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In the paper [10] the L p -realization L p of the matrix Schrödinger operator L u = d i v ( Q ∇ u ) + V u was studied. The generation of a semigroup in L p ( R d , C m ) and characterization of the domain D ( L p ) has been established. In this paper we perturb the operator L p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on L p ( R d , C m ) with domain embedded in W 2 , p ( R d , C m ) . We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schrödinger operator are investigated. [ABSTRACT FROM AUTHOR]

In this paper equilibrium measures in the presence of external fields created by fixed charges are analyzed. These external fields are a particular case of the so-called rational external fields (in the sense that their derivatives are rational functions). Along with some general results, a thorough analysis of the particular case of two fixed negative charges (“attractors”) is presented; indeed, the main result of the paper deals with this particular case. As for the main tools used, this paper is a natural continuation of [31] , where polynomial external fields were thoroughly studied, and [37] , where rational external fields with a polynomial part were considered. However, the absence of the polynomial part in the external fields analyzed in the current paper adds a considerable difficulty to solve the problem and justifies its separated treatment; moreover, it is noteworthy to point out the simplicity and beauty of the results obtained. [ABSTRACT FROM AUTHOR]

The paper revisits the classical problem of evaluating f ( A ) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A . We present several numerical examples to illustrate our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 < p < ∞ ) . [ABSTRACT FROM AUTHOR]

Geometric properties of the Jackson and Hahn–Exton q -Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q -generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre–Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn–Exton q -Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q -Bessel function and its derivatives. Some open problems are proposed at the end of the paper. [ABSTRACT FROM AUTHOR]

Abstract: We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new. [Copyright &y& Elsevier]

Abstract: In this paper, we study the number of small amplitude limit cycles in arbitrary polynomial systems. It is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating Lyapunov constants and determining the order of the corresponding Hopf bifurcation. It is well known that the difficulty in calculating the Lyapunov constants increases with the increasing of the degree of polynomial systems. So, it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree , which is denoted by . In this paper, by applying the method developed by C. Christopher and N. Lloyd in 1995, and M. Han and J. Li in 2012, we first obtain the lower bounds for , and then prove that if . Finally, we obtain that grows as least as rapidly as for all large (it is proved by M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differential Equations 252 (2012) 3278–3304 that the number of all limit cycles in arbitrary polynomial systems with degree grows as least as rapidly as ). [Copyright &y& Elsevier]

Abstract: Exponential B-splines are the most well-known non-stationary subdivision schemes. A crucial limitation of these schemes is that they can reproduce at most two exponential polynomials (Jena et al., 2003) [26]. Although interpolatory schemes can improve the reproducing property of exponential polynomials, they are usually less smooth than the (exponential) B-splines of corresponding orders. In this regard, this paper proposes a new family of non-stationary subdivision schemes which extends the exponential B-splines to allow reproduction of more exponential polynomials. These schemes can represent exactly circular shapes, spirals or parts of conics which are important analytical shapes in geometric modeling. This paper also discusses the Hölder regularities of the proposed schemes. Lastly, some numerical examples are presented to illustrate the performance of the new schemes. [Copyright &y& Elsevier]

Abstract: We say that a symmetric noncommutative (nc) polynomial is nc plurisubharmonic (nc plush) on an nc open set if it has an nc complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if an nc polynomial is nc plurisubharmonic on an nc open set then the polynomial is actually nc plurisubharmonic everywhere and has the form where the sums are finite and , , are all nc analytic. Greene et al. (2011) has shown that if is nc plurisubharmonic everywhere then has the form in Eq. (0.1). In other words, makes a global assumption while the current paper makes a local assumption, but both reach the same conclusion. We show that if is nc plurisubharmonic on an nc “open set” (local) then is, in fact, nc plurisubharmonic everywhere (global) and has the form expressed in Eq. (0.1). This paper requires a technique that is not used in . We use a Gram-like vector and matrix representation (called the border vector and middle matrix) for homogeneous degree 2 nc polynomials. We then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix. This rigid structure plus the theorems in ultimately force the form in Eq. (0.1). [Copyright &y& Elsevier]

