On the link between binomial theorem and discrete convolution

0.19 MB·English

by  Petro Kolosov

ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE
CONVOLUTION
PETRO KOLOSOV
2 Abstract. Let Pm(x) be a 2m+1-degree integer-valued polynomial in b,x∈R
2 b
0 b−1 m
2 Pm(x)= A kr(x−k)r,
b m,r

n k=0r=0
XX
a whereA isarealcoefficient. InthismanuscriptweestablisharelationbetweenBinomial
J m,r
7 theoremandpolynomialPmb (x). Furthermore,arelationshipbetweenBinomialtheoremand
2 discrete convolution in terms of polynomials is provided.

]
T
N
.
h Contents
t
a
m 1. Definitions, notations and conventions 1
[
2. Introduction and main results 2

4 3. Polynomial Pm(x) and its properties 4
2 b
v 4. Polynomial Pm(x) in terms of Binomial theorem 7
8 b
6 5. Polynomial Pm(x) in terms of Discrete convolution 8
b
4
6. Relation between Binomial theorem and Discrete convolution 10
2
0
6.1. Generalization for Multinomials 12
.
3
7. Derivation of coefficient A 13
0 m,r
6
8. Verification of the results and examples 15
1
: 8.1. Mathematica commands 15
v
i 8.2. Examples 16
X
r 9. Acknowledgements 17
a
10. Conclusion 17
References 17
1. Definitions, notations and conventions
We now set the following notation, which remains fixed for the remainder of this paper:
Date: January 28, 2022.
2010 Mathematics Subject Classification. 44A35 (primary), 11C08 (secondary).
Key words and phrases. Binomial theorem, Convolution, Discrete convolution, Polynomials.
1
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 2
• A , m ∈ N is a real coefficient defined recursively
m,r
(2r+1) 2r , if r = m;
r
Am,r := (2r+1)(cid:0)2rr(cid:1) md=2r+1Am,d 2rd+1 (−d1−)dr−1B2d−2r, if 0 ≤ r < m; (1.1)



0, (cid:0) (cid:1)P (cid:0) (cid:1) if r < 0 or r > m,

whereB are Bernoulli numbers [Wei]. It is assumed that B = 1.
t 1 2
• Pm(x), m ∈ N is a 2m+1-degree integer-valued polynomial in b,x ∈ R
b
b−1 m
Pm(x) := A kr(x−k)r (1.2)
b m,r
k=0 r=0
XX
• H (b), m,t,b ∈ N is a polynomial defined as
m,t
m j (−1)j 2j −t+1
H (b) := A B (1.3)
m,t m,j 2j−t+1−b
t 2j −t+1 b
j=t (cid:18) (cid:19) (cid:18) (cid:19)
X
• X (j), m,t ∈ N is polynomial of degree 2m+1−t in j ∈ R
m,t
2m+1−t
X (j) := (−1)m H (k)·jk (1.4)
m,t m,t
k=1
X
• L (x,k), m ∈ N is 2m degree polynomial in x,k ∈ R
m
m
L (x,k) := A kr(x−k)r (1.5)
m m,r
r=0
X
• (f∗f)[n] is discrete convolution [BDM11] of function f defined over set of integers Z
(f ∗f)[n] = f(k)f(n−k)
k
X
2. Introduction and main results
The polynomial Pm(x), m ∈ N is 2m+1-degree integer-valued polynomial in x,b ∈ R.
b
b−1 m
Pm(x) = A kr(x−k)r,
b m,r
k=0 r=0
XX
where A is real coefficient. By means of Lemma 4.1, the polynomial Pm(x) has the
m,r b
following relation with Binomial theorem [AS72]
2m+1
2m+1
Pm (x+y) = x2m+1−ryr.
x+y r
r=0 (cid:18) (cid:19)
X
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 3
From the other hand, polynomial Pm(x) might be expressed in terms of discrete convolution
b
of polynomial nj, j ∈ N
m
Pm (x) = A (nr ∗nr)[x], n ≥ 0.
x+1 m,r
r=0
X
It is of first necessity to notice that nr of discrete convolution (nr ∗nr)[x] evaluated at x is
implicit piecewise-defined polynomial such as
n·n···n, if n ≥ 0
nr =
 rtimes
0, otherwise
| {z }
Therefore, it is easy to notice the following identities in terms of Binomial theorem and

