POLYNOMIAL COEFFICIENTS AND DISTRIBUTION
OF THE SUM OF DISCRETE UNIFORM VARIABLES
CAMILA C. S. CAIADO and PUSHPA N. RATHIE
Departamento de Estatı́stica, Instituto de Ciências Exatas, Universidade de Brası́lia
Campus Universitário Darcy Ribeiro - Asa Norte, 70910-900 - Brası́lia - DF - Brasil
E-mails: rathie@unb.br , ccscaiado@yahoo.com
Abstract
In 1801, Euler first studied the expansion of the polynomial
f (x) = (1 + x + x2 + . . . + xk )n
in which coefficients are called the polynomial coefficients (Comtet, 1974). These coefficients generalize the binomial coefficients. We give alternative proofs for André (1875)
results on the distribution of the sum of discrete uniform variables. The result is also
given in terms of hypergeometric polynomials. Some properties and identities are also
obtained.
2000 Mathematics Subject Classification: 60G50, 62E15, 60E05, 05A10, 11B65.
Keywords: Polynomial Coefficients, Discrete Uniform Sum Distribution, Hypergeometric Polynomials
1. Introduction and polynomial coefficients. The polynomial coefficients are defined as
follows:
Definition 1.1. The coefficient of xm in the expansion of (1 + x + . . . + xk )n , 0 ≤ m
≤ nk,
is called polynomial coefficient (PC) (see, Comtet (1974)). It is defined by nj
for
k+1
n n = 0, j < {0, 1, . . . , nk}.Thus, we have
k, n ∈ Z+ with 0 = 1 and j
0
k+1
 k
n
!
kn
n X
X α 
n
k

x  = 1 + x + . . . + x =
xj
j
k+1
α=0
j=0
(1.1)
Lemma 1.1 The PC satisfies the recursive relation
n
q
!
k+1
=
⌊q(k−1)/k⌋
X
p=0
!
!
n
q− p
, k = 0, 1, . . .
q− p
p k
1
(1.2)
Proof. By binomial expansion, we have
!
n
X
n
(1 + x + . . . + x ) =
(x + x2 + . . . + xk )z
z
z=0
!
n
X
n z
=
x (1 + x + . . . + xk−1 )z
z
z=0
! z(k−1) !
n
X
n z X z p
=
x
x
z
pk
z=0
p=0
n z(k−1)
X
X n! z !
=
x p+z
z
p
k
z=0 p=0
k n
q(k − 1)
, so that
k
!
!
nk ⌊q(k−1)/k⌋
X
X
n
q− p
Let q = p + z. Then 0 ≤ p ≤ z(k − 1) and p ≤
(1 + x + x2 + . . . + xk )n =
q=0
p=0
q− p
p
xq
(1.3)
k
where ⌊y⌋ denotes the greatest integer less than or equal y.
Comparing (1.1) and (1.3), we obtain (1.2).
In subsequent sections, we give some properties like symmetry and recursivity. We
obtain expressions for the distribution of the sum of discrete uniform variates, moments
and identities.
2. Distribution of the sum of discrete uniform variables. Let X = (X1 , X2 , . . . , Xn ) be
a sample from the random variable X with discrete uniform distribution on {0, 1, . . . , k}.
Then the characteristic function of X is given by
k
X
√
eitp
, ∀t ∈ R, i = −1
φX (t) =
k+1
p=0
(2.1)
Define
Y=
n
X
Xi
(2.2)
i=1
By independency,
n
 k
n X eitp 
 , ∀t ∈ R
φY (t) = φX (t) = 
k + 1
(2.3)
p=0
In this section we obtain the distribution of Y in three different forms. Earlier attempts on
distribution of Y, as a convolution, are given in [2] and [4]. Our results appear to be in a
better and explicit forms as compared to that given by André (1875).
Theorem 2.1. The distribution of Y is given by
!
n
1
, y ∈ {0, 1, . . . , nk}
P(Y = y) =
(k + 1)n y k+1
2
(2.4)
Proof. Applying the inversion formula of the characteristic function for discrete distributions to (2.3), based on Fourier series, we have
Z π
1
P(Y = y) =
e−ity φY (t)dt
2π −π
 k
n
Z π
X eitp 
1
 dt
e−ity 
=
(2.5)