Abstract: Analytic signals of finite energy in signal analysis are identical with non-tangential boundary limits of functions in the related Hardy spaces. With this identification this paper studies a subclass of the analytic signals that, with the amplitude-phase representation , , satisfy the relation a.e., signals in this subclass are called mono-components, and, in that case, the phase derivative is called the analytic instantaneous frequency of s. This paper proves that when , where is real-valued, band-limited with minimal bandwidth B and is real-valued, as the restriction on the real line of some entire function, then s is an analytic signal if and only if is a linear function, and with there holds . In the case s is a mono-component. This generalizes the corresponding result obtained by Xia and Cohen in 1999 in which is assumed to be a real-valued polynomial. [Copyright &y& Elsevier]

Abstract: This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials , and through some congruence transformations for block matrices, where A is a Hermitian matrix. Then, the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and Hermitian unitary matrices. As applications, the paper establishes necessary and sufficient conditions for many matrix equalities to hold, as well as many inequalities in the Löwner partial ordering to hold. [Copyright &y& Elsevier]

The main purpose of this paper is to study the problem of simultaneous existence of two isochronous centers in some families of planar polynomial differential systems with symmetry. More precisely, we present conditions for a family of Z 2 -equivariant systems of degree 5 to have an isochronous bi-center. We also study their global phase portraits in the Poincaré disk and present considerations about the number of simultaneous centers in Z 2 -equivariant systems of degree 3 and 5. This study provides examples of quintic systems possessing three and five isochronous centers simultaneously. [ABSTRACT FROM AUTHOR]

In this paper we will discuss asymptotic properties of monic polynomials { S n λ ( x ) } n ≥ 0 orthogonal with respect to the Sobolev inner product 〈 p , q 〉 S = ∫ R p ( x ) q ( x ) d μ 0 + λ ∫ R p ′ ( x ) q ′ ( x ) d μ 1 , with λ > 0 , d μ 0 = e − x 2 d x , d μ 1 = x 2 + a x 2 + b e − x 2 d x , a , b ∈ R + and a ≠ b . It is well known that ( μ 0 , μ 1 ) is a pair of symmetric ( 1 , 1 ) -coherent measures. This means that there exist sequences { a n } n ∈ N , { b n } n ∈ N , a n ≠ b n for every n ∈ N , such that the algebraic relation H n ( x ) + b n − 2 H n − 2 ( x ) = Q n ( x ) + a n − 2 Q n − 2 ( x ) , n ≥ 2 , is satisfied, where { Q n ( x ) } n ≥ 0 is the sequence of monic orthogonal polynomials associated with μ 1 and { H n ( x ) } n ≥ 0 is the sequence of monic Hermite polynomials. We will study the relative asymptotics for Sobolev scaled polynomials and we will obtain Mehler–Heine type formulas, among others. [ABSTRACT FROM AUTHOR]

Zudilin proved some Ramanunjan-type supercongruences by the WZ method. Long proved a Ramanujan-type supercongruence conjecture due to Van Hamme by applying some hypergeometric evaluation identities. In this paper, we propose a conjecture on a complete q -analogue of this supercongruence of Van Hamme and prove it in a weaker form via the q -WZ method. Additionally, we give a q -analogue of a related congruence involving cubes of binomial coefficients due to Sun in the same way. We also present a conjecture on a q -analogue of the corresponding infinite series for 1 / π due to Ramanujan. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of heat equation with a constant memory kernel. We prove that the system does not decay for the positive kernel, and is polynomially stable but not exponentially for the negative kernel. In particular, a more detailed description of the decay property for the negative kernel is obtained by dividing initial values into two categories. Moreover, we show that the system can be exponentially stabilized by boundary control for any constant memory kernel. [ABSTRACT FROM AUTHOR]