discrete convolution, see Corollaries 6.1, 6.2
m 2m+1
2m+1
A (nr ∗nr)[x+y] = 1+ x2m+1−ryr, n ≥ 0,
m,r
r
r=0 r=0 (cid:18) (cid:19)
X X
m 2m+1
2m+1
A (nr ∗nr)[x+y] = −1+ x2m+1−ryr, n > 0.
m,r
r
r=0 r=0 (cid:18) (cid:19)
X X
Also, the following generalizations for multinomial case are discussed, see Corollaries 6.3, 6.4
m t
2m+1
A (nr ∗nr)[x +x +···+x ] = 1+ xkℓ, n ≥ 0,
m,r 1 2 t k ,k ,...,k ℓ
Xr=0 k1+k2+·X··+kt=2m+1(cid:18) 1 2 t(cid:19)Yℓ=1
m t
2m+1
A (nr ∗nr)[x +x +···+x ] = −1+ xkℓ, n > 0.
m,r 1 2 t k ,k ,...,k ℓ
Xr=0 k1+k2+·X··+kt=2m+1(cid:18) 1 2 t(cid:19)Yℓ=1
A few polynomial identities are straightforward as well by means of Theorems 5.3, 5.5.
Precisely, by the theorem 5.3 we have an odd-power identity as follows
m x−1
x2m+1 = A kr(x−k)r
m,r
r=0 k=0
X X
From the other prospective, the theorem 5.5 concludes as follows
m x
x2m+1 = A kr(x−k)r
m,r
r=0 k=1
X X
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 4
In its explicit form an identity x2m+1 = m A x−1kr(x−k)r looks like as follows
r=0 m,r k=0
x
P P
x3 = 6k(x−k)+1
k=1
X
x
x5 = 30k2(x−k)2 +1
k=1
X
x
x7 = 140k3(x−k)3 −14k(x−k)+1
k=1
X
x
x9 = 630k4(x−k)4 −120k(x−k)+1
k=1
X
x
x11 = 2772k5(x−k)5 +660k2(x−k)2 −1386k(x−k)+1
k=1
X
x
x13 = 51480k7(x−k)7 −60060k3(x−k)3 +491400k2(x−k)2 −450054k(x−k)+1
k=1
X
3. Polynomial Pm(x) and its properties
b
We continue our mathematical journey from short overview of polynomial L (x,k) that
m
is essential part of polynomial Pm(x) since that Pm(x) = b−1 L (x,k). Polynomial
b b k=0 m
L (x,k), m ∈ N is polynomial of degree 2m in x,k ∈ R, see definition (1.5). In explicit
m P
form the polynomial L (x,k) is as follows
m
L (x,k) = A km(x−k)m +A km−1(x−k)m−1 +···+A ,
m m,m m,m−1 m,0
where A are real coefficients defined by (1.1). Coefficients A are nonzero only for r
m,r m,r
within the interval r ∈ {m}∪ 0, m−1 . For example,
2
(cid:2) (cid:3)
m/r 0 1 2 3 4 5 6 7
0 1
1 1 6
2 1 0 30
3 1 -14 0 140
4 1 -120 0 0 630
5 1 -1386 660 0 0 2772
6 1 -21840 18018 0 0 0 12012
7 1 -450054 491400 -60060 0 0 0 51480
Table 1. Coefficients A . See the OEIS entries: A302971, A304042.
m,r
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 5
Thus, the polynomial L (x,k) may also be written as
m
m−1
2
L (x,k) = A km(x−k)m + A kr(x−k)r
m m,m m,r
r=0
X
For example, the polynomials L (x,k) for 0 ≤ m ≤ 3 are
m
L (x,k) = 1,
0
L (x,k) = 6k(x−k)+1 = −6k2 +6kx+1,
1
L (x,k) = 30k2(x−k)2 +1 = 30k4 −60k3x+30k2x2 +1,
2
L (x,k) = 140k3(x−k)3 −14k(x−k)+1
3
= −140k6 +420k5x−420k4x2 +140k3x3 +14k2 −14kx+1
It is worth to notice that L (x,k) is symmetrical over x
m
Property 3.1. For every x,k ∈ R
L (x,k) = L (x,x−k)
m m
This might be seen in the following table
x/k 0 1 2 3 4 5 6 7
0 1
1 1 1
2 1 7 1
3 1 13 13 1
4 1 19 25 19 1
5 1 25 37 37 25 1
6 1 31 49 55 49 31 1
7 1 37 61 73 73 61 37 1
Table 2. Values of L (x,k). See the OEIS entry: A287326.
1
Next we discuss the polynomial Pm(x). In its extended form, the polynomial Pm(x) is
b b
b−1 b−1 m m b−1
Pm(x) = L (x,k) = A kr(x−k)r = A kr(x−k)r
b m m,r m,r
k=0 k=0 r=0 r=0 k=0
X XX X X
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 6
By the binomial theorem (x−y)n = n (−1)k n xn−kyk,
k=0 k
m P b−1 (cid:0)r(cid:1) r
Pm(x) = A kr (−1)j xr−jkj
b m,r j
r=0 k=0 j=0 (cid:18) (cid:19)
X X X
m b−1 r
r
= A (−1)j xr−jkr+j
m,r
j
r=0 k=0 j=0 (cid:18) (cid:19)
X XX
m r b−1
r
= A (−1)jxr−j kr+j
m,r
j
r=0 j=0 (cid:18) (cid:19)k=0
X X X
However, by the symmetry (3.1) of L (x,k) the polynomial Pm(x) may also be written in
m b
the form
b m b m r
r
Pm(x) = A kr(x−k)r = A kr (−1)r−txt kr−t
b m,r m,r t
k=1 r=0 k=1 r=0 t=0 (cid:18) (cid:19)
XX XX X
m b m
r
= xt (−1)r−t A k2r−t
m,r
t
t=0 k=1 r=t (cid:18) (cid:19)
X XX
(−1)m−tXm,t(b)
Note that b m (−1|)r−t r A k{2rz−t is the (−}1)m−tX (b). From this formula it
k=1 r=t t m,r m,t
may be not immediately clear why X (b) represent polynomials in b. However, this
