Ann. Univ. Ferrara - Sez. VII - Sc. Mat.
Vol. XLVII, 107-115 (2001)
A Note on Chebyshev Polynomials.
G. DATTOLI (*) - D. SACCHETTI (**) - C. CESARANO (***)
SUNTO - Si derivano nuove famiglie di funzioni generatrici e di identità relative ai polinomi di Chebyshev. Si dimostra che il metodo proposto permette la derivazione di regole di somma relative a prodotti di polinomi di Chebyshev e teoremi di addizione.
La possibilità di estendere i risultati includendo funzioni generatrici di prodotti di
polinomi di Chebyshev ed altri polinomi è infine analizzata.
ABSTRACT - New families of generating functions and identities concerning the Chebyshev polynomials are derived. It is shown that the proposed method allows the
derivation of sum rules involving products of Chebyshev polynomials and addition
theorems. The possiblity of extending the results to include gnerating functions involving products of Chebyshev and other polynomials is finally analyzed.
1. – Introduction.
The link between first and second kind Chebyshev polynomials and elementary circular functions is expressed by the relations [1]
Tn (x) 4 cos (n Q arccos (x) ) ,
(1)
Un (x) 4
sin ( (n 1 1 ) arccos (x) )
.
k1 2 x 2
(*) Indirizzo dell’autore: Dipartimento Innovazione Divisione Fisica Applicata C.
R. Frascati, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy.
E-mail: dattoliHfrascati.enea.it
(**) Indirizzo dell’autore: Dipartimento di Statistica, Probabilità e Stat. Appl.,
Università Degli Studi «La Sapienza», Piazzale Aldo Moro 5, 00185 Rome, Italy
E-mail: sacchettHpow2.sta.uniroma1.it
(***) Indirizzo dell’autore: Via G. Carducci 2, Roma.
108
G. DATTOLI
-
D. SACCHETTI
- C.
CESARANO
It is therefore convenient to introduce the complex quantity
(2)
Rn (x) 4 exp (in Q arccos (x) )
and define
Tn (x) 4 D[Rn (x) ] ,
(3)
Un 2 1 (x) 4
4[Rn (x) ]
.
k1 2 x 2
The use of eq. (2) allows the derivation of the following result
Q
! j R (x) 4 1 2 j exp (i1Q arccos (x)) ,
n
(4)
NjNE 1
n
n40
which, according to eqs. (3), yields the well known identities [1]
Q
! j T (x) 4 1 2122jxjx1 j
n
n
n40
(5)
Q
!j U
n
n40
n 2 1 (x) 4
2
j
1 2 2 jx 1 j 2
,
,
NjNE 1 .
The same procedure provides the obvious relation
n
Q
! jn! R (x) 4 exp (j exp (i cos
(6)
n
n40
21
(x) ) )
leading to the further identities
n
Q
! jn! T (x) 4 exp (jx) cos (j k1 2 x ) ,
(7)
2
n
n40
n
Q
! jn! U
n40
n 2 1 (x) 4 exp (jx)
sin (j k1 2 x 2 )
,
k1 2 x 2
which, albeit trivial, do not currently appear in literature.
It is evident that the use of the complex representation of Chebyshev polynomials can be exploited to derive less trivial relations, we find indeed
Q
!j T
n
(8)
n40
n 1 l (x) 4
Q
!j U
n
n40
n 1 l 2 1 (x) 4
( 1 2 jx) Tl (x) 2 j( 1 2 x 2 ) Ul 2 1 (x)
1 2 2 jx 1 j 2
jTl (x) 1 ( 1 2 jx) Ul 2 1 (x)
1 2 2 jx 1 j 2
,
,
NjNE 1 ,
A NOTE ON CHEBYSHEV POLYNOMIALS
109
and
Q
n
! jn! T
n40
n 1 l (x) 4
4 exp (xj)[ cos (j k1 2 x 2 ) Tl (x) 2 k1 2 x 2sin (j k1 2 x 2 ) Ul 2 1 (x) ] ,
Q
n
! jn! U
(9)
n40
n 1 l 2 1 (x) 4
4 exp (xj)
[ k1 2 x 2cos (j k1 2 x 2 ) Ul 2 1 (x) 1 sin (j k1 2 x 2 ) Tl (x)
.
k1 2 x 2
Eqs. (8, 9) appear overlooked and not reported in literature, at least to the
authors’ knowledge.