The (L.2) supercongruence of Van Hamme was proved by Swisher recently. In this paper we provide a conjectural q -analogue of the (L.2) supercongruence of Van Hamme and prove a weaker form of it by using the q -WZ method. In the same way, we prove a complete q -analogue of the following congruence ∑ k = 0 n ( 6 k + 1 ) ( 2 k k ) 3 ( − 512 ) n − k ≡ 0 ( mod 4 ( 2 n + 1 ) ( 2 n n ) ) , which was conjectured by Z.-W. Sun and confirmed by B. He. We also provide a conjectural q -analogue of another congruence proved by Swisher. [ABSTRACT FROM AUTHOR]

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight ≤5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the spline interpretations of Eulerian numbers and refined Eulerian numbers are presented. Many classical results about Eulerian numbers can follow from the properties of B-splines directly, and some new results about the refined Eulerian numbers and descent polynomials are also derived. Specifically, the explicit and recurrence formulas for the refined Eulerian numbers and descent polynomials are obtained. This paper also provides a new approach to study Eulerian numbers. [Copyright &y& Elsevier]

Abstract: It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that for , then for . If p is a polynomial of degree at most n with for , then is an entire function of exponential type n with on the real axis. Hence, by the just mentioned inequality for functions of exponential type, for . Lately, many papers have been written on polynomials p that satisfy the condition . They do form an intriguing class. If a polynomial p satisfies this condition, then is an entire function of exponential type n that satisfies the condition . This led Govil [N.K. Govil, inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445–452] to consider entire functions f of exponential type satisfying and find estimates for their derivatives. In the present paper we present some additional observations about such functions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the growth of solutions of a first-order linear differential equation and that of a second-order linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to investigate some general properties of common zeros of orthogonal polynomials in two variables for any given region from a view point of invariant factor. An important result is shown that if is a common zero of all the orthogonal polynomials of degree k then the intersection of any line passing through and D is not empty. This result can be used to settle the problem of location of common zeros of orthogonal polynomials in two variables. The main result of the paper can be considered as an extension of the univariate case. [Copyright &y& Elsevier]

Abstract: In this paper some new properties and applications of modified Chebyshev polynomials and Morgan–Voyce polynomials will be presented. The aim of the paper is to complete the knowledge about all of these types of polynomials. [Copyright &y& Elsevier]

Abstract: This paper presents a study on the limit cycles of -equivariant polynomial vector fields with degree 3 or 4. Previous studies have shown that when , cubic-order systems can have 12 small amplitude limit cycles. In this paper, particular attention is focused on the cases of . It is shown that for cubic-order systems, when there exist 3 small limit cycles and 1 big limit cycle; while for , it has 4 small limit cycles and 1 big limit cycle; and when , there is only 1 small limit cycle. For fourth-order systems, the cases for even q are the same as the cubic-order systems. When it can have 10 small limit cycles; while for , there exists only 1 small limit cycle. The case is not considered in this paper. Numerical simulations are presented to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: Let be a polynomial of degree 4 with four distinct real roots . Let be the critical points of p, and define the ratios , . For notational convenience, let , , and . is called the ratio vector of p. We prove necessary and sufficient conditions for to be a ratio vector of a polynomial of degree 4 with all real roots. Most of the necessary conditions were proven in an earlier paper. The main results of this paper involve using the theory of Groebner bases to prove that those conditions are also sufficient. [Copyright &y& Elsevier]

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single-, double-, and triple-integral representations. Some known or new consequences of the general results presented here, involving such classical orthogonal polynomials as the Jacobi, Laguerre, Hermite, and Bessel polynomials, and various other relatively less familiar hypergeometric polynomials, are also considered. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [Copyright &y& Elsevier]

