On the location of ratios of zeros of special trinomials

2025-04-27 0 0 371.27KB 17 页 10玖币

侵权投诉

A. S. Bamunobaa,1, I. Ndikubwayoa,1,∗

aMakerere University, Department of Mathematics, Kampala, 256, Uganda

Abstract

Given coprime integers k, ℓ with k> ℓ ⩾1 and arbitrary complex polynomials A(z),B(z) with deg(A(z)B(z)) ⩾1,

we consider the polynomial sequence {Pn(z)}satisfying a three-term recurrence Pn(z)+B(z)Pn−ℓ(z)+A(z)Pn−k(z)=0

subject to the initial conditions P0(z)=1, P−1(z)=··· =P1−k(z)=0 and fully characterize the real algebraic curve Γ

on which the zeros of the polynomials in {Pn(z)}lie. In addition, we show that, for any (randomly chosen) n∈Z⩾1

and zero z0of Pn(z) with A(z0),0, at-least two of the distinct zeros of the trinomial D(t;z0) :=A(z0)tk+B(z0)tℓ+1

have a ratio that lies on the real line and /or on the unit circle centred at the origin. This reveals a previously unknown

geometric property exhibited by the zeros of trinomials of the form tk+atℓ+1 where a∈C−{0}is such that ak∈R.

Keywords: trinomial, null-collinearity, q-discriminant, real-rootedness, reciprocal polynomials

2010 MSC: Primary 12D10, 30C15, Secondary 26C10, 30C10

1. Basic notions and main result

Recursively deﬁned polynomials have always been a subject of interest since they can be used to explain phe-

nomena frequently occurring in mathematics, statistics, physics and engineering. For example, (recursive) poly-

nomial sequences arise in physics and approximation theory as the solutions of certain ordinary diﬀerential equations.

In particular, the Hermite polynomials are used in quantum and statistical mechanics to describe solutions of the

Schr¨

odinger equation for a harmonic oscillator while Laguerre polynomials are used to describe the eigenfunctions

for the Schr¨

odinger operator associated with the hydrogen atom, [1]. In combinatorics, these recursive polynomials

can be shown to represent certain combinatorial objects like graphs and therefore can be used to come up with new

identities and generating functions, the validity of which can be proved using the combinatorial interpretation.

Some of the above mentioned polynomial sequences are examples of sequences of orthogonal polynomials satisfy-

ing three-term recursive formulae. These three-term recursion formulae are famous since they provide a necessary

condition for polynomial sequences to be orthogonal, a highly sought for property in functional representations and

∗Corresponding author

Email addresses: alex.bamunoba@mak.ac.ug (A. S. Bamunoba), innocent.ndikubwayo@mak.ac.ug (I. Ndikubwayo)

1This research was funded by the NORHED-II project “Mathematics for Sustainable Development (MATH4SD), 2021-2026” at Makerere

University in collaboration with the University of Dar es Salaam and University of Bergen in Norway.

Preprint submitted to Quaestiones Mathematicae November 8, 2024

arXiv:2210.06403v3 [math.CV] 7 Nov 2024

numerical computations. Typically, orthogonal polynomials are used as basis functions in which to expand other

more complicated functions to be used in either interpolation, approximation or numerical quadrature. In addition,

interesting properties such as real-rootedness of the generated polynomials can be deduced from the orthogonality

property, for details [2]. Now, it is almost a tradition in mathematics that, whenever one encounters a family of poly-

nomials (preferably generated by recursions), it is of interest to study properties such as orthogonality, unimodularity,

real-rootedness, stability or interlacing roots among others, see [3] for further reading.

In recent years however, several authors for example [4, 5, 6, 7, 8, 9] have interested themselves in the problem of the

location of zeros of polynomials in polynomial sequences, especially those that are generated by linear recurrences.

This is partly because, knowing the zeros of some polynomials in the polynomial sequence may give information

concerning the zeros of other polynomials in the same sequence. In all of these studies, the authors either search for

a criterion for real-rootedness of the polynomial sequence (for applications in numerical analysis and combinatorics)

or explicit determination of the limiting curve on which the zeros of the polynomials lie, see for example [4, 5, 7, 8].

