ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS

ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS

Hac` ene Belbachir and Farid Bencherif USTHB, Faculty of Mathematics,

P.O.Box 32, El Alia, 16111, Algiers, Algeria e-mail: hbelbachir@usthb.dz

or hacenebelbachir@gmail.com e-mail: fbencherif@usthb.dz

or fbencherif@gmail.com

Abstract

Letting T

(resp. U

) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences X

T

k

and X

U

k

for n − 2 bn/2c ≤ k ≤ n − bn/2c are two basis of the Q-vectorial space E

[X] formed by the polynomials of Q [X]

having the same parity as n and of degree ≤ n. Also T

and U

admit remarkableness integer coordinates on each of the two basis.

Keywords: Chebyshev polynomials, integer coordinates.

2000 Mathematics Subject Classification: 11B83, 33C45, 11B39.

1. Introduction and main results

For each integer n ≥ 0, T

= T

(X) and U

= U

(X) denote the unique polynomials, with integer coefficients, satisfying

cos nx = T

(cos x) and sin((n + 1)x) = sin x U

(cos x), (x ∈ R) .

The well known Simpson’s formulae [2], for all n ∈ N and x ∈ R : cos nx = 2 cos x cos (n − 1) x − cos (n − 2) x sin (n + 1) x = 2 cos x sin nx − sin (n − 1) x give the recurrence relations

T

= 2XT

− T

, with T

= 1, T

= X, (1)

U

= 2XU

− U

, with U

= 1, U

= 2X.

(2)

We deduce (see [1]), for n ≥ 1, the following relations

T

=

bn/2c

X

k=0

(−1)

2

n n − k

n − k k

 X

, (3)

U

=

bn/2c

X

k=0

(−1)

2

n − k k

 X

. (4)

These relations allows to state that, for any n ≥ 0, T

and U

belong to E

[X] , (3) and (4) are their decompositions in the canonical basis B

:= X

0≤k≤bn/2c

For instance, the first fifth values of thus polynomials are

T

= 1 U

= 1

T

= X U

= 2X

T

= 2X

− 1 U

= 4X

− 1 T

= 4X

− 3X U

= 8X

− 4X

T

= 8X

− 8X

+ 1 U

= 16X

− 12X

+ 1

The main goal of this paper is to prove that the families T

:= X

T

k

and U

:= X

U

k

for n − 2 bn/2c ≤ k ≤ n − bn/2c constitute two other basis of E

[X] (Theorem 1) for which T

and U

admit remarkableness integer coordinates.

Our first result is the following

Theorem 1. For any n ≥ 0, T

and U

are two basis of E

[X] .

So we can decompose T

and U

(resp. T

and U

) over each of the basis T

and U

(resp. T

and U

). Decompositions of T

over T

and U

over U

are trivial, it remains to examine the six decompositions:

1. The decomposition of T

over T

and U

over U

in Theorem 2.

2. The decomposition of T

over U

and U

over T

in Theorem 3.

3. The decomposition of T

over U

and U

over T

in Theorem 4.

Let us define the families of integers (α

) , (β

) and (γ

) by the following equalities

− (1 − 2X)

= X

k≥0

α

X

, (5)

(X − 1) (1 − 2X)

= X

k≥0

β

X

, (6)

(2X − 1)

+ 2 

1 + (2X − 1) + · · · + (2X − 1)



= X

k≥0

γ

X

, (7)

we deduce then, for (n, k) ∈ N

, the following relations α

= (−1)

2

n

k



,

(8)

β

= α

− α

, (9)

γ

= (−1)

α

+ 2

n−1

X

j=0

(−1)

α

. (10)

Theorem 2. For each n ≥ 0, we have

T

=

n+1

X

k=1

β

X

T

, (11)

U

=

n+1

X

k=1

α

X

U

. (12)

Theorem 3. For each n ≥ 0, we have

T

= U

− XU

, n ≥ 1, (13)

U

=

n

X

k=0

γ

X

T

. (14)

Theorem 4. For each n ≥ 0, we have

T

=

n+1

X

k=1

(α

− δ

) X

U

, (15)

U

=

n+1

X

k=1

(γ

+ β

) X

T

, (16)

where δ

denotes the Kronecker symbol.