Wavelets of Haar type of higher order m on self-similar fractals were introduced by the author in J. Fourier Anal. Appl. 4 (1998) 329–340. These are piecewise polynomials of degree m instead of piecewise constants. It was shown that for certain totally disconnected fractals, spaces of functions defined on the fractal may be characterized by means of the magnitude of the wavelet coefficients of the functions. In this paper, the study of these wavelets is continued. It is shown that also in the case when the fractals are not totally disconnected, the wavelets can be used to study regularity properties of functions. In particular, the self-similar sets considered can be, e.g., an interval in R or a cube in Rn. It turns out that it is natural to use Haar wavelets of higher order also in these classical cases, and many of the results in the paper are new also for these sets. [Copyright &y& Elsevier]

Three extremal problems for trigonometric polynomials are studied in this paper. The first was initiated by Maiorov. It relates to the trigonometric polynomials with n nonzero harmonics. The Lp-norm of the Weyl derivative is compared with the Lq-norm of the polynomial. The other two problems have appeared in the recent paper by Oswald. They deal with the polynomials of degree n. The minimum of Lp-norm with respect to the choice of phases is compared with lq-norm of its coefficients. [Copyright &y& Elsevier]

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation. The pantograph equation contains a linear functional argument. In this paper we generalize this functional argument to include nonlinear polynomials. In contrast to the entire solutions generated by the pantograph equation for the retarded case, we show that in the nonlinear case natural boundaries occur. These boundaries are linked to the Julia set of the polynomial functional argument. [Copyright &y& Elsevier]

In this paper, we study a type of polynomial Liénard system of degree m ( m ≥ 2 ) with polynomial perturbations of degree n . We prove that the first order Melnikov function of such system has at most n + 1 − [ n + 1 m + 1 ] independent perturbation parameters which can be used to simplify this kind of systems. As an application, we study a type of Lienard systems for m = 4 , n = 19 , 28 and obtain the new lower bounds of the maximal number of limit cycles. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the behavior of holomorphic mappings on A -compact sets. Motivated by the recent work of Aron, Çalişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ -Banach operator ideal A) under which A -compact sets are preserved. Appealing to the notion of tensor stability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A ; B) -compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p , r) -compact set under any holomorphic function (defined on any open set of a Banach space) is again (p , r) -compact. [ABSTRACT FROM AUTHOR]

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators. In an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in Complex Analysis – and, also, taking into account recent results in the field – in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional subspace of C ( X ; R ) , where X is a locally compact Hausdorff space, and A = { f 1 , … , f m } a basis of A . A sequence s = ( s j ) j = 1 m is called a moment sequence if s j = ∫ f j ( x ) d μ ( x ) , j = 1 , … , m , for some positive Radon measure μ on X . Each moment sequence s has a finitely atomic representing measure μ . The smallest possible number of atoms is called the Carathéodory number C A ( s ) . The largest number C A ( s ) among all moment sequences s is the Carathéodory number C A . In this paper the Carathéodory numbers C A ( s ) and C A are studied. In the case of differentiable functions methods from differential geometry are used. The main emphasis is on real polynomials. For a large class of spaces of polynomials in one variable the number C A is determined. In the multivariate case we obtain some lower bounds and we use results on zeros of positive polynomials to derive upper bounds for the Carathéodory numbers. [ABSTRACT FROM AUTHOR]

In this paper we present several characterizations of ordinary polynomials as the solution sets of certain functional equations related to the equation ∑ i = 0 m f i ( b i x + c i y ) = ∑ i = 1 n a i ( y ) v i ( x ) , where x , y ∈ R d and b i , c i ∈ GL d ( C ) , whose solution set is, typically, formed by exponential polynomials. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory. [ABSTRACT FROM AUTHOR]

