Applications of Waring's formula to some identities of Chebyshev polynomials

0.09 MB·English

by  Jiang Zeng

Applications of Waring’s formula to some
identities of Chebyshev polynomials
Jiang Zeng1,2 and Jin Zhou2
1 Institut Girard Desargues, Universit´e Claude Bernard (Lyon I)
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69622 Villeurbanne Cedex, France
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[email protected]
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2 and
n 2 Center for Combinatorics, LPMC, Nankai University
a Tianjin 300071, People’s Republic of China
J
[email protected]
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AMS Math Subject Classification Numbers: 11B39, 33C05, 05E05

]
O
Abstract. Someidentities ofChebyshev polynomialsarededuced fromWar-
C
ing’sformulaonsymmetricfunctions. Inparticular,theseformulaegeneralize
.
h
some recent results of Grabner and Prodinger.
t
a
m
[ 1 Introduction

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v Given a set of variables X = {x ,x ,...}, the kth (k ≥ 0) elementary sym-
6 1 2
1 metric polynomial ek(X) is defined by e0(X) = 1,
2
1
e (X) = x ...x , for k > 1,
0 k i1 ik
5 i1<X...<ik
0
h/ and the kth (k ≥ 0)power sum symmetric polynomial pk(X) is defined by
t p (X) = 1,
a 0
m p (X) = xk, for k > 1.
k i
v: Xi
Xi Let λ = 1m12m2 ...be a partition of n, i.e., m11+m22+...+mnn = n, where
r mi ≥ 0 for i = 1,2,...n. Set l(λ) = m1 +m2 +...+mn. According to the
a
fundamental theorem of symmetric polynomials, any symmetric polynomial
can be written uniquely as a polynomial of elementary symmetric polynomi-
als e (X) (i ≥ 0). In particular, for the power sum p (x), the corresponding
i k
formula is usually attributed to Waring [1, 4] and reads as follows:
k(l(λ)−1)!
p (X) = (−1)k−l(λ) e (X)m1e (X)m2 ..., (1)
k 1 2
m !
Xλ i i
Q
where the sum is over all the partitions λ = 1m12m2 ... of k.
In a recent paper [3] Grabner and Prodinger proved some identities about
Chebyshev polynomials using generating functions, the aim of this paper is
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to show that Waring’s formula provides a natural generalization of such kind
of identities.
Let U and V be two sequences defined by the following recurrence rela-
n n
tions:
U = pU −U , U = 0,U = 1, (2)
n n−1 n−2 0 1
V = pV −V , V = 2,V = p. (3)
n n−1 n−2 0 1
Hence U and V are rescaled versions of the first and second kind of Cheby-
n n
shev polynomials U (x) and T (x), respectively:
n n
1
U (x) = U (2x), T (x) = T (x).
n n+1 n n
2
Theorem 1 For integers m,n ≥ 0, let W = aU +bV and Ω = a2+4b2−
n n n
b2p2. Then the following identity holds
k
W2k +W2k = θ (m)Ωk−rWrWr , (4)
n n+m k,r n n+m
Xr=0
where
k(k −j −1)!
θ (m) = (−1)j Vr−2jU2k−2r.
k,r j!(k −r)!(r−2j)! m m
06X2j6k
NotethattheidentitiesofGrabnerandProdinger[3]correspondtothem = 1
and implicitly m = 2 cases of Theorem 1 (cf. Section 3).
2 Proof of Theorem 1
We first check the k = 1 case of (4):
W2 +W2 = V W W +U2Ω. (5)
n n+m m n n+m m
Set α = (p+ p2 −4)/2 and β = (p− p2 −4)/2 then it is easy to see that
p p
αn −βn
U = , V = αn +βn,
n n
α−β
it follows that
W = aU +bV = Aαn +Bβn,
n n n
where A = b+a/(α−β) and B = b−a/(α−β). Therefore
V W W +U2Ω = (αm +βm)(Aαn +Bβn)(Aαn+m +Bβn+m)
m n n+m m
αm −βm 2
+ (a2 +4b2 −b2p2),
(cid:18) α−β (cid:19)
2
which is readily seen to be equal to W2 +W2 .
n n+m
Next we take the alphabet X = {W2,W2 }, then the left-hand side of
n n+m
(4) is the power sum p (X). On the other hand, since
k
e (X) = W2 +W2 , e (X) = W2W2 , e (X) = 0 if i > 3,
1 n n+m 2 n n+m i
the summation at the right-hand side of (1) reduces to the partitions λ =
(1k−2j2j), with j ≥ 0. Now, using (5) Waring’s formula (1) infers that
W2k +W2k
n n+m
k(k −j −1)!
= (−1)j (V W W +U2Ω)k−2j(W2W2 )j
j!(k −2j)! m n n+m m n n+m
06X2j6k
k−2j
k(k −j −1)!
= (−1)j Vk−2j−iU2iΩi(W W )k−i
j!i!(k −2j −i)! m m n n+m
06X2j6k Xi=0
Setting k −i = r and exchanging the order of summations yields (4).
3 Some special cases
When m = 1 or 2, as U = 1, V = p and U = p, V = p2 −2 the coefficient
1 1 2 2
θ (r) of Theorem 1 is much simpler.
k,r
Corollary 1 We have
k(k −1−j)!
θ (1) = (−1)j pr−2j, (6)
k,r
(k −r)!j!(r−2j)!
06X2j6r
k(k −j −1)!
θ (2) = (−1)j (p2 −2)r−2jp2k−2r. (7)
k,r
j!(k −r)!(r−2j)!
06X2j6k
Wenoticethat(6)isexactlytheformulagivenbyGrabnerandProdinger[3]
for θ (1), while for θ (2) they give a more involved formula than (7) as
k,r k,r
follows:
Corollary 2 (Grabner and Prodinger [3]) There holds
λ
k(k −⌊λ⌋−1)!2⌈λ2⌉ ⌊2⌋−1 λ
θ (2) = (−1)λp2k−2λ 2 (2k−2⌈ ⌉−1−2i). (8)
k,r
(k −r)!λ!(r−λ)! 2
06Xλ6k Yi=0
In order to identify (7) and (8), we need the following identity.
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Lemma 2 We have
j/2
(k −i−1)!2j−2i
(−1)i
(j −2i)!i!
Xi=0
⌊j/2⌋−1
(k −⌊j/2⌋−1)!
= 2⌈j/2⌉ (2k −2⌈j/2⌉−1−2i). (9)
j!
Yi=0
Proof: For n ≥ 0 let (a) = a(a + 1)...(a + n − 1), then the Chu-
n
Vandermonde formula [2, p.212] reads:
(−n) (a) (c−a)
k k n
F (−n,a;c;1) := = . (10)
2 1
(c) k! (c)
Xk>0 k n
Note that n! = (1) , so using the simple transformation formulae:
n
a a+1 a a+1
(a) = 22n, (a) = 22n+1,
2n 2n+1
2 (cid:18) 2 (cid:19) 2 (cid:18) 2 (cid:19)
(cid:16) (cid:17)n n (cid:16) (cid:17)n+1 n
and
(a) (a)
(a) = N = (−1)n N ,
N−n
(a+N −n) (−a−N +1)
n n
we can rewrite the left-hand side of identity (9) as follows:
(k−1)! F (−m,−m+ 1;−k +1;1) if j = 2m,
(12)m(1)m 2 1 2