In this setting, one considers arbitrary non-zero complex polynomials A(z),B(z) with deg(A(z)B(z)) ⩾1, two coprime

integers k, ℓ with k> ℓ ⩾1 and the polynomial sequence {Pn(z)}satisfying a three-term recurrence

Pn(z)+B(z)Pn−ℓ(z)+A(z)Pn−k(z)=0,(1)

subject to the initial conditions P0(z)=1, P−1(z)=··· =P1−k(z)=0. If we take a solution of the form Pn(z)=(t(z))n

where n∈Z⩾0and t(z) is a nonzero complex rational function, then substituting it in the recurrence (1) yields

0=(t(z))n=B(z)(t(z))n−ℓ+A(z)(t(z))n−k=((t(z))k+B(z)(t(z))k−ℓ+A(z))(t(z))n−k.

Since t(z).0, it follows that, (t(z))k+B(z)(t(z))k−ℓ+A(z)=0 except probably for ﬁnitely many z∈C. The equation

tk+B(z)tk−ℓ+A(z)=0 (with the variable zof tdropped) is called the characteristic equation of recurrence in (1),

while the polynomial ∆(t;z) :=tk+B(z)tk−ℓ+A(z) is called the characteristic polynomial of recurrence in (1). In [5,

Lemma 1], the ordinary generating function of the polynomial sequence {Pn(z)}generated by (1) is shown to be

G(t;z) :=∞

n=0

Pn(z)tn=1

1+B(z)tℓ+A(z)tk.

We shall denote the denominator of G(t;z) by D(t;z) :=A(z)tk+B(z)tℓ+1, the reciprocal polynomial of ∆(t;z).

Remark 1.1. We consider k and ℓto be coprime since the case for non-coprime integers, does not provide any new

information about the distribution of the zeros of non-zero polynomials in its sequence. To see this, we take r,s∈Z⩾1

with r =ℓd, s =kd, gcd(k, ℓ)=1and k > ℓ ⩾1. The ordinary generating function GR(t;z)for the polynomial

sequence {Rm(z)}generated by the three-term recurrence Rn(z)+B(z)Rn−r(z)+A(z)Rn−s(z)=0, subject to the initial

conditions R0(z)=1, R−1(z)=··· =R1−s(z)=0is related to G(t;z)as follows:

GR(t;z)=∞

m=0

Rm(z)tm=1

1+B(z)tr+A(z)ts=1

1+B(z)wℓd+A(z)wkd =∞

n=0

Pn(z)tnd =G(td;z).

Therefore, Rm(z)=Pn(z)if m =nd and Rm(z)=0if d ∤m, i.e., the sequence {Rm(z)}is “essentially” {Pn(z)}.

Of all the results in the literature regarding the location of (the algebraic curve containing) the zeros of {Pn(z)}, the

most general and crucial to us is [4, Theorem 1.1] due to B¨

ogvad, et. al. To state it, we need the following notation:

For a non-empty subset Xof nonzero complex polynomials, we set ZX, (or ZA(z)if X={A(z)}) to be the union of the

set of zeros of all the polynomials in X, i.e., ZX:={z∗∈C:f(z∗)=0 for some f∈ X} =[

f(z)∈X Zf(z).

Theorem 1.2 ([4, Theorem 1.1]).If {Pn(z)}is the polynomial sequence generated by (1) and z0∈ Z{Pn(z)}− ZA(z),

then z0lies on the real algebraic curve Γ:=(z∈C: Im (−1)kBk(z)

Aℓ(z)!=0).

Remark 1.3. .

1. Theorem 1.2 generalizes the speciﬁc cases k =2,3and 4with ℓ=1proved in [7, Theorems 1, 3, 5] respectively.

2. The zeros of Pn(z)become dense in H:= Γ ∩nz∈C: 0 ⩽Re (−1)kBk(z)

Aℓ(z)!<kk

(k−ℓ)k−ℓoas n tends to inﬁnity

for the cases (k, ℓ)∈ {(2,1),(3,1),(4,1)}, as proved in [7, Theorems 1, 3, 5]. For k ⩾5and ℓ=1, Tran

established the density of zeros of Pn(z)in Hfor suﬃciently large n, see [8, Theorem 1] for details.

3. The case ℓ > 1is of a diﬀerent ﬂavour, (see Theorem 1.4)and will be established in Subsection 2.2.

Theorem 1.2 gives a partial characterisation of the real algebraic curve on which the zeros of the generated polynomials

lie. This motivated the authors to search for the full characterisation of the curve Γ, as stated in Theorem 1.4.