In the next sections it will be shown that the simple method we have proposed in these introductory remarks offers a fairly important tool of analysis
for a wide classes of properties of the Chebyshev polynomials. We will indeed
employ the complex representation to derive generating functions involving
products of Chebyshev polynomials and products of Chebyshev with other
type of polynomials.
The plane of the paper is the following. In section 2 generating functions
involving products of Chebyshev polynomials are derived. Addition theorems
are discussed in section 3. Finally Section 3 is devoted to concluding remarks
concerning the possibility of extending the obtained results to products of
Chebyshev, Hermite and Laguerre polynomials.
2. – Generating functions for products of Chebyshev polynomials.
The following two consequences of eqs. (2,3) are trivial but crucial for our
purposes
NRn (x) N2 4 [Tn (x) ]2 1 ( 1 2 x 2 )[Un 2 1 (x) ]2 4 1 ,
(10)
D( [Rn (x) ]2 ) 4 [Tn (x) ]2 2 ( 1 2 x 2 )[Un 2 1 (x) ]2 ,
4( [Rn (x) ]2 ) 4 2 k12 x 2 Tn (x) Un 2 1 (x) .
It is also evident that
Q
(11)
n
! jn! NR (x) N 4 exp (j) ,
! jn! [R (x)] 4 exp (j exp ( 2 i cos
Q
n40
2
n
n40
n
n
2
21
(x) ) ) .
110
G. DATTOLI
-
D. SACCHETTI
- C.
CESARANO
The last two identities can be combined to get
n
Q
(12)
! jn! NT (x)N 4 12 [exp (j)1exp (j(2x 21)) cos (2jx k12x )] ,
1
[exp (j)2exp (j(2x 21)) cos (2jx k12x )]
! jn! NU (x)N 4 2(12x
)
2
n
n40
2
2
n
Q
2
n21
n40
2
2
2
and also
exp (j( 2 x 2 2 1 ) )
n
Q
(13)
! jn! T (x) U
n
n40
n 2 1 (x) 4
2 k1 2 x
2
sin ( 2 xj k1 2 x 2 ) .
Furthermore by noting that
Q
! j NR (x) N 4 1 21 j , NjNE 1
! j [R (x)] 4 12 j exp ( 21i cos (x))
n
n40
(14)
2
n
Q
n
n40
2
n
21
we end up with the identities
! j [T (x)] 4 12 1 21 j y1 1 ( 1122 j)(2 j(122x j(221x) 12j1 )) z ,
! j [U (x)] 4 2( 1 21 x ) 1 21 j y1 2 ( 1122j)(2 j(122x j(221x) 12j1 )) z
2
Q
n
(15)
2
n
n40
2
2
2
Q
n
2
n21
n40
2
2
2
and
Q
(16)
! j T (x) U
n
n40
n
n 2 1 (x) 4
xj
( 1 2 2 j( 2 x 2 2 1 ) 1 j 2 )
.
It is also easily argued that (the * denotes complex conjugation)
(17)
D[Rn (x) R*n (y) ] 4 Tn (x) Tn (y) 1 k( 1 2 x 2 )( 1 2 y 2 ) Un 2 1 (x) Un 2 1 (y) ,
4(Rn (x) R*n (y) ) 4 k1 2 x 2 Un 2 1 (x) Tn (y) 2 k1 2 y 2 Un 2 1 (y) Tn (x)
and
(18)
D[Rn (x) Rn (y) ] 4 Tn (x) Tn (y) 2 k( 1 2 x 2 )( 1 2 y 2 ) Un 2 1 (x) Un 2 1 (y) ,
4(Rn (x) Rn (y) ) 4 k1 2 x 2 Un 2 1 (x) Tn (y) 2 k1 2 y 2 Un 2 1 (y) Tn (x) .