The paper aims to examine convexity and concavity conditions for homogeneous orthogonally additive polynomials in quasi-Banach lattices. We characterize ( p , q ) -convex quasi-Banach lattice by the property that every positive orthogonally additive polynomial defined on such lattice is ( p , q ) -convex as well as obtain some versions of Krivine factorization result and Grothendieck's inequality for orthogonally additive polynomials in quasi-Banach lattices. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the planar polynomial system: x ˙ = a x − y + P n ( x , y ) , y ˙ = x + a y + Q n ( x , y ) , where a ∈ R and P n , Q n are homogeneous polynomials of degree n ≥ 2 . Denote ψ ( θ ) = cos ⁡ ( θ ) ⋅ Q n ( cos ⁡ ( θ ) , sin ⁡ ( θ ) ) − sin ⁡ ( θ ) ⋅ P n ( cos ⁡ ( θ ) , sin ⁡ ( θ ) ) . We prove that the system has at most 1 limit cycle surrounding the origin provided ( n − 1 ) a ψ ( θ ) + ψ ˙ ( θ ) ≠ 0 . Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ , for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin. [ABSTRACT FROM AUTHOR]

In this paper we consider a system of Timoshenko-thermoelasticity with second sound in the presence of an infinite history term. We prove, under suitable conditions on the coefficients and the relaxation function, a general decay result from which the exponential and polynomial decay estimates are only special cases. We also establish a lack of exponential decay result and discuss the optimality for some situation. [ABSTRACT FROM AUTHOR]

Let A be an expanding integer matrix with characteristic polynomial f ( x ) = x 2 + p x + q , and let D = { 0 , 1 , … , | q | − 2 , | q | + m } v be a collinear digit set where m ⩾ 0 , v ∈ Z 2 . It is well known that there exists a unique self-affine fractal T satisfying A T = T + D . In this paper, we give a complete characterization of connectedness of T . That generalizes the previous result for | q | = 3 . [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles in the family d H − ε ω = 0 , where H = y 2 2 − ∫ 0 x g ( u ) d u , ω = y f ( x ) d x , with g ( x ) = x ( x 2 − 1 ) ( x 2 − 1 4 ) 2 , and f ( x ) an even polynomial of degree 10. We will consider mainly the bifurcation of limit cycles near the eye-figure loop and the center of d H = 0 . Our investigation focuses on the lower bound of the maximal number of limit cycles for these systems. In particular, we show that the perturbed system can have at least 8 limit cycles when deg ⁡ ( f ( x ) ) = 10 . [ABSTRACT FROM AUTHOR]

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n . It proves that for n = 3 , 4 and 5, there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively, and in each case this upper bound is sharp. Moreover, for any n > 5 , there are at most [ n − 1 2 ] critical periods induced by periodic orbits of the unperturbed cubic system. An example is given to show that the upper bound in the case of n = 11 can be reached. [ABSTRACT FROM AUTHOR]

In this paper, we show how to deduce Askey–Wilson type integrals by the method of q -difference equation. In addition, we construct the relation between bilinear generating functions and solutions of q -difference equation. More over, we generalize Bailey's ψ 6 6 summation from the perspective of q -integral by the method of q -difference equation. At last, we deduce U ( n + 1 ) type generating functions for Al-Salam–Carlitz polynomials by the method of q -difference equation. [ABSTRACT FROM AUTHOR]

The non-vanishing finite order entire solutions of nonlinear difference equation f ( z ) n + a n − 1 f ( z ) n − 1 + ⋯ + a 1 f ( z ) + q ( z ) e Q ( z ) f ( z + c ) = P ( z ) , are studied in this paper for n ≥ 2 and n = 1 , respectively, where q ( z ) , Q ( z ) , P ( z ) are polynomials and a n − 1 , … , a 1 , c ∈ C . We classified these solutions according to growth and zero distribution. Particularly, we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms when n = 2 and a 1 ≠ 0 . Our results are improvements and complements of Wen et al. [13] . [ABSTRACT FROM AUTHOR]

