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Alternatively, our aim is to give an explicit expression of the mode of the L-th convolution powers of the discrete uniform distribution (section two) by means of the unimodality of the ordinary multinomials for which we study also the strong unimodality (section one), we end the paper (section three) by giving the generating functions for the two sequences z nz of generalized ordinary multinomials: and ,z C, and thus of c 2 , { n q}n { n q}n ∈ { q, n/q}n when q is even.

1 Unimodality of ordinary multinomials

The ordinary multinomials are a natural extension of binomial coefficients (see [1, 2007] for a recent overview on ordinary multinomials). Letting q,L N, for an integer k = L ∈ 0, 1, . . . , qL, the ordinary multinomial k q is the coefficient of the k-th term of the following multinomial expansion
L L 1+ x + x2 + + xq = xk. (1) ··· k k≥0  q
X L L L with k 1 = k (being the usual binomial coefficient) and k q = 0 for k > qL. Using the classical binomial coefficient, one has


L L j1 j 1 = q− . (2) k j1 j2 ··· jq q j1+j2+···+j =a   X q      Readily established properties are the symmetry relation L L = (3) k qL k  q  − q and the recurrence relation q L L 1 = − . (4) k k m q m=0 q   X  −  As an illustration of the latter recurrence relation, we give the triangles of pentanomial and hexanomial coefficients which are just an extension, well known in the combinatorial literature, of the standard Pascal triangle.

L Table 1: Triangle of pentanomial coefficients: k 4
L k 012345 6 7 8 9 10 11 12 13 0\ 1 1 1 1 1 1 1 2 12 3 4 5 4 3 2 1 3 1 3 6 10 15 18 19 18+ 15+ 10+ 6+ 3+ 1 4 1 4 10 20 35 52 68 80 85 80 68 =52 35 20 5 1 5 15 35 70 121 185 255 320 365 381 365 320 255 ··· ··· Determining the mode for convolution powers of discrete uniform distribution. 3

L Table 2: Triangle of hexanomial coefficients: k 5 L k 012345 6 7 8 9 10 11
12 13 14 0\ 1 1 1 1 1 1 1 1 2 123 4 5 6 5 43 2 1 3 1 3 6 10 15 21 25 27 27 25 21 15 10 6 3 1 4 1 4 10 20 35 56 80 104 125 140 146 140 125 104 80 5 1 5 15 35 70 126 205 305 420 540 651 735 780 780 735 ··· ··· L m Let us investigate the unimodality of the sequence k q k=0. A finite sequence of real m { } numbers ak k=0 (m 1) is called unimodal if there exists an integer l 0,...,m { } ≥ l
m ∈ { } such that the subsequence a increases, while a decreases. If a0 a1 { k}k=0 { k}k=l ≤ ≤···≤ al0−1 < al0 = = al1 > al1+1 am then the integers l0,...,l1 are the modes of m ··· ≥···≥ ak k=0. In the case where l0 = l1 we talk about a peak, otherwise the set of the values {of the} mode is called plateau. For positive non increasing and non decreasing sequences, m unimodality is implied by log-concavity. A sequence ak k=0 is said to be logarithmically { 2} concave ( log-concave for short) or strongly unimodal if al al−1al+1, 1 l m 1. Also if the sequence is strictly log-concave (SLC for short), i.e.≥ if the previous≤ inequalities≤ − are strict, then the sequences have at most two consecutive modes (a peak or a plateau). For these notions, one can see Belbachir & Bencherif [3, 2007], Belbachir & al [2, 2007], Bertin & Theodorescu [4, 1984], Brenti [5, 1994], Comtet [6, 1970], Dharmadhikari & Joak-Dev [7, 1988], Keilson & Gerber [8, 1971], Medgyessy [10, 1972], Sagan [14, 2007], Stanley [17, 1986] and [16, 1989] and Tanny & Zuker [18, 1974]. In the following, a denotes the greatest integer in a. ⌊ ⌋

The first main result of this article is the following. Theorem 1 Let q 1 and L 0 be integers. Then the sequence L qL is unimodal ≥ ≥ { k q}k=0 and its smallest mode is given by
L kL := arg max = (qL + 1) /2 , k k ⌊ ⌋  q Furthermore, we have the following recurrence relation L L 1 = − , k k −1 + i L q ∈ L q   iXIq   where q/2,...,q/2 if q is even, {− } I = (q + 1) /2,..., (q 1) /2 if q and L are odd, q  {− (q 1) /2,..., (q +− 1) /2} otherwise.  {− − } Proof. It suffices, for each one of the two cases: q odd and q even, to proceed by induction over L using the recurrence relation (4).  Remark 2 For odd qL we have a plateau of two modes: qL/2 and qL/2+1. Otherwise we have a peak: (qL + 1) /2. Determining the mode for convolution powers of discrete uniform distribution. 4

2 Determining the maximal probability for convolution powers of discrete uniform distribution

We are now able to achieve our purpose: the expression of the maximal probability of the L- th convolution powers of the discrete uniform distribution. Let Uq be the random variable ⋆L of the discrete uniform distribution on 0, 1, ..., q and let Uq be its L-th convolution powers: { } 1 U := (δ0 + δ1 + + δ ) (δ is the Dirac measure). q q +1 ··· q a In [1, 2007], Belbachir and al. established a link between the ordinary multinomials and the density probability of convolution powers of discrete uniform distribution. With respect to the counting measure, such a density is given by

