Self-reciprocal Polynomials Research Articles

Self-reciprocal polynomials connecting unsigned and signed relative derangements

In this paper, we introduce polynomials (in x) of signed relative derangements that track the number of signed elements. Note that relative derangements are those without any signed elements, i.e. the evaluations of the polynomials at x=0. Also, the numbers of all signed relative derangements are given by the evaluations at x=1. Then the coefficients of the polynomials connect unsigned and signed relative derangements and reveal how putting elements with signs affects the formation of derangements. We first prove a recurrence satisfied by these polynomials which results in a recurrence satisfied by the coefficients. A combinatorial proof of the latter is provided next. Thanks to the recurrence, we also show that the sequences of the coefficients are unimodal. Moreover, other results are obtained, for instance, a kind of dual of a relation between signed derangements and signed relative derangements previously proved by Chen and Zhang is presented.

Curious congruences for cyclotomic polynomials

Let \(\Phi _n^{(k)}(x)\) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values \(\Phi _n^{(k)}(1)\) for fixed positive integers n. D. H. Lehmer proved that \(\Phi _n^{(k)}(1)/ \Phi _n(1)\) is a polynomial of the Euler totient function \(\phi (n)\) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that \(\Phi _n^{(k)}(1)/\Phi _n(1)\) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: \(2\Phi ^{(3)}_n(1)\) is divisible by \(\phi (n)-2\). Moreover, if k is greater than 1, then \(\Phi ^{(2k+1)}_n(1)\) is divisible by \(\phi (n)-2k\). The proof depends on a new combinatorial identity for general self-reciprocal polynomials over \({{\mathbb {Z}}}\), which gives rise to a formula that expresses the value \(\Phi _n^{(k)}(1)\) as a \({{\mathbb {Z}}}\)-linear combination of the coefficients in the minimal polynomial of \(2\cos (2\pi /n)-2\). As a supplement, we show the monotonic increasing property of \(\Phi _n(x)\) on \([1,\infty )\) in two ways.

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

More permutation polynomials with Niho exponents which permute [formula omitted

Constructions of permutation polynomials over finite fields have attracted much interests in recent years, especially those with few terms, such as trinomials, due to their simple form and additional properties. In this paper, we construct several classes of permutation trinomials over Fp2k with Niho exponents of the form f(x)=x+λ1xs(pk−1)+1+λ2xt(pk−1)+1; some necessary and sufficient conditions for the polynomial f(x) to permute Fp2k are provided. Specifically, for p=5, new permutation trinomials are presented. We also give recursive constructions of permutation polynomials using self-reciprocal polynomials.

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Let $n_1 \lt n_2 \lt \cdots \lt n_N$ be nonnegative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials $T_N(\theta ) = \sum _{j=1}^N {\cos (n_j\theta )}$ tends to $\infty $ as a funct

An inverse problem for a class of canonical systems and its applications to self-reciprocal polynomials

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.

Expected Number of Real Zeros of Gaussian Self-Reciprocal Random Algebraic Polynomials

Let $$Q_{n} (x) = \sum\nolimits_{i = 0}^{n} A_{i} x^{i}$$ be a random algebraic polynomial where the coefficients $$A_{0} ,A_{1} , \cdots$$ form a sequence of centered Gaussian random variables. Moreover, we assume that the increments $$\Delta_{j} = A_{j} - A_{j - 1}$$ , $$j = 0,1,2, \ldots ,n$$ are independent normal random variables with mean zero, finite variances $$\sigma_{j}^{2}$$ and the conventional notation of $$A_{ - 1} = 0$$ . The coefficients can be considered as $$n$$ consecutive observations of a Brownian motion. Assuming the symmetric property of $$A_{j}$$ ’s, we investigate some new and considerable results about the distribution of zeros. We prove that the expected number of real zeros of $$Q_{n} (x)$$ is asymptotically of order $$logn.$$

Self-reciprocal polynomials and coterm polynomials

We classify all self-reciprocal polynomials arising from reversed Dickson polynomials over $$\mathbb {Z}$$ and $$\mathbb {F}_p$$ , where p is prime. As a consequence, we also obtain coterm polynomials arising from reversed Dickson polynomials.

Generalizations of self-reciprocal polynomials

A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.