2π −π
k
+
1
p=0
n
 k
Z 2π
ip(t−π) 
X



1
e

 dt
e−iy(t−π) 
=

2π 0
k
+
1
p=0
The transformation w = eit , due to residue theory, yields
 k
n
X

1
y −y−1 
 (−1) p w p  dw
(−1)
w
P(Y = y) =


2πi(k + 1)n |w|=1
p=0
Applying Cauchy’s theorem, we have

 k
n 
 dy X

(−1)y

 (−1) p w p  
lim
P(Y = y) =

y!(k + 1)n w→0  dwy  p=0


 nk
!

 dy X
n
(−1)y
q
 , using Definition 1.1

 (−1)q


w
lim
=
q k+1 
y!(k + 1)n w→0  dwy  q=0
 nk

!
X

(−1)y
n
q!
q−y 
 (−1)q

=
lim
w

y!(k + 1)n w→0  q=y
q k+1 (q − y)!
!
y!
(−1)y
y n
(−1)
=
n
y!(k + 1)
y k+1 (y − y)!
!
n
1
=
n
(k + 1) y k+1
(2.6)
(2.7)
Next, we give two alternate expressions for P(Y = y).
Theorem 2.2. P(Y = y) for y ∈ {0, 1, . . . , nk} has the following form

⌊y/(k+1)⌋
p

 X
n
Γ(n
+
y
−
p(k
+
1))(−1)


P(Y = y) =
(k + 1)n  p=0 Γ(p + 1)Γ(n − p + 1)Γ(y − p(k + 1) + 1) 
3
(2.8)
Proof. Starting from (2.7), we have

 dy
(−1)y

P(Y = y) =
lim
y!(k + 1)n w→0  dwy
 k
n 
X
 
 (−1) p w p  
p=0
!n
d 1 + (−1)k+2 wk+1
(−1)
lim
=
y!(k + 1)n w→0 dwy
1+w

 n
!
p 
dy X n (−1)y

(−1)k+2 wk+1  ×
lim
=
y!(k + 1)n w→0 dwy  p=0 p
∞

X n + z − 1!

z
× 
(−w) 
z
z=0

∞ n
!
!

(−1)y
dy X X n n + z − 1
p(k+2)+z p(k+1)+z 


(−1)
w
=
lim


z
y!(k + 1)n w→0 dwy  z=0 p=0 p
X X n ! n + z − 1!
(−1)y
=
×
lim
z
y!(k + 1)n w→0 z p p
y
y
(p(k + 1) + z)!
(−1) p(k+2)+z w p(k+1)+z−y
(p(k + 1) + z − y)!
⌊y/(k+1)⌋ !

!

(−1)y  X n n + y − p(k + 1) − 1
p+y 
,

=
y!(−1)


y!(k + 1)n 
p
y − p(k + 1)
×
p=0
on using the transformations p(k + 1) + z − y = 0 and z = y − p(k + 1) ≥ 0
n
=
(k + 1)n
⌊y/(k+1)⌋

