VIETA’S TRIANGULAR ARRAY AND A RELATED FAMILY OF POLYNOMIALS

" If n> 1, let the n th row of an infinite triangular array consist of entries B n j) -if-k-in n J where 0 _< j _< [1/2n]. We develop some properties of this array, which was discovered by Vieta. In addition, we prove some irreducibility properties of the family of polynomials These polynomials, which we call Vieta polynomials, are related to Chebychev polynomials of the first kind.

Equations (1.4), (1.7) and (1.8)illustrate that the [B(n,j)] have properties similar to those of the binomial coefficients.
In this paper, we develop additional properties associated with this array as well as prove some irreducibility properties of the family of polynomials Vn(x _, (_ 1)B(n,j)xn-2j (1.9) 3-o which we call Vieta polynomials.Jacobstahl [6] proved that {V.(x)} is a family of permutable polynomials, that is, V.,(V.(x))=V.(V,.(x))=V,..(x) for all natural numbers m,n.He also proved that if {/.(z)} is a family of permutable polynomials such that the degree of y.(x) is n for all n, then/.(x) is similar to either v.(x) or x", that is, there exists a polynomial g(x) ax + such that/.(x)g(::) *V.(x) *g 1() or /'.(x)= g()*" .g-l(z),where denotes composition.Jacobstahl also noted that the Vieta polynomials are related to the Chebychev polynomials of the first kind (denoted T.(x)) by the identity: In addition, he gave an inductive definition for the V,(z), namely: The Lucas polynomials, defined by Bicknell in [7] as Lo(x) 2, Ll(x x, Ln(x xL l(x) + Ln_ 2(x),n >_ 2 are no__At permutable and satisfy [lnl as well as (1.12) where (-1).
In conclusion, we point out that Bergum and Hoggatt [8] prove that if k > and p is an odd prime, then Lk(:c) and Lr,()/: are irreducible over the rationals.Their proofs adapt easily to show that the same properties hold respectively for vk(,) and Vp(:c)/x.In this paper, we generalize the latter result.Specifically, we prove that if m > 1, p and n are odd, and p is prime, then both Vm,(x.)/v,(z and V3(2,,,)(:c)/v,.,(a:are irreducible over the rationals.

PRELIMINARIES.
Below is a list of identities, some of which are used to develop the results in section 3 of this paper, while others are of interest in their own right.
In Cohn [10, p. 156], we find x 2-5y --4 if there exists an odd k such that x Lk, y F k while Vmn(X Vrn(Vn(x)) for all m,n appears in [6].

THE MAIN RESULTS.
We begin with some divisibility properties of the B(n,j).
PROOF.This follows from the hypothesis and (2.3).
PROOF.Equation (1.2) implies n ljB(n,j).The conclusion now follows from the hypothesis and Euclid's Lemma.
p is an odd prime and _< j _< P-, then p lB(p, j).
COROLLARY 1.If PROOF.This follows directly from Theorem 2.
THEOREM 3.For all j such that < j < [_n_] there exists a prime, p, such that pin and io B(n,j).
PROOF.Assume the contrary.Then there exists j such that _< j _< [-] and for all primes p where ioln we have io][B(n,j), so op(B(n,j))=O.But then (2.3) implies ot,(n)<_ot,(j ).Since this inequality holds for all prime divisors of n, we have nil which is contrary to our hypothesis.k COROLLARY 2. If < j _< [io ], then iolB(io ', j).
The next theorem locates the largest entry or entries in the n th row of Vieta's triangular array.
If r is not an integer, then B(n,[r]) > B(n,j) for all j It].
If r is an integer, then B(n,r-1) B(n,r) > B(n,j) for all j # r-1,r.
B(n,j) where < j <[1/2n].Equation (1.2)implies that PROOF.Let f,(j)= B(n,j-1) Now B(n,j), considered as a function of for a fixed value of n, is increasing, decreasing, or not changing accordingly as y,(j) is greater than, less than, or equal to 1.These possibilities occur accordingly as 5j2-(5n+6)j+(n+l)(n+2) is positive, negative, or zero, or accordingly as < j < r,j r, or r < j < [lnl If r is not an integer, then B(n,j) is increasing for < j < [r] and decreasing for [r] < j < [1/2n].
Hence, B(n,j) assumes a unique maximum at j [r].If r is an integer, then B(n,r-1)= B(n,r) and B(n,j) is increasing for < j < r-1, decreasing for r < j < [!nl Hence, B(n,j) assumes its maximum value when j or r.
The following two theorems give necessary and sufficient conditions for r to be an integer.
We now turn our attention to the Vieta polynomials.
THEOREM 10.If n is odd and p is an odd prime, then vp,(x)/v,,(:) is irreducible over the rational numbers.
REMARK.Lemma says that constant term in Vt,,,(z)/v,,(z is if n is even, so that Eisenstein criterion is not immediately applicable to prove irreducibility.THEOREM 11.Let g,,(z)-V2m(Z)--1, where m > 0. Then g,(z) is irreducible over the rational numbers.