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Parity is an important and easy to recognise property for meromorphic functions. On the parity of meromorphic functions, Liu, Liu and Korhonen [J Math Anal Appl 512(2022):126129] obtained some meaningful results. In this paper, we investigate the parity of a meromorphic function y(z) under the hypothesis that y(z)2n-2y(z)n is even. In addition, we discuss the relationship on the parity of a meromorphic function with its q-difference polynomials and differential expressions. For instance, we consider the parity of a meromorphic function y(z) under the assumption that y'(z)/ y(z)n and y(qz)/y(z)n are odd or even functions, where n is a positive integer. [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

Consider a polynomial f (z) = a n z n +... + a 1 z + a 0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the p th Hadamard power of f : f [ p ] (z) : = a n p z n +... + a 1 p z + a 0 p is stable if p is a positive integer number. However, it turns out that f [ p ] does not need to be stable for all real p > 1. A counterexample is known for n = 8 and p = 1.139. On the other hand, f [ p ] is stable for n = 1 , 2 , 3 , 4 , and every p > 1. In this paper we fill the gap by showing that f [ p ] is stable for n = 5 and constructing counterexamples for n ≥ 6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the unicity of meromorphic functions and their differential-difference polynomials. Our results improve some results due to Chen-Yi [Results Math 63 (2013):557-565], Chen-Xu [Open Math 18 (2020):211-215], Banerjee-Maity [Bull Korean Math Soc 58 (2021):1175-1192], and Narasimha-Shilpa [Adv Pure Appl Math 13 (2022):53-61] [ABSTRACT FROM AUTHOR]

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

In this paper, we derive a Mittag-Leffler function for real index and establish solutions of special type of fractional order differential equations (FDEs). The same concept is extended to discrete case by replacing polynomials into factorial polynomials and differentiation into l-difference operator. Moreover, numerical examples of our results are stated to validate our findings. The acquired results here have the ability to generate a wide range of formulas in relation to newer results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new function class called n-fractional polynomial convex function and their some algebric properties. We obtain some refinements of the right-hand side of Hermite-Hadamard inequality for the class of functions whose derivatives in absolutely value at certain powers are n-fractional polynomial convex. Also, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved-power-mean integral inequalities and show that the result obtained with Hölder-İşcan and improved power-mean inequalities give better approach than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

This paper finds new quasi-polynomials over Z for the number p k (n) of partitions of n with parts at most k. Methods throughout are elementary. We derive a small number of polynomials (e.g., one for k = 3 , two for k = 4 or 5, six for k = 6 ) that, after addition of appropriate constant terms, take the value p k (n) . For example, for 0 ≤ r < 6 and for all q ≥ 0 , p 3 (6 q + r) = p 3 (r) + π 0 (q , r) , a polynomial of total degree 2 in q and r. In general there are M ⌊ k / 2 ⌋ = lcm { 1 , 2 , ... , ⌊ k / 2 ⌋ } such polynomials. In two variables q and s, they take the form ∑ a i , j q i s j with a i , j ∈ Z , which we call the proper form for an integer-valued polynomial. They constitute a quasi-polynomial of period M ⌊ k / 2 ⌋ for the sequence (p k (n) - p k (r)) with n ≡ r (mod M k) . For each k the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on k. A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over Q for p k (n) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At k = 6 we find serious errors in Sylvester's calculation of his "waves." Sylvester JJ (Q J Pure Appl Math 1:141–152, 1855). The results are generalized to the (not very different) problem called "making change," where significant improvements to existing approaches are found. We find an infinitude of new congruences for p k (n) for k = 3 , 4 , and one new one for k = 5 . Reduced modulo m the periodic sequence (p k (n)) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: "Given a generic polynomial P of degree at least 2 , there exist positive integers N , M such that Re (P k (z)) = Re (P k (w)) for all 0 ≤ k ≤ N implies P M (z) = P M (w) ". [ABSTRACT FROM AUTHOR]

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of |x-a| (with |a|<1) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95% range of [-1,1] except for a small neighbourhood near the singular point x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of \Phi-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired \log n-factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates. [ABSTRACT FROM AUTHOR]

We define a q-linear path in a hypergraph H as a sequence (e1,. . ., eL) of edges of H such that |ei nei+1| [[1, q]] and ei n ej = Ø if |i-j| > 1. In this paper, we study the connected components associated to these paths when q = k - 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

We are concerned with the stability of a 1-D coupled Rayleigh beam-string transmission system. We obtain the polynomial decay rate t - 1 or the exponential decay rate for the given transmission system whether the frictional damping is only effective in the beam part or the string part, respectively. This paper generalizes the recent result in [Y.-F. Li, Z.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett. 78 2018, 51–58]. The main ingredient of the proof is some careful analysis for the Rayleigh beam and string transmission system. [ABSTRACT FROM AUTHOR]

This paper is devoted to the isochronicity of two kinds of planar piecewise polynomial potential systems separated by x-axis and y-axis, respectively. By means of the expansion of the period function near the infinity, we obtain in this paper that for these two cases, the piecewise potential system has an isochronous center at the origin if and only if the subsystems are linear. It is however not true in the analytic scenario; one can find in the paper by F. Mañosas and P. J. Torres [Proc. Amer. Math. Soc. 133 (2005), pp. 3027–3035] that the isochronicity of the center of the piecewise analytic potential system cannot imply the linearity of the two subsystems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of unit-trace n × n PSD matrices, denoted by D, using at most N number of k × k PSD constraints?" In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any k = o (n) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have S + k -extension complexity at least exp (C · min { n , n / k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S + k -extension complexity at least exp (C · min { n 1 / 3 , n / k }) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any k = o (n / log 2 n) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of S + n , which corresponds to the special case with k = 1 . [ABSTRACT FROM AUTHOR]