 X

Γ(n
+
y
−
p(k
+
1))
p

(−1) 
Γ(p + 1)Γ(n − p + 1)Γ(y − p(k + 1) + 1)
p=0
Theorem 2.3. For y ∈ {0, 1, . . . , nk}, P(Y = y) is given by
!
1
y+n−1
P(Y = y) =
×
(k + 1)n n − 1
!
−y + k −y − n + 1
−y − n + 1 + k
−y
,...,
;
,...,
;1
× k+2 Fk+1 −n,
k+1
k+1
k+1
k+1
j y k
!
y
(−n)⌊ k+1
+n−1
y
−
(k
+
1)
1
+
+1
⌋
k+1
− j y k
×
n−1
Γ k+1 + 2 (k + 1)n
y y −y
−y + k y × k+3 Fk+2 1, −n +
+
+
+ 1,
+ 1, . . . ,
+ 1;
k+1
k+1
k+1
k+1
k+1
!
y −y − n + 1 y −y − n + 1 + k y + 2,
+ 1, . . . ,
+ 1; 1
+
+
k+1
k+1
k+1
k+1
k+1
(2.9)
4
Proof: From (2.8), we get
⌊y/(k+1)⌋

p
 X

Γ(n
+
y
−
p(k
+
1))(−1)
n


P(Y = y) =
(k + 1)n  p=0 Γ(p + 1)Γ(n − p + 1)Γ(y − p(k + 1) + 1) 


k
Y


−y
+
a


(−n)
Γ
p
+
p

⌊y/(k+1)⌋
n−1
X
k
+
1
(−1)
a=0


=
! 
k
(n − 1)!(k + 1)n  p=0
Y
−y − n + 1 + a 

Γ p+
p!(k + 1)−n+1


k
+
1
a=0

k
k −y + a Y
Y

−y + a ⌊y/(k+1)⌋ (−n) p
Γ
 X
k+1
k+1 p
(−1)n−1
a=0
a=0

=
!
!
k
k
(n − 1)!(k + 1) Y
Y
−y − n + 1 + a  p=0
−y − n + 1 + a
Γ
p!

k+1
k+1
a=0
=
(−1)n−1
k
(n − 1)!(k + 1) Y
a=0
k
−y + a Y
Γ
k+1
a=0
−y − n + 1 + a
k+1
a=0
k Y
−y + a Γ
!×

k Y

−y + a 
(−n) p
(−n) p
X
∞
∞
X
k+1 p
k+1 p
a=0
a=0

× 
−
!
!
k
k

Y
Y
−y − n + 1 + a
−y − n + 1 + a
p=⌊y/(k+1)⌋+1
 p=0 p!
p!

k+1
k+1
p
a=0
=
(−1)n−1
k
(n − 1)!(k + 1) Y
a=0
k
−y + a Y
Γ
k+1
a=0
−y − n + 1 + a
Γ
k+1
a=0
k Y
−y + a 


(−n) p
X
∞
k+1 p
a=0

× 
!
k

Y
−y − n + 1 + a
 p=0 p!

k+1
a=0
p








!×
p
k Y


−y + a 
(−n)
q+⌊y/(k+1)⌋+1

∞
X
k
+
1
q+⌊y/(k+1)⌋+1
a=0

−

!
k
Y

−y − n + 1 + a
q=0

(q + ⌊y/(k + 1)⌋ + 1)!
k
+
1
q+⌊y/(k+1)⌋+1
a=0
5
p








(2.10)
Using multiplication for Gamma function [5, page11]
nz+b− 12
Γ(nz + b) = n
(2π)
1−n
2
n−1
Y
k=0
!
b+k
Γ z+
, n∈N
n
and the definition of hypergeometric function [5, page 41]
Qp
k
∞
X
j=1 (a j )k z
Qq
p F q (a1 , . . . , a p ; b1 , . . . , bq ; z) =
j=1 (b j )k k!
k=0
(2.11)
(2.12)
(2.10) yields (2.9). It may be observed that (2.12) is a terminating series (polynomial)
when any ai is a negative integer or zero.
3. Special Cases. It is interesting to obtain the distribution of X1 + X2 by various methods.
We first obtain the distribution of X1 + X2 in a different manner, and then obtain it as a
special case of Theorems 2.1, 2.2 and 2.3 respectively.
Lemma 3.1. We have
P(X1 + X2 = y) =
k − |k − y| + 1
, y ∈ {0, 1, . . . , 2k}
(k + 1)2
(3.1)
Proof.
P(X1 + X2 = y) =
X
P(X1 + X2 = y, X1 = z)
X
P(X2 = y − z, X1 = z)
X
P(X2 = y − z)P(X1 = z)
z
=
z
=
z
 y
X