The sequences of integers (α

) , (β

) and (γ

) satisfy the following recurrence relation



 

 

 

 

α

= −1, for n ≥ 0, α

= 0, for k ≥ 1,

α

= α

− 2α

, for n, k ≥ 1,



 

 

 

 

β

= −1, for n ≥ 0,

β

= 1 and β

= 0, for k ≥ 2, β

= β

− 2β

, for n, k ≥ 1,



 

 

 

 

γ

= 1, for n ≥ 0, γ

= 0, for k ≥ 1,

γ

= −γ

+ 2γ

, for n, k ≥ 1.

The following tables give the values of α

, β

and γ

for 0 ≤ n ≤ 4

n α

α

α

α

α

0 −1

1 −1 2

2 −1 4 −4

3 −1 6 −12 8

4 −1 8 −24 32 −16

n β

β

β

β

β

β

0 −1 1

1 −1 3 −2

2 −1 5 −8 4

3 −1 7 −18 20 −8

4 −1 9 −32 56 −48 16

n γ

γ

γ

γ

γ

0 1

1 1 2

2 1 0 4

3 1 2 −4 8

4 1 0 8 −16 16

Notice that α

= γ

= 0 for k > n and β

= 0 for k > n + 1.

According to these tables, one obtains

• Using Theorem 2 T

= XT

T

= 3XT

−2X

T

T

= 5XT

−8X

T

+4X

T

T

= 7XT

−18X

T

+20X

T

−8X

T

T

= 9XT

−32X

T

+56X

T

−48X

T

+16X

T

U

= 2XU

U

= 4XU

−4X

U

U

= 6XU

−12X

U

+8X

U

U

= 8XU

−24X

U

+32X

U

−16X

U

• Using Theorem 3

T

= U

T

= U

−XU

T

= U

−XU

T

= U

−XU

T

= U

−XU

U

= T

U

= T

+2XT

U

= T

+0XT

+4X

T

U

= T

+2XT

−4X

T

+8X

T

U

= T

+0XT

+8X

T

−16X

T

+16X

T

• Using Theorem 4

T

= XU

T

= 3XU

−4X

U

T

= 5XU

−12X

U

+8X

U

T

= 7XU

−24X

U

+32X

U

−16X

U

T

= 9XU

−40X

U

+80X

U

−80X

U

+32X

U

U

= 2XT

U

= 4XT

U

= 6XT

−8X

T

+8X

T

U

= 8XT

−16X

T

+16X

T

U

= 10XT

−32X

T

+64X

T

−64X

T

+32X

T

2. Proofs of Theorems 2.1. Proof of Theorem 1

T

and U

are two families of polynomials of E

[X] with cardT

= cardU

= dim E

[X] = bn/2c + 1. Using the following Lemma, we prove that the determinant of T

and U

relatively to the canonical basis B

of E

[X] are not zero. Theorem 1 follows.

Lemma 5. For any integer n ≥ 0, by setting m = bn/2c , we have det

(T

) = 2

and det

(U

) = 2

.

P roof. For any integer m ≥ 0 and for 1 ≤ k ≤ m + 1, set V

= (2X)

T

and W

= (2X)

U

. Notice that V

and W

are polynomials of E

[X] with dominant coefficient 2

and 2

respectively. Using the recurrence equations (1) and (2), we obtain for m ≥ 1

V

− V

= V

and W

− W

= W

.

