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This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…
We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family…
In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well…
Koiran's real $\tau$-conjecture claims that the number of real zeros of a structured polynomial given as a sum of $m$ products of $k$ real sparse polynomials, each with at most $t$ monomials, is bounded by a polynomial in $m,k,t$. This…
Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their…
We describe a relation between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M.~Kobayashi. We show that this relation continues to hold for the extension of Arnold's strange duality found…
It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the…
DAHA-Jones polynomials of torus knots $T(r,s)$ are studied systematically for reduced root systems and in the case of $C^\vee C_1$. We prove the polynomiality and evaluation conjectures from the author's previous paper on torus knots and…
New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the…
Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…
Iterated Geronimus transformations generate Sobolev-type orthogonal polynomials from classical families. We establish a direct equivalence between a Sobolev inner product involving point evaluation and the first derivative at a point a…
We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials,…
The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…
The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(\lambda)$ by Klav\v{z}ar et. al. by defining $S_0(\lambda) = 0$, $S_1(\lambda) = 1$, and $$S_{2n}(\lambda) = \lambda S_n(\lambda),\quad…
We investigate asymptotic behavior of polynomials $p^{\omega}_n(z)$ satisfying varying non-Hermitian orthogonality relations $$ \int_{-1}^{1} x^kp^{\omega}_n(x)h(x) e^{\mathrm{i} \omega x}\mathrm{d} x =0, \quad k\in\{0,\ldots,n-1\}, $$…
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1- |x|)^{\g_{d+1}}$ when all $\g_i > -1$ and they are eigenfunctions of a second order partial…
Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Dur\'an discovered a gap in the original proof of completeness for exceptional Hermite…
Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…