A method of constructing Krall's polynomials

We propose a method of constructing orthogonal polynomials P_n(x) (Krall's polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krall's polynomials P_n(x) are rational functions of n. Let P_n^(a,b;M)(x) be polynomials obtained from the Jacobi polynomials P_n^(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials P_n^(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials P_n^(a,b;M)(x) are Krall's polynomials satisfying a differential equation of order 2a+4.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials

The paper describes some modifications of Newton??s method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x _i ,f(x _i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

For a matrix polynomial P(λ) and a given complex number μ, we introduce a (spectral norm) distance from P(λ) to the matrix polynomials that have μ as an eigenvalue of geometric multiplicity at least κ, and a distance from P(λ) to the matrix polynomials that have μ as a multiple eigenvalue. Then we compute the first distance and obtain bounds for the second one, constructing associated perturbations of P(λ).     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

A new expansion of the confluent hypergeometric function in terms of modified Bessel functions

An asymptotic expansion of the confluent hypergeometric function U(a,b,x) for large positive 2a − b is given in terms of modified Bessel functions multiplied by Buchholz polynomials, a family of double polynomials in the variables b and x with rational coefficients.     

Approximation by (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Lorentz Polynomials on a Compact Disk

The (p, q)-factors were introduced in order to generalize or unify several forms of q-oscillator algebras well known in the physics literature related to the representation theory of single parameter quantum algebras. This notion has been recently used in approximation by positive linear operators via (p, q)-calculus which has emerged a very active area of research. In this paper, we introduce a new analogue of Lorentz polynomials based on (p, q)-integers. We obtain quantitative estimate in the Voronovskaja’s type theorem and exact orders in simultaneous approximation by the complex (p, q)-Lorentz polynomials of degree \(n\in \mathbb {N}\) (\(q>p>1)\), attached to analytic functions on compact disks of the complex plane. In this way, we put in evidence the overconvergence phenomenon for the (p, q)-Lorentz polynomial, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.     

Enumeration of arrays of a given size

Enumeration of arrays whose row and column sums are specified have been studied by a number of people. It has been determined that the function that enumerates square arrays of dimension n, whose rows and columns sum to a fixed non-negative integer r, is a polynomial in r of degree (n ? 1)².In this paper we consider rectangular arrays whose rows sum to a fixed non-negative integer r and whose columns sum to a fixed non-negative integer s, determined by ns = mr. in particular, we show that the functions which enumerate 2 × n and 3 × n arrays with fixed row sums $\text{nr}\text{(2, n)}$ and $\text{nr}\text{(3, n)}$, where the symbol (a, b) denotes the greatest common divisor of a and b, and fixed column sums, are polynomials in r of degrees (n ? 1) and 2(n ? 1) respectively. We have found simple formulas to evaluate these polynomials for negative values, - r, and we show that for certain small negative integers our polynomials will always be zero. We also considered the generating functions of these polynomials and show that they are rational functions of degrees less than zero, whose denominators are of the forms (1 ? y)ⁿ and (1 ? y)²ⁿ?1 respectively and whose numerators are polynomials in y whose coefficients satisfy certain properties. In the last section we list the actual polynomials and generating functions in the 2 × n and 3 × n cases for small specific values of n.     

A special class of polynomials orthogonal on the unit circle including the associated polynomials

Let (P _ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ω_υ) the monic associated polynomials of (P _v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (AP_υλ+BΩ_υλ)/A^λ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP _ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP _ν and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p(x) on the interval (a,b). We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.     

Moment systems and orthogonal polynomials in several variables

The first part of this paper deals with general moment (“Appell”) systems on $\text{R}$^N generated by a Hamiltonian function H(x, D) and also with representations of GL(N) on the associated spaces of polynomials. The second part discusses the theory of Bernoulli generators on $\text{R}$^N determining systems of orthogonal polynomials that are extensions of the Meixner polynomials to several variables. Linear actions for these spaces are discussed. Some tensors related to the general Bernoulli generators are considered.     

On difference polynomials and hereditarily irreducible polynomials

A difference polynomial is one of the form P(x, y) = p(x) ? q(y). Another proof is given of the fact that every difference polynomial has a connected zero set, and this theorem is applied to give an irreducibility criterion for difference polynomials. Some earlier problems about hereditarily irreducible polynomials (HIPs) are solved. For example, P(x, y) is called a HIP (two-variable case) if P(a(x), b(y)) is always irreducible, and it is shown that such two-variable HIPs actually exist.     

Derivation of the Real-Rootedness of Coordinator Polynomials from the Hermite–Biehler Theorem

By using the Hermite–Biehler theorem, we give a new proof of the real-rootedness of the coordinator polynomials of type D, which was recently established by Wang and Zhao. As a consequence, we also obtain the compatibility between the coordinator polynomials of type D and those of type C.     

The numerical solution of fredholm integral equations with rapidly varying kernels

Approximating a solution to the Fredholm integral equation ø(x)=α(x) + ∫ ^b_aK(x, y)ø(y) dy by the Nyström method involves some numerical quadrature for approximating the integral, producing a linear system satisfied by approximate function values of ø. This paper discusses the use of generalized product-interpolatory formulas which model ø as one mth-degree polynomial on each subinterval and model K as a (possibly large) sequence of nth-degree polynomials. In cases where K is varying much more rapidly than ø this allows for ø to be sampled much less often than K. Since K is modeled as a sequence of polynomials, its frequent sampling does not require a prohibitive increase in the degree of the interpolating polynomials. Coefficient formulas and examples are given for the (m,n) cases (1,1), (1,2), (2,1) and (2,2).     

On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following “polynomial moment problem”: for a given complex polynomial P(z) and distinct a,b∈C to describe polynomials q(z) orthogonal to all powers of P(z) on [a,b]. We show that for given P(z), q(z) the condition that q(z) is orthogonal to all powers of P(z) is equivalent to the condition that branches of the algebraic function Q(P⁻¹(z)), where , satisfy a certain system of linear equations over Z. On this base we provide the solution of the polynomial moment problem for wide classes of polynomials. In particular, we give the complete solution for polynomials of degree less than 10.     

Remarks on a generalized beta function

We show that a certain generalized beta function B(x,y;b) which reduces to Euler's beta functions B(x,y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B(x,y;b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.     

DAHA-Jones polynomials of torus knots

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 extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for \(C^\vee C_1\) depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots \(T(2p+1,2)\) is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of \(C^\vee C_1\), developing the author’s previous results for \(A_1\).