1
y+1



=
,
0≤y≤k


2

(k + 1)2

 z=0 (k + 1)
=
k

X

1
2k − y + 1



=
, k + 1 ≤ y ≤ 2k


2

(k + 1)2
z=y−k (k + 1)
=
k − |k − y| + 1
(k + 1)2
The result (3.1) can be obtained from the convolution result given in [2] or [4] after
some simplifications. It can be obtained from Theorem 2.1 as a special case. From (2.5),
6
we have
 k
2
X eipt (−1) p 
 dt
e−ity (−1)y 

k
+
1
0
p=0
!
2
1
(k + 1)2 y k+1
 k

k Z
iqt
q 
ipt
p
X


1 X 2π −iyt
e
(−1)
e
(−1)

 dt
e (−1)y

2π q=0 0
k + 1 p=0 k + 1 
 k

k Z 2π
X
X eipt (−1) p 
1
−i(y−q)t
y−q 
 dt
e
(−1) 
2π(k + 1)2 q=0 0
k
+
1
p=0
!
k
X 1
1
(k + 1)2 q=0 y − q k+1
1
P(X1 + X2 = y) =
2π
=
=
=
=
=
=
=
=
Z
2π
 y
X 1 !





,
0≤y≤k




1
 q=0 y − q
!

k
X

(k + 1)2 
1



, k < y ≤ 2k



q=y−k y − k k+1
 y
X





(1), 0 ≤ y ≤ k




1
 q=0

k
X

(k + 1)2 



(1), k < y ≤ 2k



q=y−k



0≤y≤k
1
y + 1,


2
(k + 1) 2k − y + 1, k < y ≤ 2k
k − |k − y| + 1
(k + 1)2
The Lemma 3.1. can be verified from Theorem 2.2 also. From (2.6), we have
⌊y/(k+1)⌋ !

!

(−1)y  X 2 2 + y − p(k + 1) − 1
p+y


P(X1 + X2 = y) =
y!(−1)


y!(k + 1)2  p=0 p
y − p(k + 1)
⌊y/(k+1)⌋ !


(−1)y  X 2
p+y 


(y
−
p(k
+
1)
+
1)y!(−1)
=


y!(k + 1)2  p=0 p
We have that
y 


0, 0 ≤ y ≤ k
=

1, k < y ≤ 2k
k+1
7
So that
!


(−1)y 2



(y + 1)y!(−1)y ,


 y!(k + 1)2 0!
!
!
P(X1 + X2 = y) = 

2
2
1




 (k + 1)2 0 (y + 1) − 1 (y − (k + 1) + 1) ,

y+1



,
0≤y≤k


 (k + 1)2
=

2k − y + 1



, k < y ≤ 2k

(k + 1)2
k − |k − y| + 1
=
(k + 1)2
0≤y≤k
k < y ≤ 2k
The Lemma 3.1 can also be obtained from Theorem 2.3 as follows. From (2.9), we
have
!
y+1
−y + k −y − 1
P(Y = y) =
;
;1
2 F 1 −2,
(k + 1)2
k+1 k+1
y (−2)⌊ y ⌋+1
y − (k + 1) 1 +
+1 ×
− j y k k+1
k+1
Γ k+1 + 2 (k + 1)2
y −y + k y × 3 F2 1, −2 +
+
+ 1,
+ 1;
k+1
k+1
k+1
!
y −y − 1 y + 2,
+ 1; 1
+
k+1
k+1
k+1
For 0 ≤ y ≤ k,
!
y+1
−y + k −y − 1
P(Y = y) =
;
;1
2 F 1 −2,
(k + 1)2
k+1 k+1
!
k−y
2k − y + 1
y−k
; 2,
;1
+
3 F 2 1, −1,
(k + 1)2
k+1
k+1
y+1
=
(k + 1)2
For k < y ≤ 2k,
!
−y + k −y − 1
y+1
;
;1
P(Y = y) =
2 F 1 −2,
(k + 1)2
k+1 k+1
!
(−2)2
−y + k
−y − 1
−
(y − 2(k + 1) + 1)3 F2 1, 0,
+ 2; 3.
+ 2; 1
Γ3(k + 1)2
k+1
k+1
Thus,
P(X1 + X2 = y) =
8
k − |k − y| + 1
(k + 1)2
4. Further properties and identities of PC. In this section we give two more properties
of PC and two identities.
Property 4.1. We have,
n
q
!
k+1
!
k
X
n−1
=
q − z k+1
z=0
(4.1)
Proof. From (2.4) and (2.5), we have that
Thus,
 k
n
!
Z π
X eitp 
1
n
1
 dt
e−itq 
=