The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

We study the number of real zeros of trigonometric polynomials in a period and the number of zeros of self-reciprocal algebraic polynomials on the unit circle under the assumption that their coefficients are in a fixed finite set of real numbers.

Some properties of classes of real self-reciprocal polynomials

The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials Rn(λ)(z)=1+λ(z+z2+⋯+zn−1)+zn, n≥2 and λ∈R. Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials Sn(λ)(z)=∑k=0nsn,k(λ)zk, n≥2, with sn,0(λ)=sn,n(λ)=1, sn,n−k(λ)=sn,k(λ)=1+kλ, k=1,2,…,⌊n/2⌋ when n is odd, and sn,n−k(λ)=sn,k(λ)=1+kλ, k=1,2,…,n/2−1, sn,n/2(λ)=(n/2)λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008.

The Concept of q-Cycle and Applications

The concept of q-cycle is investigated for its properties and applications. Connections with irreducible polynomials over a finite field are established with emphases on the notions of order and degree. The results are applied to deduce new results about primitive and self-reciprocal polynomials.

S-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

Enumeration of linear transformation shift registers

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.

A new subclass of cyclic Goppa codes

We propose a subclass of cyclic Goppa codes given by separable self-reciprocal Goppa polynomials of degree two. We prove that this subclass contains all reversible cyclic codes of length n, n | (q m ± 1), with a generator polynomial g(x), g(? ±i ) = 0, i = 0, 1, ? n = 1, ? ? GF(q 2m ).

SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS

For each real number <TEX>$n$</TEX> > 6, we prove that there is a sequence <TEX>$\{pk(n,z)\}^{\infty}_{k=1}$</TEX> of fourth degree self-reciprocal polynomials such that the zeros of <TEX>$p_k(n,z)$</TEX> are all simple and real, and every <TEX>$p_{k+1}(n,z)$</TEX> has the largest (in modulus) zero <TEX>${\alpha}{\beta}$</TEX> where <TEX>${\alpha}$</TEX> and <TEX>${\beta}$</TEX> are the first and the second largest (in modulus) zeros of <TEX>$p_k(n,z)$</TEX>, respectively. One such sequence is given by <TEX>$p_k(n,z)$</TEX> so that <TEX>$$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$</TEX>, where <TEX>$q_0(n)=1$</TEX> and other <TEX>$q_k(n)^{\prime}s$</TEX> are polynomials in n defined by the severely nonlinear recurrence <TEX>$$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$</TEX> for <TEX>$m{\geq}1$</TEX>, with the usual empty product conventions, i.e., <TEX>${\prod}_{j=0}^{-1}\;b_j=1$</TEX>.

On Two Sequences of Polynomials Satisfying Certain Recurrence

Bae and Kim displayed a sequence of 4th degree self-reciprocal polynomials whose maximal zeros are related in a very nice and far from obvious way. Kim showed that the auxiliary polynomials in their results are related to Chebyshev polynomials. In this paper, we study two sequences of polynomials satisfying the recurrence of the auxiliary polynomials with generalized initial conditions. We obtain same results with the auxiliary polynomials from a sequence, and some interesting conjectural properties about resultants and discriminants from another sequence.

ON SOME COMBINATIONS OF SELF-RECIPROCAL POLYNOMIALS

Let <TEX>$\mathcal{P}_n$</TEX> be the set of all monic integral self-reciprocal poly-nomials of degree n whose all zeros lie on the unit circle. In this paper we study the following question: For P(z), Q(z)<TEX>${\in}\mathcal{P}_n$</TEX>, does there exist a continuous mapping <TEX>$r{\rightarrow}G_r(z){\in}\mathcal{P}_n$</TEX> on [0, 1] such that <TEX>$G_0$</TEX>(z) = P(z) and <TEX>$G_1$</TEX>(z) = Q(z)?.

ON THE ZEROS OF SELF-RECIPROCAL POLYNOMIALS SATISFYING CERTAIN COEFFICIENT CONDITIONS

Kim and Park investigated the distribution of zeros around the unit circle of real self-reciprocal polynomials of even degrees with five terms, where the absolute value of middle coefficient equals the sum of all other coefficients. In this paper, we extend some of their results to the same kinds of polynomials with arbitrary many nonzero terms.