111
A NOTE ON CHEBYSHEV POLYNOMIALS
Which allow to get the further generating functions
n
Q
! jn! T (x) T (y) 4 12 [e
n
n40
n
Q
(19)
n
! jn! U
n40
n 2 1 (x)
Un 2 1 (y) 4
jF1
cos (jF 2 ) 1 e jF2 cos (jF 1 ) ] ,
21 [e jF2 cos (jF 1 ) 2 e jF1 cos (jF 2 ) ]
2
,
k( 1 2 x 2 )( 1 2 y 2 )
F6 4 xy 6 k( 1 2 x 2 )( 1 2 y 2 ) ,
F 6 4 y k1 2 x 2 6 k1 2 y 2 .
and the following, involving T 2 U products
n
Q
! jn! U
n 2 1 (x) Tn (y) 4
n40
(20)
n
Q
! jn! U
n40
n 2 1 (y) Tn (x) 4
1 [e jF1 sin (jF 2 ) 1 e jF2 sin (jF 1 ) ]
2
,
k1 2 x 2
1 [e jF2 sin (jF 1 ) 2 e jF1 sin (jF 2 ) ]
2
,
k1 2 y 2
We complete this section by presenting the two-variable extensions of eqs.
(15, 16) (NjNE 1 ):
! j T (x) T (y) 4 12 k 1 2122jFjF1 j 1 1 2122jFjF1 j
2
k
l
1
Q
n
n40
1
n
2
2
1
2
1 2 jF1
Q
2
l
,
1 2 jF2
! j U (x) U (y) 4 2 k( 1 2 x )( 1 2 y ) ,
j
! j U (x) T (y) 4 12 k12x
g 122jFF 1j 1 122jFF 1j h ,
j
! j U (y) T (x) 4 12 k12y
g 122jFF 1j 1 122jFF 1j h .
n
(21)
n
n40
1 2 2 jF1 1 j 2
n21
n21
1 2 2 jF2 1 j 2
2
2
Q
n
n40
n21
n
2
2
1
2
1
2
2
Q
n
n40
n21
n
1
2
2
2
2
2
1
These last results complete this section devoted to the generating functions involving products of Chebyshev polynomials. Further comments can be
found in the concluding section.
112
G. DATTOLI
-
D. SACCHETTI
- C.
CESARANO
3. – Addition theorems.
In the previous section we have noted that
2
Q
(22)
! j D(R (x) ) 4 1 2122j(j(2 x2 x221 )11) j
n
n40
n
2
2
2
according to eq. (5) we can write
(23)
1 2 j( 2 x 2 2 1 )
Q
1 2 2 j( 2 x 2 2 1 ) 1 j 2
4
! j T (2x 21)
n
2
n
n40
which, combined with the second of eqs. (10), yields the identity
(24)
Tn ( 2 x 2 2 1 ) 4 [Tn (x) ]2 2 ( 1 2 x 2 )[Un 2 1 (x) ]2 .
This fairly useful result can be used, along with the first of eqs. (10) to get
the further relations
Tn (x) 4
(25)
Un 2 1 (x) 4
o
o
1 1 Tn ( 2 x 2 2 1 )
2
1 2 Tn ( 2 x 2 2 1 )
2( 1 2 x 2 )
,
.
which can also viewed as a consequence of the ordinary bisection formulas of
circular functions.
It is evident that eq. (24) can also be recast in the form
y uo a 12 1 vz 2 1 22 a yU uo a 12 1 vz .
2
(26)
Tn (a) 4 Tn
2
n21
The same procedure allows to get the two variable extension of eq. (24),
namely
(27)
Tn (xy 2 k( 1 2 x 2 )( 1 2 y 2 ) ) 4
4 Tn (x) Tn (y) 2 k( 1 2 x 2 )( 1 2 y 2 ) Un 2 1 (x) Un 2 1 (y)
113
A NOTE ON CHEBYSHEV POLYNOMIALS
which can be written in the form of an addition theorem. From eq. (27) we get,
indeed
Tn (a 2 b) 4 Tn (A) Tn (B) 2 bUn 2 1 (A) Un 2 1 (B) ,
(28)
k(a 1 1 )2 2 b 2 1 k(a 2 1 )2 2 b 2
A4
2
k(a 1 1 )2 2 b 2 2 k(a 2 1 )2 2 b 2
B4
2
,
.