P P (cid:0) (cid:1) m,t
can be seen if we change the summation order and use Faulhaber’s formula n kp =
k=1
1 p p+1 B np+1−j to obtain
p+1 j=0 j j P
P (cid:0) (cid:1) m r (−1)r 2r−t 2r−t+1
X (b) = (−1)m A B b2r−t+1−ℓ
m,t m,r ℓ
t 2r −t+1 ℓ
r=t (cid:18) (cid:19) ℓ=0 (cid:18) (cid:19)
X X
Introducing k = 2r−t+1−ℓ we further get the formula
2m−t+1 m r (−1)r 2r −t+1
X (b) = (−1)m bk A B
m,t m,r 2r−t+1−k
t 2r−t+1 k
k=1 r=t (cid:18) (cid:19) (cid:18) (cid:19)
X X
Hm,t(k)
Polynomials X (b), 0 ≤ t ≤ 3 are | {z }
3,t
X (j) = 7b2 −28b3 +70b5 −70b6 +20b7,
3,0
X (j) = 7b−42b2 +175b4 −210b5 +70b6,
3,1
X (j) = −14b+140b3 −210b4 +84b5,
3,2
X (j) = 35b2 −70b3 +35b4
3,3
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 7
Polynomials H (k) are defined by (1.3) and examples for m = 3, 0 ≤ t ≤ 3 are
3,t
1 14 3 7
H (k) = B + B −20B ,
3,0 1−k 3−k 7−k
k 3 k k
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
2 6
H (k) = 7B −70B ,
3,1 2−k 6−k
k k
(cid:18) (cid:19) (cid:18) (cid:19)
5
H (k) = −84B ,
3,2 5−k
k
(cid:18) (cid:19)
4
H (k) = −35B
3,3 4−k
k
(cid:18) (cid:19)
It gives us an opportunity to overview the polynomial Pm(x) from the different prospective,
b
for instance
m m 2m−r+1
Pm(x) = (−1)m−rX (b)·xr = (−1)2m−rH (ℓ)·bℓ ·xr (3.1)
b m,r m,r
r=0 r=0 ℓ=1
X X X
Equation (3.1) clearly states why Pm(x) is polynomial in x,b. For example,
b
P0(x) = b,
b
P1(x) = 3b2 −2b3 −3bx+3b2x,
b
P2(x) = 10b3 −15b4 +6b5
b
−15b2x+30b3x−15b4x
+5bx2 −15b2x2 +10b3x2,
P3(x) = −7b2 +28b3 −70b5 +70b6 −20b7
b
+7bx−42b2x+175b4x−210b5x+70b6x
+14bx2 −140b3x2 +210b4x2 −84b5x2
+35b2x3 −70b3x3 +35b4x3
The following property also holds for Pm(x)
b
Property 3.2. For every m ∈ N, x,b ∈ R
Pm (x) = Pm(x)+L (x,b)
b+1 b m
4. Polynomial Pm(x) in terms of Binomial theorem
b
Lemma 4.1. For every m ∈ N, x,y ∈ R
2m+1
2m+1
Pm (x+y) = x2m+1−ryr
x+y r
r=0 (cid:18) (cid:19)
X
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 8
By Lemma 4.1 and equation (3.1) the following polynomial identities straightforward
m 2m−r+1 m
x2m+1 = (−1)2m−rH (ℓ)·xℓ+r = (−1)m−rX (x)·xr
m,r m,r
r=0 ℓ=1 r=0
X X X
For instance,
P2 (x+y) = (x+y)(x4 +4x3y +6x2y2+4xy3 +y4).
x+y
In addition, the following identities hold
m 2m−r+1
(x+y)2m+1 = (−1)2m−rH (ℓ)·(x+y)ℓ+r
m,r
r=0 ℓ=1
X X
m
= (−1)m−rX (x+y)·(x+y)r
m,r
r=0
X
Obviously, Multinomial expansion of t-fold sum (x +x +···+x )2m+1 can be reached by
1 2 t
Pm(x +x +···+x ) as well
b 1 2 t
Corollary 4.2. For all x ,x ,...,x ∈ R, m ∈ N
1 2 t
t
2m+1
Pm (x +x +···+x ) = xks
x1+x2+···+xt 1 2 t k ,k ,...,k t
k1+k2+·X··+kt=2m+1(cid:18) 1 2 t(cid:19)Ys=1
Moreover, the following multinomial identities hold
m 2m−r+1
(x +x +···+x )2m+1 = (−1)2m−rH (ℓ)·(x +x +···+x )ℓ+r
1 2 t m,r 1 2 t
r=0 ℓ=1
X X
m
= (−1)m−rX (x +x +···+x )·(x +x +···+x )r
m,r 1 2 t 1 2 t
r=0
X
5. Polynomial Pm(x) in terms of Discrete convolution
b
In this section we discuss the relation between Pm(x) and discrete convolution of polyno-
b
mials. To show that Pm(x) involves the discrete convolution of polynomial nr let’s remind
b
the definition of Pm(x)
b
b−1 m m b−1
Pm(x) = A kr(x−k)r = A kr(x−k)r
b m,r m,r
k=0 r=0 r=0 k=0
XX X X
A discrete convolution of defined over set of integers Z function f is
(f ∗f)[n] = f(k)f(n−k)
k
X
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 9
General formula of discrete convolution for polynomials f(n) = nj, n ≥ a ∈ R may be
derived immediately
(nj ∗nj)[x] = kj(x−k)j[k ≥ a][x−k ≥ a]
k
X
= kj(x−k)j[k ≥ a][k ≤ x−a]
k
X
= kj(x−k)j[a ≤ k ≤ x−a]
k
X
x−a
= kj(x−k)j,
k=a
X
where [a ≤ k ≤ x−a] is Iverson’s bracket [Ive62].
Lemma 5.1. For every n ∈ N, x ∈ R
x
(nr ∗nr)[x] = kr(x−k)r, n ≥ 0.
k=0
X
Itisoffirstimportancetokeepinmindthatnr ofdiscreteconvolution(nr∗nr)[x]evaluated
at x is implicit piecewise-defined polynomial such as
n·n···n, if n ≥ 0
nr =
 rtimes
0, otherwise
| {z }
Thus, the corollary follows 

