =qm−r∑l=0((q + 1)m− l− rl)q=qm∑k=0(m+ k(q + 1) k + r)q.For q = 1, we find the classical Fibonacci sequence:F−1 = 0, F0 = 1, Fn+1 = Fn + Fn−1, for n > 0.Thus, we obtain the well known identityFn =⌊n/2⌋∑l=0(n− ll).Connection between ordinary multinomials, Fibonacci numbers,. . . 27Recently, in [5], the first author and Szalay prove the unimodality of the se-quence uk =(n−kk)qassociated to generalized Fibonacci numbers. More generally,they establish the unimodality for all rays of generalized Pascal triangles by showingthat the sequence wk =(n+αkm+βk)qis log-concave, then unimodal.4. Exponential partial Bell partition polynomialsIn this section, we establish a connection of the ordinary multinomials withexponential partial Bell partition polynomials Bn,L (t1, t2, . . .) which are defined(see Comtet [8, p. 144]) as follows1L!∑m>1tmm!xmL=∑n>LBn,Lxnn!, L = 0, 1, 2, . . . . (4.1)An exact expression of such polynomials is given byBn,L (t1, t2, . . .) =∑k1+2k2+···=nk1+k2+···=Ln!k1!k2! · · · (1!)k1 (2!)k2 · · ·tk11 tk22 · · · .In this expression, the number of variables is finite according to k1+2k2+ · · · =n.Next, we give some particular values of Bn,L :Bn,L (1, 1, 1, . . .) ={nL}Stirling numbers of second kind,Bn,L (0!, 1!, 2!, . . .) =[nL]Stirling numbers of first kind,Bn,L (1!, 2!, 3!, . . .) =n!L!(n− 1n− L). (4.2)In [1], Abbas and Bouroubi give several extended values of Bn,L.The connection with ordinary multinomials is given by the following result:Theorem 4.1. We have the following identityBn,L (1!, 2!, . . . , (q + 1)!, 0, . . .) =n!L!(Ln− L)q. (4.3)Proof. Taking in (4.1) tm = m! for 1 6 m 6 q + 1 and zero otherwise, we obtain(x+ · · ·+ xq+1)L= L!∑n−L>0Bn,L (1!, 2!, . . . , (q + 1)!, 0, . . .)xnn!,28 H. Belbachir, S. Bouroubi, A. Khelladifrom which it follows∑a>0(La)qxa =∑n−L>0L!n!Bn,L (1!, 2!, . . . , (q + 1)!, 0, . . .)xn−L.Corollary 4.2. Let q > 1, L > 0 be integers, and a ∈ {0, 1, . . . , qL} . For q > a,we have the following identity(La)q=(L+ a− 1a).Proof. Using the fact that Bn,L (1!, 2!, . . . , (q + 1)!, 0, . . .) = Bn,L (1!, 2!, 3!, . . .) forq + 1 > n − L + 1, we obtain(Ln−L)q=(n−1n−L)for q > n − L. We conclude witha = n− L. This is simply a combination with repetition permitted (i.e. multi combination).5. Convolution powers of discrete uniform distribu-tionThis section gives a connection between the ordinary multinomials and theconvolution power of the discrete uniform distribution. The right hand side ofidentity (2.2) is a very well known expression. Indeed for q, L ∈ N, let us denoteby U⋆Lq the Lth convolution power of the discrete uniform distributionUq :=1q + 1(δ0 + δ1 + · · ·+ δq) (δa is the Dirac measure),then for a ∈ N (see de Moivre [13] or [10]), with respect to the counting measure,its density is given byP(U⋆Lq = a)=1(q + 1)L⌊a/(q+1)⌋∑j=0(−1)j(Lj)(a+ L− (q + 1) j − 1L− 1). (5.1)Combining Theorem 2.1 and relation (5.1), we have the following result:Corollary 5.1. Using the above notations, we obtain the following identityP(U⋆Lq = a)=(La)q(q + 1)L.Connection between ordinary multinomials, Fibonacci numbers,. . . 29It should be noted that the multinomials may be seen as the number of favorablecases to the realization of the elementary event {a} .It is easy to show that the distribution of U⋆Lq is symmetric by relation (1.3).Corollary 5.2. We have the following identitiesqL∑k=0k(Lk)q= (q + 1)LqL2,qL∑k=0k2(Lk)q= (q + 1)LqL2(qL2+q + 26),qL∑k=0k3(Lk)q= (q + 1)L(qL2)2 (qL2+q + 22),More generally, for m > 1, the following identity holdsqL∑k=0km(Lk)q= (q + 1)L∑i1+i2+···+iL=m(mi1, i2, . . . , iL)ui1ui2 · · ·uiL ,where ui is the i-th moment of the random variable