Further identities along the lines developed in the paper can be obtained,
either concerning the generating functions and the addition theorems and will
provide the matter of forthcoming investigations.
The results obtained so far are just a consequence of the link between
Chebyshev polynomials, circualr functions and of the complex representation
we have used, which also yields
(29)
Rn (x) 4 [x 1 i k1 2 x 2 ]n .
In the forthcoming concluding section we will present further interesting
consequences of the point of view developed in the paper.
4. – Concluding remarks.
In the previous section we have dealt with generating functions involving
Chebyshev polynomials only or product of Chebyshev polynomials. In the following we will show that families of generating functions including products of
Chebyshev, Hermite, Laguerre... can be obtained too.
We consider, for example, the case
(30)
S(y , x ; j) 4
Q
jn
n40
n!
!
Tn (y) Hn (x) ,
according to the previous discussion and to the fact that the Hermite polynomials are generated by [1]
(31)
Q
jn
n40
n!
!
2
Hn (x) 4 e 2 jx 2 j ,
114
G. DATTOLI
-
D. SACCHETTI
- C.
CESARANO
we end up with
S(y , x ; j)4 D
(32)
k!
Q
jn
n40
n!
4 D[e 2 jxe
l
Rn (y) Hn (x)
i cos21 (y)
21 (y)
2 j 2 e 2 i cos
].
Which allows to conclude that
Q
jn
n40
n!
!
(33)
Tn (y) Hn (x) 4 e 2 jyx 2 j
2
(2y 221)
cos [ 2 j(x 2 jy) k1 2 y 2 ]
and
jn
Q
(34)
!
n40
n!
Un 2 1 (y) Hn (x) 4
e 2 jyx 2 j
2
(2y 221)
k1 2 y
2
sin [ 2 j(x 2 jy) k1 2 y 2 ] .
It is also worth emphasizing that
(35)
Q
1
n40
n!
!
Rn (y) Hn (y) 4 e 1 1 i .
Furthermore, by recalling that two variable Hermite are defined by [2]
Q
(36)
!
n40
jn
n!
2
Hn (x , y) 4 e jx 1 j y ,
we can use eq. (32) to state the «Theorem»
Tn (y) Hn (x) 4 D[Hn ( 2 x(y 1 ik1 2 y 2 ), 2(y 1 i k1 2 y 2 )2 ) ] ,
(37)
Un 2 1 (y) Hn (x) 4
4[Hn ( 2 x(y 1 i k1 2 y 2 ), 2(y 1 i k1 2 y 2 )2 )
.
k1 2 y 2
A further example of exotic generating function is offered by the case involving products of Laguerre and Chebyshev polynomials. According to the
well known identity [1]
Q
(38)
! j L (x) 4 1 21 j e
n
n40
n
2
xj
12j
115
A NOTE ON CHEBYSHEV POLYNOMIALS
we get e.g.
Q
(39)
e
2
xj(y 2 j)
! j T (y) L (x) 4 1 2 2 yj 1 j
n
n
n40
y
1 2 2 yj 1 j 2
n
Q ( 1 2 yj) cos
u
Q
2
xj k1 2 y 2
1 2 2 yj 1 j 2
v
1 j k1 2 y 2 sin
u
xj k1 2 y 2
1 2 2 yj 1 j 2
vz
.
It is evident that the results of this section can be further generalized, we
believe however that in absence of a specific application the derivation of new
identities may be useless.
The method we have developed indicates that Chebyshev polynomials can
be exploited in a wealth of flexible ways, in a forthcoming investigations we
will discuss further implications and applications of the lines developed in this
paper.
REFERENCES
[1] L. C. ANDREWS, Special functions for applied mathematicians and engineers, Mc
Millan, New York (1985); L. GATTESCHI, Funzioni Speciali, UTET, Torino (1973); T.
J. RIVLIN, The Chebyshev Polynomials, J. Wiley & Sons, New York (1974).
[2] G. DATTOLI - A. TORRE, Theory and applications of generalized Bessel functions,
Aracne (Rome) (1996).
Pervenuto in Redazione il 29 marzo 2000.
In forma definitiva l’8 novembre 2000.