By using the generalization of the neutrosophic q-Poisson distribution series, we introduce a new subclass of analytic and bi-univalent functions defined in the open unit disk. We then apply the q-Gegenbauer polynomials to investigate the estimates for the Taylor coefficients and Fekete–Szegö type inequalities of the functions belonging to this new subclass. In addition, we consider several corollaries and the consequences of the results presented in this paper. The neutrosophic q-Poisson distribution is expected to be significant in a number of areas of mathematics, science, and technology. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

Let p 1 < p 2 < ⋯ < p n be positive numbers, then for any integer m the Loewner matrix associated with the function x m is given by L m = (p i m − p j m) / (p i − p j) i , j = 1 n. A question was left open in a paper [Bhatia R, Friedland S, Jain T. Inertia of loewner matrices. Indiana Univ Math J. 2016;65:1251–1261] by Bhatia, Friedland and Jain to find formulas for the determinants of the matrices L m in the case n = 3. We aim to answer this question firmly in terms of the Schur polynomials in this paper. [ABSTRACT FROM AUTHOR]

Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers K of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota's unit formula without cohomological results of [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175]. [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we have characterized the nature and form of solutions of the following nonlinear delaydifferential equation where bi C, L(z, f) are a linear delay-differential polynomial of f; n is positive integers; q, Q and P respectively are nonzero, nonconstant and any polynomials. Different special cases of our result will accommodate all the results of [J. Math. Anal. Appl., 452(2017), 1128-1144; Mediterr. J. Math., 13(2016), 3015-3027; Open Math., 18(2020), 1292-1301]. Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occur and consequently the same are automatically applicable to the previous results. [ABSTRACT FROM AUTHOR]

Let P(z) := Σnv=0 avzv be a univariate complex coefficient polynomial of degree n. It was shown by Malik [J London Math Soc, 1 (1969), 57-60] that if P(z) has all its zeros in |z| ≤ k, k ≤ 1, then max |z|=1 |P′(z)| ≥ n 1 + k max |z|=1 |P(z)|. In this paper, we prove an inequality for the polar derivative of a polynomial which besides give extensions and refinements of the above inequality also produce various inequalities that are sharper than the previous ones known in very rich literature on this subject. [ABSTRACT FROM AUTHOR]

We point out a aw in a result proved by Singh and Shah [Kyungpook Math. J., 57(2017), 537-543] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form. [ABSTRACT FROM AUTHOR]

A flaw in the main definition of the original paper has been corrected. Other typos have been fixed. An error in a lemma proof has been amended. With these changes, all the results in the aforementioned paper are now valid. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019). [ABSTRACT FROM AUTHOR]

Let f be a transcendental entire function defined in the open complex plane ℂ. A difference-monomial generated by f is an expression of the form F = fn(fm -1) ... (f (z+cj))..., where n, m and Vj are all non-negative integers. Now for the sake of definiteness let us take, Mi[f ] = fn(fm -1) ... (f (z+cj))..., where 1 ≤ i ≤ d. If M1 [f], M1 [f ],..., Mn [f] are such monomials in f as defined above, then ψ[f] = a1M1[f] + a2M2[f] + ... + anMn [f] where ai ≠ 0 (i = 1, 2,..., n) is called a difference-polynomial generated by f. In this paper, we compare the Valiron defect with the relative Nevan-linna defect of a particular type of differential-difference polynomial generated by a transcendental entire function with respect to integrated moduli of logarithmic derivative. Some examples are provided in order to justify the results obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle. [ABSTRACT FROM AUTHOR]

Let N be one of the 38 distinct square-free integers such that the arithmetic group \Gamma _0(N)^+ has genus one. We constructed canonical generators x_N and y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x_N, which we express as a polynomial in y_N with coefficients that are rational functions in x_N. As a corollary, we prove that for any point e in the upper half-plane which is fixed by an element of \Gamma _0(N)^+, one can explicitly evaluate x_N(e) and y_N(e). As it turns out, each value x_N(e) and y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x_N(\tau) and y_N(\tau) at any CM point \tau, including elliptic points, for any genus one group \Gamma _0(N)^+. Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity. [ABSTRACT FROM AUTHOR]

Let us say that a three-qubit state u 000|000âź© + u 001|001âź© + â‹Ż + u 111|111âź© is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | Ď• âź©, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)| Ď• âź© is a three-qubit of the form λ 1|000âź© + λ 2|011âź© + λ 3|101âź© + λ 4|110âź© + λ 5|111âź© with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (AcĂ-n et al 2000 Phys. Rev. Lett. 85 1560; AcĂ-n et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643). [ABSTRACT FROM AUTHOR]

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence (P n (x)) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two P m (x) , P n (x) is a conical combination of the polynomials P | m - n | (x) , ... , P m + n (x) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs (α , β) ∈ (- 1 , ∞) 2 for which the corresponding Jacobi polynomials (R n (α , β) (x)) n ∈ N 0 , normalized by R n (α , β) (1) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials (T n (α , β) (x)) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure (1 - x 2) α | x | 2 β + 1 χ (- 1 , 1) (x) d x . In this paper, we give the solution and show that (T n (α , β) (x)) n ∈ N 0 satisfies nonnegative linearization of products if and only if (α , β) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative properties of some polynomials generated by a q-shift difference and differentials of a meromorphic function. Our results generalize and supplement the results of Zheng and Xu (J. Math. Res. Appl., 42(2022), No. 1, 31-40). [ABSTRACT FROM AUTHOR]

In this paper we prove that, after an appropriate rescaling, the sum of moments of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments. [ABSTRACT FROM AUTHOR]

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada's method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods. [ABSTRACT FROM AUTHOR]

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green's function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation. [ABSTRACT FROM AUTHOR]