L ⋆L k q P Uq = k = L , k =0, 1, . . . , qL. (q+
1)
Remark 3 From Odlyzko and Richmond [13, 1985] we know that for L sufficiently large, the sequence of probabilities P U ⋆L = k is strongly unimodal, from which we easily { q }k deduce that the sequence L is also asymptotically strongly unimodal. { k q}k

Conjecture 4 For each positive integer q, the sequence L is SLC. { k q}k

L L From Theorem 1, as second main result, we give the values of cq,L := maxk k q/ (q + 1) .
Theorem 5 The maximal probability of the Lth convolution power of the discrete uniform distribution over 0, 1,...,q is { } 1 L cq,L = . (q + 1)L (qL + 1) /2 ⌊ ⌋q 3 Some generating functions

As a third main result, we give the generating functions for the sequence of generalized ordinary multinomials, the sequences z and nz , z C, and the extended { n q}n { n q}n ∈ sequence of maximal probabilities for convolution power of discrete uniform distribution:

c 2 . { q, n/q}n Definition 6 For z C, we define the generalized ordinary multinomials, as follows ∈ z z (z 1) (z k1 + 1) := − ··· − . (5) k (k1 k2)! (k2 k3)! (kq−1 kq)!kq! q 1+ 2+···+ =   k k X kq k − − ··· − This definition is motivated by the relation (2). Determining the mode for convolution powers of discrete uniform distribution. 5

Lemma 7 We have the following inequality

z (z 1) (z k1 + 1) z (z 1) (z k1 + 1) − ··· − = − ··· − . (k1 k2)! (kq−1 kq)!kq! h1!h2! hq−1!hq! k1+···+kq =k − ··· − h1+2h2+···+qhq =k ··· X h1+h2+X···+hq =k1

Theorem 8 Let z C, the generating function for generalized ordinary multinomials is given by ∈ z α tn = 1+ t + t2 + + tq . n ··· n≥0  q X
Proof. Using the Lemma, we have z tn = z m! tn. n≥0 n q m h1!h2!···hq−1 !hq! h1+2h2+···+qhq =n P
h1+h2+P···+hq =m
On the other hand z m 1+ t + t2 + + tq = z t + t2 + + tq ··· m≥0 m ···
P

! = z m tl1+2l2+···+qlq . m≥0 m l1!l2!···lq−1!lq ! l1+2l2+···+qlq =n P
l1+l2+P···+lq =m We conclude by summation over n 0 is equivalent to summation over m 0.  ≥ ≥ Remark 9 Problem 19 of Comtet [6], Vol.1, p. 172, states that

1 n 2 n 2 − 2 x ∁ n 1+ t + t = 1 2x 3x , t − − n≥0 X

n 2 n 2 n using the fact that the coefficient of t in the development of 1+ t + t : ∁tn 1+ t + t = n is max n , we obtain the following combinatorial identity n 2 k k 2



−1/2 n t −1/2 G2 (t) := c2 t = 1+ (1 t) . ,n 3 − ≥0 nX  

This last identity can be shown as the generating function of the sequence c2 . { ,n}n Theorem 10 Let z C, the generating function of the sequence nz is given by ∈ { n q}n −1
nz u +2u2 + + quq tn = u 1 z ··· , n − 1+ u + u2 + + uq ≥0 q nX    ···  −z where u is a solution of the equation t = u 1+ u + u2 + + uq . ···
Proof. Use Hermite’s Theorem [6] for the function t t (1 + t + + tq)−z .  7→ ··· Determining the mode for convolution powers of discrete uniform distribution. 6

Theorem 11 For q even, the generating function of the sequence c 2 is given by { q, n/q}n

2 q −1 q/2 −k k n 2 u +2u + + qu q 1+ k=1 u + u Gq (t) := t cq,2n/q = 1 ··· = , − q 1+ u + u2 + + uq 2 q/2 −k k n≥0 kP=1 k (u u )
X  ···  − where u is a solution of the equation P

2 2/q q +1 /q q +1 t = u = . 1+ u + u2 + + uq q/2 −k k  ···  1+ k=1 (u + u )! Proof. Use the above Theorem for z =2/q, and the changeP of variable t (q + 1) t.  →

Remark 12 The sequence c 2 contains strictly the subsequence c . { q, n/q}n { q,L}L

Corollary 13 For q =4, the generating function of c4 2 is given for t √5, 1 by { ,n/ }n ∈ − 1 2 3 − /   n 1 2 1 4 1 2 2 G4 (t) := t c4 2 = 1 t t t 5t + 20 . ,n/ − 4 − 8 − 200 n≥0   X
Corollary 14 We have the following identities

−n n/2 n ( 5) =2 and ( 1) c4 n =2/√5. − n − , 2 ≥0 4 ≥0 nX   nX

Remark 15 The generating function of the sequence c4 is given for t ] 1, 1[ by { ,n}n ∈ − n t c4 = (G4 t + G4 t )/2. ,n | | − | | n≥0 X p   p  Acknowledgement The author is grateful to Pr. A. Aknouche for useful suggestions.

References

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Hac`ene Belbachir USTHB/ Facult´ede Math´ematiques. BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria. [email protected] and [email protected]