 (k−1)! F (−m,−m− 1;−k +1;1) if j = 2m+1,
(12)m+1(1)m 2 1 2



which is clearly equal to the right-hand side of (9) in view of (10).
Now, expanding the right-hand side of (7) by binomial formula yields
r−2j
k(k −j −1)! r −2j
(−1)j p2i(−2)r−2j−ip2k−2r.
j!(k −r)!(r −2j)! (cid:18) i (cid:19)
06X2j6k Xi=0
Writing λ = r −i, so λ ≤ r ≤ k, and exchanging the order of summations,
the above quantity becomes
k (k −j −1)!2λ−2j
(−1)λp2k−2λ (−1)j ,
(k −r)!(r −λ)! (λ−2j)!j!
06Xλ6k 06Xj6k/2
which yields (8) by applying Lemma 2.
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References
[1] William Y. C. Chen, Ko-Wei Lih, and Yeong-Nan Yeh: Cyclic Tableaux
and Symmetric Functions,Studies inApplied Math., 94(1995),327-339.
[2] Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete
Mathematics, Addion-Wesley Pubilshing Co. 1989.
[3] Peter J. Grabner and Helmut Prodinger: Some identities for Chebyshev
polynomials, Portugalia Mathematicae 59 (2002), 311-314.
[4] P. A. MacMahon: Combinatory analysis, Chelsea Publishing Co. New
York, 1960.
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