n
(k + 1) q k+1 2π −π
k
+
1
p=0
n
 k
!
Z π
X 
1
n
=
e−itq  eitp  dt
q k+1 2π −π
p=0
n−1
 k

Z π
k
X
X 


1


=
e−itq  eitr   eitp  dt
2π −π
p=0
r=0

n−1
Z π
k
k
X
X

1
=
e−it(q−r)  eitp  dt
2π −π
r=0
p=0
!
k
X n−1
=
q − r k+1
r=0
(4.2)
Property 4.2. We have
!
!
n
n
=
nk − y k+1
y k+1
Proof. From (4.2), we get
 k
n
!
Z π
X 
1
n
e−it(nk−y)  eitp  dt
=
2π −π
nk − y
p=0
 k
n
Z π
X

1
eit(nk−y)  e−it(k−p)  dt
=
2π −π
p=0
 k
n
Z π
X 
1
e−ity  eitp  dt
=
2π −π
p=0
!
n
=
y k+1
9
(4.3)
Property 4.3. We have
!
nk
X
n
n(k + 1)n k
q=
.
q k+1
2
q=0
(4.4)
Proof. We obtain E(Y) by two diffrent methods. It is easy to see that
E(Y) = nE(X)
k
n X
x
=
k + 1 x=0
=
nk
2
(4.5)
Also
d
φY (t)
t→0 dt
 k
n
X eitp 
d

= i−1 lim 

t→0 dt
k
+
1
p=0
 nk !

X n

d
= i−1 lim 
eitq 
t→0 dt
q k+1
q=0
 nk !

X n

1



q
=
(k + 1)n  q=0 q k+1 
E(Y) = i−1 lim
(4.6)
The expressions (4.5) and (4.6) result into the identity (4.4).
Property 4.4. We have
!
nk
X
n
n(k + 1)n k
(k(3n + 1) + 2)
q2 =
12
q k+1
q=0
(4.7)
Proof. We have
Var(Y) = nVar(X)
n
= k(k + 2)
12
(4.8)
Thus,
E(Y 2 ) = Var(Y) + (EY)2
nk
(k(3n + 1) + 2)
=
12
10
(4.9)
Also
d2
φY (t)
t→0 dt2
n
 k
d2 X eitp 

= − lim 2 
t→0 dt
k
+
1
p=0
 nk !

2 
X n


d

= −(k + 1)−n lim 2 
eitq 
t→0 dt
q k+1
q=0
 nk !