A well-known result of Montel states that for a family F of meromorphic functions in a domain D ⊂ C , if there exist three distinct points a 1 , a 2 , a 3 in C ˆ and positive integers ℓ 1 , ℓ 2 , ℓ 3 such that 1 ℓ 1 + 1 ℓ 2 + 1 ℓ 3 < 1 and all zeros of f − a i have multiplicity at least ℓ i for all f ∈ F and i ∈ { 1 , 2 , 3 } , then F is normal in D . Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established for the case where meromorphic functions (or differential polynomials generated by the members of the family) meet some distinct points with sufficiently large multiplicities. This means that these criteria strictly apply only to the case in which derivatives of functions (differential polynomials, respectively) vanish on respective zero sets. In this paper, we generalize some normality criteria of Montel, Grahl–Nevo, Gu, and Bergweiler–Langley to the case where derivatives are bounded from above on zero sets. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large time behavior of solutions to the one-dimensional degenerate p -degree Fisher equation, where the initial data are assumed to be asymptotically front-like and to decay to zero non-exponentially at one end. By applying sub–super solution method we first prove the Lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space. Further, by using semigroup estimates, sub–super solution method and the nonlinear stability of the traveling waves, we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite, which is solely determined by the detailed decaying rate of the initial data. [ABSTRACT FROM AUTHOR]

Inverse and direct balancing methods for the construction of kink solitary solutions to generalized Riccati equations with polynomial coefficients are presented in this paper. Necessary and sufficient conditions for the existence of kink solitary solutions are derived in terms of the equation's parameters and initial conditions. The proposed technique enables a straightforward derivation of parameters of solitary solutions. Computational experiments are used to demonstrate the efficiency of the proposed analytical approach. [ABSTRACT FROM AUTHOR]

The Hermitian monogenic system is an overdetermined system of two Dirac type operators in several complex variables generalizing both the holomorphic system and the real Dirac system. Due to the fact that it is overdetermined, the Cauchy–Kowalevski extension problem only has a solution if the Cauchy data satisfy certain constraints. There is however a subsystem, called Hermitian submonogenic system, for which theses constraints are no longer necessary, while, if the constraints hold, the Cauchy–Kowalevski extension will still be Hermitian monogenic. In this paper we focus on Cauchy–Kowalevski extensions of general polynomials, in the case of the Hermitian submonogenic system, and we compute the corresponding dimensions and reproducing kernels. [ABSTRACT FROM AUTHOR]

This paper deals with a unified matrix representation for the Sheffer polynomials. The core of the proposed approach is the so-called creation matrix, a special subdiagonal matrix having as nonzero entries positive integer numbers, whose exponential coincides with the well-known Pascal matrix. In fact, Sheffer polynomials may be expressed in terms of two matrices both connected to it. As we will show, one of them is strictly related to Appell polynomials, while the other is linked to a binomial type sequence. Consequently, different types of Sheffer polynomials correspond to different choices of these two matrices. [ABSTRACT FROM AUTHOR]

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L 2 ( − 1 , 1 ) which has the Legendre polynomials { P n } n = 0 ∞ as eigenfunctions. As a consequence, they explicitly determined the domain D ( A 2 ) of the self-adjoint operator A 2 . However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L 2 ( − 1 , 1 ) so D ( A 2 ) should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman–Krein–Naimark) theory. In addition, we determine a new characterization of D ( A 2 ) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory. [ABSTRACT FROM AUTHOR]

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions gives rise to a sequence of polynomials {Pm (z) }m=0∞ that is eventually hyperbolic. Moreover, the real zeros of the polynomials Pm (z) form a dense subset of an interval I⊂ R+, whose length depends on the particular values of the parameters in the generating function. [ABSTRACT FROM AUTHOR]

This paper investigates the problem for limit cycle bifurcations of system x ˙ = y F ( x , y ) + ε p ( x , y ) , y ˙ = − x F ( x , y ) + ε q ( x , y ) , where F ( x , y ) consists of multiple circles and p ( x , y ) , q ( x , y ) are polynomials of degree n . The upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order Melnikov function. Furthermore, Hopf bifurcation for a cubic system of this type is discussed. [ABSTRACT FROM AUTHOR]