Let ∆

= det



V

, V

, . . . , V

 and

D

= det



W

, W

, . . . , W

 . We have

∆

= det



V

, V

− V

, V

− V

, . . . , V

− V



= det

 V

, V

, V

, . . . , V



= 2

∆

= 2

∆

= 2

,

and similarly, we obtain D

= 2

D

= 2

D

= 2

. For n = 2m + r, with m = bn/2c and r ∈ {0, 1} , we have

det

(T

) = det

X

T

, X

T

, . . . , X

T

= det

(T

, XT

, . . . , X

T

)

= 2

∆

= 2

,

and similarly det

(U

) = 2

D

= 2

.

2.2. Proof of Theorems 2, 3 and 4

Let us denote E denote the shift operator on Q [X]

defined by

E ((W

)

) = (W

)

,

or in a more simple form

EW

= W

, (n ≥ 0) .

For any m ≥ 0, define the operators A

= − (E − 2X)

, B

= (X − E) (E − 2X)

,

C

= (2X − E)

+ 2

m

X

k=1

E

(2X − E)

.

Using relations (5) , (6) and (7) , we have also

A

=

m

X

k=0

α

X

E

,

B

=

m+1

X

k=0

β

X

E

,

C

=

m

X

k=0

γ

X

E

.

Lemma 6. For any integer n, we have

(a) (2X − E)

T

= T

and (2X − E)

U

= U

, (m ≥ n ≥ 0) . (b) T

= U

− XU

(n ≥ 1) .

(c) 2T

= U

− U

(n ≥ 2) . (d) U

= 1 + 2 P

k=1

T

(n ≥ 0) .

P roof.

(a) For m ≥ 1, one has (2X − E) T

= 2XT

− T

= T

and (2X − E) U

= 2XU

− U

= U

. We conclude by induction.

(b) Letting for n ≥ 1, W

= T

− U

+ XU

. The sequence (W

)

satisfies the following relation

W

= 2XW

− W

, (n ≥ 3) , with W

= W

= 0,

leading to W

= 0, for n ≥ 1.

(c) For n ≥ 2, one has 2T

= U

+ (U

− 2XU

) from (b) and thus 2T

= U

− U

from (2) .

(d) For n ≥ 0, one has U

= U

+ P

k=1

(U

− U

) = 1 + 2 P

T

from (c).

P roof of T heorem 2. Relations (11) and (12) are respectively equivalent to

B

T

= 0 and A

U

= 0, for n ≥ 0.

These last relations follows from Lemma 6 (a). We have, for any integer n ≥ 0

B

T

= (−1)

(X − E) (2X − E)

T

= (−1)

(X − E) T

= (−1)

(XT

− T

) = 0, and

A

U

= (−1)

(2X − E) (2X − E)

U

= (−1)

(2X − E) U

= (−1)

(2XU

− U

) = 0.

P roof of T heorem 3. Relation (13) follows from Lemma 6 (b).

Relation (14) is equivalent to U

= C

T

, for n ≥ 0. One has indeed,

for any n ≥ 0

C

T

= 

(2X − E)

+ 2 P

k=1

E

(2X − E)

 T

= T

+ 2

n

X

k=1

T

= U

, (from (d) of Lemma 6).

P roof of T heorem 4. For any n ≥ 0, we have

T

= U

− XU

, (from (b) of Lemma 6)

=

n+1

X

k=1

α

X

U

− XU

(from (12) )

=

n+1

X

k=1

(α

− δ

) X

U

.

We have also, for any n ≥ 0

U

= XU

+ T

, (from (b) of Lemma 6)

=

n

X

k=0

γ

X

T

+

n+1

X

k=1

β

X

T

, (from (11) and (14) )

=

n+1

X

k=1

(γ

+ β

) X

T

.

References

[1] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square matrix, Integers 6 (A12) (2006), 1–17.

[2] E. Lucas, Th´eorie des Nombres, Ghautier-Villars, Paris 1891.

[3] T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, second edition, Wiley Interscience 1990.

Received 30 April 2007

Revised 24 July 2007