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

U_{n} = pU_{n−1} − U_{n−2},  U_{0} = 0, U_{1} = 1,
V_{n} = pV_{n−1} − V_{n−2},  V_{0} = 2, V_{1} = p,

and conjectured the formula

U_{n}^{2k} + U_{n+1}^{2k} = ∑_{r=0}^{k} \frac{D^{r}V_{k}}{r!} U_{n}^{k−r}U_{n+1}^{k−r},

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

U_{n}(p) = U_{n−1}(\frac{p}{2}),
V_{n}(p) = 2T_{n}(\frac{p}{2}),

where T_{n} and U_{n} denote the classical Chebyshev polynomials of first and second kind, respectively.

The aim of this paper is to prove a general identity that contains Melham’s conjecture as a special case: Set W_{n} = aU_{n} + bV_{n} and Ω = a^{2} + 4b^{2} − b^{2}p^{2}, then

W_{n}^{2k} + W_{n+1}^{2k} = ∑_{r=0}^{k} Ω^{k−r} λ_{k,r} W_{n}^{r}W_{n+1}^{r},  (1)

Date: January 15, 2001.

1991 Mathematics Subject Classification. Primary: 33C45; Secondary: 05A15, 11B37.

Key words 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.

2 P. J. GRABNER AND H. PRODINGER

with

λ_{k,r} = ∑_{0≤2j≤r}(−1)^{j} \frac{k(k − 1 − j)!}{(k − r)!j!(r − 2j)!}p^{r−2j}

and λ_{0,0} = 2.

From [1, 2] we know explicit expansions for Chebyshev polynomials:

V_{j} = ∑_{0≤2k≤j}(−1)^{k}\binom{j − k}{k}\frac{j}{j − k} p^{j−2k}

for j ≥ 1 and V_{0} = 2. Then we have

λ_{k,r} = \frac{D^{k−r}V_{k}}{(k − r)!},

which links Melham’s conjecture and (1).

2. Proof of the Formula

In order to prove (1) we form the generating function

g(z) = ∑_{k=0}^{∞} z^{k} ∑_{r=0}^{k} Ω^{k−r} λ_{k,r} σ^{r}

with σ = W_{n}W_{n+1}. We reorder this to obtain (setting r − k = t)

g(z) = ∑_{j≥1} ∑_{k≥2j} ∑_{t≥0} z^{t}Ω^{t}(σz)^{k}(−1)^{j} \frac{(k + t)(k + t − 1 − j)!}{t!j!(k − 2j)!}p^{k−2j}

+ ∑_{k≥0} ∑_{t≥0} z^{t}Ω^{t}(σz)^{k} \frac{(k + t)!}{t!k!}p^{k} + 1

= ∑_{j≥1} ∑_{k≥2j}(−1)^{j}p^{k−2j}(σz)^{k}(1 − Ωz)^{j−k−1} \frac{(k − j − 1)!}{j!(k − 2j)!}(k − jΩz)

+ ∑_{k≥0}(pσz)^{k}(1 − Ωz)^{−k−1} + 1

= ∑_{j≥1}(−1)^{j}(σz)^{2j}(1 − Ωz)^{−j−1}(2 − (Ω + pσ)z)

+ \frac{1}{1 − (Ω + pσ)z} + 1

= −\frac{(σz)^{2}(2 − (Ω + pσ)z)}{(1 − (Ω + pσ)z)(1 − (Ω + pσ)z + σ^{2}z^{2})} + \frac{1}{1 − (Ω + pσ)z} + 1

= \frac{2 − (Ω + pσ)z}{1 − (Ω + pσ)z + σ^{2}z^{2}} .

The generating function of the left hand side is

\frac{1}{1 − W_{n}^{2}z} + \frac{1}{1 − W_{n+1}^{2}z} = \frac{2 − (W_{n}^{2} + W_{n+1}^{2})z}{1 − (W_{n}^{2} + W_{n+1}^{2})z + W_{n}^{2}W_{n+1}^{2}z^{2}}

and the assertion follows from

W_{n}^{2} + W_{n+1}^{2} = pW_{n}W_{n+1} + Ω,

which is easily proved e. g. by using the explicit forms

U_{n} = \frac{α^{n} − β^{n}}{α − β} , V_{n} = α^{n} + β^{n},

with

α = \frac{p + \sqrt{p^{2} − 4}}{2}, β = \frac{p − \sqrt{p^{2} − 4}}{2}.

3. Further identities

Many other similar formulæ seem to exist; we just give one other example; set

a_{k,r} = ∑_{0≤λ≤r}(−1)^{λ}p^{2k−2λ} \frac{k(k − ⌊\frac{λ}{2}⌋ − 1)!2^{⌈ \frac{λ}{2} ⌉}}{(k − r)!λ!(r − λ)!} ∏_{i=0}^{⌊ \frac{λ}{2} ⌋−1} (2k − 2⌈\frac{λ}{2}⌉ − 1 − 2i)

and a_{0,0} = 2, then

W_{n}^{2k} + W_{n+2}^{2k} = ∑_{r=0}^{k} Ω^{k−r} a_{k,r} W_{n}^{r}W_{n+2}^{r}.

The proof is as before.

References

1. G. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, no. 71, Cambridge University Press, 1999.

2. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.

3. R. S. Melham, 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 address: grabner@weyl.math.tu-graz.ac.at

(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 address: helmut@gauss.cam.wits.ac.za