X n

1
2


q
=
(k + 1)n  q=0 q k+1 
E(Y 2 ) = i−2 lim
(4.10)
The expressions (4.9) and (4.10) yield the identity (4.7).
5. Multinomial Triangles. Multinomial triangles are number triangles formed by
polynomial coefficients. Analogously to [9], they can be built by beginning the first row
with the number one and filling the second with k+1 number ones. The p-th row elements
are computed by applying Property 4.1 or simply by summing the k + 1 elements above
to the left including the one directly above from (p − 1)-th row. For example, for the
quadrinomial triangle first 6 rows are given in Table 5.1.
1
1
1
1
1
1
1
2
3
4
5
1
3
6
10
15
1
4
10
20
35
3
12
31
65
2
12
40
101
1
10
44
135
6
40
155
3
31
135
1
20 10
101 65
4
35
1
15
5 1
Table 5.1
As expected, the multinomial triangles have properties similar to Pascal’s triangle.
= 0 for k ≤ 0 and F1(k+1) = F2(k+1) = 1. The Fibonacci (k + 1)-step
Property 5.1. Let F (k+1)
p
sequence linear recurrence equation is given by
F (k+1)
p
=
k+1
X
F (k+1)
p−i , k > 2
(5.1)
i=1
The shallow diagonals of multinomial triangles sum to Fibonacci (k + 1)-step numbers
[7].
Proof. The sum of the elements of the q-th diagonal is given by
(k+1)
Dq+1
!
q
X
q− p
=
p k+1
p=0
11
(5.2)
For q = 0,
!
0
=
= 1 = F1(k+1)
0 k+1
(5.3)
!
!
1
0
=
+
= 1 + 0 = 1 = F2(k+1)
0 k+1
1 k+1
(5.4)
!
m−1− p
, m≤n
p
k+1
(5.5)
D1(k+1)
and for q = 1,
D2(k+1)
Assume
Fm(k+1)
=
X
m−1
p=0
Then, using (4.1), we have
(k+1)
Dq+1
!
!
n
n X
n
X
X
n− p
n− p−1
=
=
p k+1 p=0 z=0
p − z k+1
p=0
!
k X
n−1
k n−z−1
X
X
X n − w − z − 1!
n− p−1
=
=
p − z k+1 z=0 w=0
w
k+1
z=0 p=z
=
k+1
X
(k+1)
(k+1)
Fn+1−z
= Fn+1
z=1
Property 5.2. The sum of the elements of the q-th row is
!
qn
X
q
= (k + 1)q
p
k+1
p=0
(5.6)
Proof. By making x = 1 in (1.1) we get
!
kq
X
q
1j
(1 + 1 + . . . + 1 ) =
j k+1
j=0
!
qn
X
q
q
(k + 1) =
j k+1
j=0
k q
Property 5.3. The sum of the first m + 1 rows are repunits in base k + 1 [8]. A repunit is
a number whose digits are copies of digit 1. In base b, we have
bn − 1
b−1
With b = 2, we obtain Mersenne numbers.
R(b)
n =
(5.7)
Proof. By using (5.6), we get
m
X
(k + 1)m+1 − 1
(k+1)
(k + 1)q =
= Rm+1
k
q=0
12
(5.8)
References
[1] D. André, Mmoire sur les combinaisons rgulires et leurs applications, Annales Scientifiques de l’. N. S., 2, tome 5 (1876), p. 155-198.
[2] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, 1974.
[3] L. Euler, “De evolutione potestatis polynomialis cuiuscunque
(1 + x + x2 + x3 + x4 + etc.)n ”, Nova Acta Academiae Scientarum Imperialis
Petropolitinae 12 (1801), 47-57.
[4] W. Feller, An Introduction to Probability Theory and Their Approximations, Vol.
I, Mathematics in Science and Engineering, vol. 53, Academic Press, New York,
1969.
[5] Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Academic
Press, New York, 1969.
[6] M. D. Springer, The Algebra of Random Variables, John Wiley & Sons, 1979.
[7] Eric W. Weisstein. “Fibonacci n-Step Number.” From MathWorld - A Wolfram
Web Resource.
[8] Eric W. Weisstein. “Repunit.” From MathWorld - A Wolfram Web Resource.
[9] Eric W. Weisstein. “Trinomial Coefficient.” From MathWorld - A Wolfram Web
Resource.
[10] Eric W. Weisstein. “Trinomial Triangle.” From MathWorld - A Wolfram Web
Resource.
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