Corollary 5.2. By Lemma 5.1 the polynomial Pm(n) might be expressed in terms of discrete
b
convolution as follows
m
Pm (x) = A (nr ∗nr)[x], n ≥ 0.
x+1 m,r
r=0
X
Therefore, another polynomial identity follows
Theorem 5.3. By Lemma 4.1, Corollary 5.2 and property 3.2, for every m ∈ N, x ∈ R
m
1+x2m+1 = A (nr ∗nr)[x], n ≥ 0.
m,r
r=0
X
NowwenoticethefollowingidentityintermsofpolynomialPm(x)anddiscreteconvolution
b
(nj ∗nj)[x]
ON THE LINK BETWEEN BINOMIAL THEOREM AND DISCRETE CONVOLUTION 10
Proposition 5.4. For every m ∈ N, x ∈ R
m x−1
Pm(x) = A 0rxr + kr(x−k)r
x m,r
!
r=0 k=1
X X
m m
= A 0rxr + A (nr ∗nr)[x]
m,r m,r
r=0 r=0
X X
m
= 1+ A (nr ∗nr)[x], n ≥ 1.
m,r
r=0
X
Since that for all r in A 0rxr we have
m,r
1, if r = 0
A 0rxr =
m,r

0, if r > 0

Above is true because A = 1 for every m ∈ N, and x0 = 1 for every x, [GKP94]. Hence,
m,0 
the following identity between Pm(x) and discrete convolution (nj ∗nj)[x] holds
b
Theorem 5.5. By Lemma 4.1 and Proposition 5.4, for every m ∈ N, x ∈ R
m
−1+x2m+1 = A (nr ∗nr)[x], n > 0.
m,r
r=0
X
Corollary 5.6. By Theorem 5.5, for all m ∈ N
m
A = 22m+1 −1
m,r
r=0
X
Corollary 5.6 holds since that convolution (nj ∗nj)[x] = 1, n > 0 for each r and x = 2.
6. Relation between Binomial theorem and Discrete convolution
Corollary 6.1. (Generalization of Theorem 5.3 for Binomials.) For every m ∈ N, x,y ∈ R
m 2m+1
2m+1
A (nr ∗nr)[x+y] = 1+ x2m+1−ryr, n ≥ 0.
m,r
r
r=0 r=0 (cid:18) (cid:19)
X X