PORTUGALIAE MATHEMATICA
Vol. 59 Fasc. 3 – 2002
Nova Série
SOME IDENTITIES FOR CHEBYSHEV POLYNOMIALS
Peter J. Grabner • and Helmut Prodinger ◦
Abstract: We prove a generalization of a conjectured formula of Melham and
provide some background about the involved (Chebyshev) polynomials.
1 – Introduction
In [3] Melham considered the two sequences
Un = pUn−1 − Un−2 ,
U0 = 0, U1 = 1 ,
Vn = pVn−1 − Vn−2 ,
V0 = 2, V1 = p ,
and conjectured the formula
2k
Un2k + Un+1
=
k
X
D r Vk
r=0
r!
k−r
Unk−r Un+1
,
where D means differentiation with respect to p. We remark here that up to
simple changes of variable these polynomials are Chebyshev polynomials. More
precisely
Un (p) = Un−1 ( p2 ) ,
Vn (p) = 2Tn ( p2 ) ,
where Tn and Un denote the classical Chebyshev polynomials of first and second
kind, respectively.
Received : January 15, 2001; Revised : March 19, 2001.
AMS Subject Classification (2000): Primary 33C45; Secondary 05A15, 11B37.
Keywords and Phrases: Chebyshev polynomials; identities.
•
This author is supported by the START-project Y96-MAT of the Austrian Science
Foundation.
◦
This author’s research was done during a visit in Graz.
312
P.J. GRABNER and H. PRODINGER
The aim of this paper is to prove a general identity that contains Melham’s
conjecture as a special case: Set Wn = aUn + bVn and Ω = a2 + 4b2 − b2 p2 , then
2k
Wn2k + Wn+1
=
(1)
k
X
r
Ωk−r λk,r Wnr Wn+1
,
r=0
with
X
λk,r =
(−1)j
0≤2j≤r
k(k − 1 − j)!
pr−2j
(k − r)! j! (r − 2j)!
and λ0,0 = 2.
From [1, 2] we know explicit expansions for Chebyshev polynomials:
Vk =
X
(−1)j
0≤2j≤k
µ
k−j
j
¶
k
pk−2j
k−j
for k ≥ 1 and V0 = 2. Then we have
Dk−r Vk
,
(k − r)!
λk,r =
which links Melham’s conjecture and (1).
2 – Proof of the Formula
We will make use of the identity
∞
X
(2)
yt
t=0
³
´
(a + t)!
(bt + c) = a! (1 − y)−a−2 c + y(ab + b − c) ,
t!
which follows from
¶
∞µ
X
a+t
(3)
t=0
y t = (1 − y)−a .
t
In order to prove (1) we form the generating function
g(z) =
∞
X
zk
k
X
Ωk−r λk,r σ r
r=0
k=0
with σ = Wn Wn+1 . We reorder this to obtain (setting k − r = t)
g(z) =
XX
z k Ωk−r σ r λk,r =
r≥0 k≥r
=
XX
r≥1 t≥0
XX
z r+t Ωt σ r λr+t,r
r≥0 t≥0
z
r+t
t
r
Ω σ λr+t,r + 1 +
X
t≥0
z t Ωt
(using λ0,0 = 2)
313
SOME IDENTITIES FOR CHEBYSHEV POLYNOMIALS
=
XX
X
z t Ωt (σz)r (−1)j
r≥1 t≥0 0≤2j≤r
=
X X X
z t Ωt (σz)r (−1)j
j≥1 r≥2j t≥0
+1 +
XX
z t Ωt (σz)r
r≥0 t≥0
=
X X
X
(r + t) (r + t − 1 − j)! r−2j
p
+1+
z t Ωt
t! j! (r − 2j)!
t≥0
(r + t) (r + t − 1 − j)! r−2j
p
t! j! (r − 2j)!
(r + t)! r
p
t! r!
(terms for j = 0 plus the last sum)
(−1)j pr−2j (σz)r (1 − Ωz)j−r−1
j≥1 r≥2j
+
X
(p σ z)r (1 − Ωz)−r−1 + 1
(r − j − 1)!
(r − j Ωz)
j! (r − 2j)!
by (2)
by (3)
r≥0
=
X (−1)j (σz)2j X µ p σ z ¶s (j + s − 1)! ³
j≥1
(1−Ωz)j+1
1−Ωz
s≥0
j! s!
j(2−Ωz) + s
1
+1
1 − (Ω + p σ) z
X
(−1)j (σz)2j
´
(r = 2j +s)
+
=
j≥1
+
1 − (Ω + p σ) z
³
´
´j+1 2 − (Ω + p σ) z
by (2)
1
+1
1 − (Ω + p σ) z
= −³
=
³
³
(σz)2 2 − (Ω + p σ) z
1 − (Ω + p σ) z
´³
´
1 − (Ω + p σ) z + σ 2 z 2
2 − (Ω + p σ)z
.
1 − (Ω + p σ)z + σ 2 z 2
´ +
1
+1
1 − (Ω + p σ)z
The generating function of the left hand side is
2 )z
2 − (Wn2 + Wn+1
1
1
+
=
2 z
2 )z + W 2 W 2 z 2
1 − Wn2 z 1 − Wn+1
1 − (Wn2 + Wn+1
n n+1
and the assertion follows from
2
Wn2 + Wn+1
= pWn Wn+1 + Ω ,
which is easily proved e. g. by using the explicit forms
Un =
αn − β n
,
α−β
Vn = α n + β n ,
with
α=
p+
p
p2 − 4
,
2
β=
p−
p
p2 − 4
.
2
314
P.J. GRABNER and H. PRODINGER
3 – Further identities
Many other similar formulæ seem to exist; we just give one other example;
set
ak,r =
X
λ 2k−2λ
(−1) p
0≤λ≤r
³
´
λ
k k − b λ2 c − 1 ! 2d 2 e
(k − r)! λ! (r − λ)!
bλ
c−1³
2
Y
2k − 2
i=0
lλm
2
− 1 − 2i
´
and a0,0 = 2, then
2k
Wn2k + Wn+2
=
k
X
r
Ωk−r ak,r Wnr Wn+2
.
r=0
The proof is as before.
ACKNOWLEDGEMENT – We used Mathematica to perform the hypergeometric summations used in the proof and Maple to guess several formulæ.
REFERENCES
[1] Andrews, G.; Askey, R. and Roy, R. – Special functions, Encyclopedia of
Mathematics and its Applications, No. 71, Cambridge University Press, 1999.
[2] Erdélyi, A.; Magnus, W.; Oberhettinger, F. and Tricomi, F.G. – Higher
Transcendental Functions, vol. I, McGraw–Hill Book Company, Inc., New York–
Toronto–London, 1953.
[3] Melham, R.S. – On sums of powers of terms in a linear recurrence, Portugaliæ
Math., 56 (1999), 501–508.
P. Grabner,
Institut für Mathematik A, Technische Universität Graz,
Steyrergasse 30, 8010 Graz – AUSTRIA
E-mail: grabner@weyl.math.tu-graz.ac.at
and
H. Prodinger,
The John Knopfmacher Centre for Applicable Analysis and Number Theory,
Department of Mathematics, University of the Witwatersrand, P. O. Wits,
2050 Johannesburg – SOUTH AFRICA
E-mail: helmut@gauss.cam.wits.ac.za