Uq.Proof. It suffices to compute the expectation of U⋆Lq using, first the density distri-bution and second the summation of uniform distributions. It also comes from theapplication of the generating function of the distribution given by Corollary 5.1. Acknowledgements. The authors are grateful to Professor Miloud Mihoubi forpointing our attention to Bell polynomials. The authors are also grateful to thereferee and would like to thank him/her for comments and suggestions which im-proved the quality of this paper.References[1] Abbas, M., Bouroubi, S., On new identities for Bell’s polynomials, Disc. Math.,293 (2005) 5–10.[2] Andrews, G.E., Baxter, J., Lattice gas generalization of the hard hexagon modelIII q-trinomials coefficients, J. Stat. Phys., 47 (1987) 297–330.[3] Barry, P., On Integer-sequences-based constructions of generalized Pascal triangles.Journal of integer sequences, Vol. 9 (2006), Art. 06.2.4.[4] Belbachir, H., Bencherif, F., Linear recurrent sequences and powers of a squarematrix. Integers, 6 (2006), A12, 17 pp.[5] Belbachir, H., Szalay, L., Unimodal rays in the regular and generalized Pascaltriangles, J. of Integer Seq., Vol. 11, Art. 08.2.4. (2008).30 H. Belbachir, S. Bouroubi, A. Khelladi[6] Bollinger, R.C., A note on Pascal T -triangles, Multinomial coefficients and PascalPyramids, The Fibonacci Quarterly, 24 (1986) 140–144.[7] Brondarenko, B.A., Generalized Pascal triangles and Pyramids, their fractals,graphs and applications, The Fibonacci Association, Santa Clara 1993, Translatedfrom Russian by R.C. Bollinger.[8] Comtet, L., Analyse combinatoire, Puf, Coll. Sup. Paris, (1970), Vol. 1 & Vol. 2.[9] Graham, R.L., Knuth, D.E., Patashnik, O., Concrete mathematics, Addison-Wesley, 1994.[10] Hald, A., A history of mathematical statistics from 1750 to 1930, John Wiley, N.Y., 1998.[11] Kallos, G., A generalization of Pascal triangles using powers of base numbers;Annales mathématiques Blaise Pascal, Vol. 13, no. 1 (2006) 1–15.[12] de Moivre, A., The doctrine of chances, Third edition 1756 (first ed. 1718 andsecond ed. 1738), reprinted by Chelsea, N. Y., 1967.[13] de Moivre, A., Miscellanca Analytica de Scrichus et Quadraturis, J. Tomson andJ. Watts, London, 1731.[14] Noe, T.D., On the divisibility of generalized central trinomial coefficients, Journalof Integer sequences, Vol. 9 (2006) Art. 06.2.7.[15] Philippou, A.N., A note of the Fibonacci sequence of order k and the Multinomialcoefficients, The Fibonacci Quarterly, 21 (1983) 82–86.[16] Schilling, A., Warnaar, S.O., Supernomial coefficients, Polynomial identitiesand q-series, The Ramanujan J., 2 (1998) 459–494.[17] Sloane, N.J.A., The online Encyclopedia of Integer sequences, Published electron-ically at http://www.research.att.com/~njas/sequences, 2008.[18] Smith, C., Hogatt, V.E., Generating functions of central values of generalizedPascal triangles, The Fibonacci Quarterly, 17 (1979) 58–67.[19] Warnaar, S.O., The Andrews-Gordon Identities and q-Multinomial coefficients,Commun. Math. Phys, 184 (1997) 203–232.[20] Warnaar, S.O., Refined q-trinomial coefficients and character identities, J. Statist.Phys., 102 (2001), no. 3–4, 1065–1081.Hacène BelbachirSadek BouroubiAbdelkader KhelladiUSTHB, Faculté de MathématiquesBP 32, El Alia16111 Bab EzzouarAlger, Algériee-mail:hbelbachir@usthb.dz, hacenebelbachir@gmail.comsbouroubi@usthb.dz, bouroubis@yahoo.frakhelladi@usthb.dz, khelladi@wissal.dz

Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution

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Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution

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