Theorem 1.4. If {Pn(z)}is the polynomial sequence generated by (1) with ℓ > 1and z0∈ Z{Pn(z)}− ZA(z), then z0lies

on the curve

Γ1:=







nz∈Γ: Re (−1)k−ℓBk(z)

Aℓ(z)⩽0o,where Z{B(z)}− ZA(z),∅

nz∈Γ: Re (−1)k−ℓBk(z)

Aℓ(z)<0o,where Z{B(z)}⊂ ZA(z)

Another problem that we deal with in the present paper concerns the location of zeros of trinomials. In general, the

problem of the number and location of zeros of an arbitrary trinomial atk+btℓ+c(where a,b,c∈C−{0}and k, ℓ ∈Z

such that k⩾3 and k> ℓ ⩾1) has a long history of study starting with J. Lambert in 1758, followed by L. Euler

in 1777 and many others to the present, see [10, 11] and the references therein. Of all the results in the literature

about this problem, we are interested in those that take the form of bounds on the magnitude of the zeros or of sectors

in the complex plane containing the zeros. These stem from J. Egerv`

ary’s observation that, the zeros of trinomials

can be interpreted as the equilibrium points of a force ﬁeld created by unit masses that are located at the vertices of

two regular concentric polygons centered at the origin in the complex plane. Utilising the symmetry and continuity

properties of this force ﬁeld, the roots of a trinomial can be separated according to their argument and moduli, see [12]

for details. From these results, A. Melman in 2012 obtained ﬁner results involving smaller annular sectors containing

the zeros of a trinomial that take into account the magnitude of the coeﬃcients. For further details, see [10].

Most of the approaches in the above studies could be characterised as being algebraic and/or geometric. However,

most recently, in 2019, D. Belki´

c employed some analytic tools that primarily focus upon derivations of the analytical

formulae for all the roots of trinomials through series developments using the Bell polynomials and the Fox-Wright

function. These are based on the fact, that all the roots of the general nth degree trinomial admit certain convenient

representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial

equation, respectively. As an application of their work, these analytical solutions are numerically illustrated in the

genome multiplicity corrections for survival of synchronous cell populations after irradiation, see [11] for details.

Despite the above eﬀorts, the problem of locating the zeros of a general trinomial is still unresolved. In the current

paper, we take a diﬀerent approach and instead study the location of ratios of zeros of trinomials. In particular, we

choose an arbitrary n∈Z⩾1, then a z0∈ ZPn(z)− ZA(z)and study the location of ratios of the zeros of the trinomil

D(t;z0)=A(z0)tk+B(z0)tℓ+1. We obtain the following results, namely: Theorem 1.5 and Corollary 1.6.

Theorem 1.5 (Main result).If {Pn(z)}is the polynomial sequence generated by (1) and z0∈ Z{Pn(z)}− ZA(z), then

there exists at-least two zeros of D(t;z0)=A(z0)tk+B(z0)tℓ+1whose ratio is real and /or has modulus 1.

Corollary 1.6. If {Pn(z)}is the polynomial sequence generated by (1) and z0∈ Z{Pn(z)}− ZA(z), then D(t;z0)has

at-least two equimodular zeros and /or exactly three “null-collinear”2zeros for k odd, or two null-collinear zeros for

k even.

The remaining sections of the paper are devoted to proving Theorem 1.4 and Theorem 1.5 in reverse order.

2. Proofs

2.1. Some lemmata and important results

Let C:={z∈C:|z|=1}and h:X→C,w7→ (1 −wk)k

(1 −wℓ)ℓ(wℓ−wk)k−ℓwhere Xis the domain of h.

Lemma 2.1. If z ∈X∩(R∪ C), then h(z)∈R.

Proof. The domain of his X=C−(µℓ∪µk−ℓ∪ {0}) where µndenotes the set of nth complex zeros of unity. Now,

there are two cases to consider, namely:

(i) if z∈X∩R, then z∈R, hence h(z)∈Ras his a quotient of two polynomials with real coeﬃcients, i.e., h∈R(w).

(ii) if z∈X∩ C, then h(z)∈Rby [5, Lemma 2].

2Null-collinear means two points (for keven) or three (for kodd) points lie on a line via the origin Owith at-least one point on either side of O.

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OnthelocationofratiosofzerosofspecialtrinomialsA.S.Bamunobaa,1,I.Ndikubwayoa,1,∗aMakerereUniversity,DepartmentofMathematics,Kampala,256,UgandaAbstractGivencoprimeintegersk,ℓwithk>ℓ⩾1andarbitrarycomplexpolynomialsA(z),B(z)withdeg(A(z)B(z))⩾1,weconsiderthepolynomialsequence{Pn(z)}satisfyingathree-term...

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