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Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness of meromorphic functions with q-shift difference-differential polynomials F = [P(f)L(z, f)s](k) and G = [P(g)L(z, g)s](k), where P(z) is a non-constant polynomial with degree n sharing a finite value. The results of this paper are an extension of the previous theorems given by Harina P. Waghamore and Rajeshwari S [19]. [ABSTRACT FROM AUTHOR]

This research centers on the application of the discrete-time Taylor–Fourier transform (DTTFT) algorithmic implementation for phasor estimation on a field-programmable gate array board. The system employs a finite impulse response structure of a digital Taylor–Fourier filter to extract amplitude and phase information. The hardware description utilizes a multiply accumulator architecture with only forty embedded 9-bit multiplier elements, achieving an 18-bit input–output resolution. Performance assessment involves signal analysis through FPGA-in-the-loop simulation in MATLAB/Simulink. Findings demonstrate that the DTTFT-based phasor estimator can be effectively characterized using VHDL code and implemented on an Intel D2-115 board. [ABSTRACT FROM AUTHOR]

Let p (z) be a polynomial of degree n having no zero in | z | < k , k ≤ 1 , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ (z) | ≤ n 1 + k n max | z | = 1 | p (z) | , provided | p ′ (z) | and | q ′ (z) | attain their maxima at the same point on the circle | z | = 1 , where q (z) = z n p (1 z ‾) ‾. In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results. [ABSTRACT FROM AUTHOR]

A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form Sc(z) = azn + b/zm + c, where a; b; c ∈ C with |a| > 1 and n;m ∈ N with n > 1. By applying viscosity approximation-type iteration processes extended with s-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters. [ABSTRACT FROM AUTHOR]

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients a 2 and a 3 for functions in these subclasses. Using the values of a 2 and a 3 , we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two novel subclasses of bi-univalent functions, MΣ(α, B(x, ξ)) and HΣ(α, µ, B(x, ξ)), utilizing Lucas-Balancing polynomials. Within these function classes, we established bounds for the Taylor-Maclaurin coefficients |a2| and |a3|, addressing the Fekete-Szegö functional problems specific to functions within these new subclasses. Moreover, we illustrated how our primary findings could lead to various new outcomes through parameter specialization. [ABSTRACT FROM AUTHOR]

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel–Darboux formula. These results should be viewed as the direct analogue of the nearest neighbour recurrence relations from the theory of multiple orthogonal polynomials on the real line. [ABSTRACT FROM AUTHOR]

In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing t = 0 Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of U q ′ (sl ^ 2) -modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials. [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function x M (x + 1) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

In terms of the creative microscoping method, El Bachraoui's lemmas, and the Chinese remainder theorem for coprime polynomials, we establish some q-supercongruences for double and triple basic hypergeometric series. Several related conjectures are also proposed for further research. [ABSTRACT FROM AUTHOR]

This study focused on "Extension of synthetic division and its applications". The study was designed to show synthetic division and its extension and to point out the applications of synthetic division and its extension. The study found out that the concepts of polynomial and rational expressions in single variables are basic concepts to deal extension of synthetic division and its applications. Using the preliminary concepts, the concept of synthetic division is extended in this study. Also, the study found out that an extension of synthetic division is used for finding the oblique asymptote of the graph of a rational function, evaluating the integration of some rational functions, representing polynomial expression by factorial function in numerical analysis, and so on. [ABSTRACT FROM AUTHOR]

Let ... be a polynomial of degree n having all its zeros in ..., then Aziz [Proc. Am. Math. Soc., 89, (1983) 259--266] proved ... . In this paper, we prove a polar derivative extension which sharpens the above inequality. As a consequence, we also derive a result on Bernstein type inequality for the class of polynomials having all its zeros in ... . [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of sharing values between entire functions f(z) and f1(z) = b-1 + ... bi f(ki)(z+iη) share two distinct values a with counted and b with ignoring multiplicities, where b-1 and bi(i = 0,1· · ·, n) are small meromorphic functions of f(z), ki ≥ 0 (i = 0,1, · · ·, n) are integers. In relation to previous research, we obtain results that improve and generalise the findings conducted by Yang and Qi [CMFT, 20.1 (2020): 159-178]. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new polynomial inequalities involving various operators and their relationship with the existing results. All these inequalities have been proved under the assumption that the underlying polynomial is being constrained with respect to its zeros. [ABSTRACT FROM AUTHOR]

Let us consider f as being an entire solution of the differential-difference equation G(z, f) + h(z)f m(z) = 0, (m ∈ ℕ), where h(z) is a transcendental entire function and G(z,f) is a differential-difference polynomial in f with entire coefficients. By considering the order and deficiency of h(z) and such coefficients, we mainly study the radial distribution of f, and establish a lower bound of measure for the set of common limiting directions of the Julia sets of derivatives and primitives of its shifts. [ABSTRACT FROM AUTHOR]

In the present study, we use several identities from the q-calculus to define the concept of q-Hermite polynomials with three variables and present their associated formalism. Many properties and new results of q-Hermite polynomials of three variables are established, including their generation function, series description, summation equations, recurrence relationships, q-differential formula and operational rules. [ABSTRACT FROM AUTHOR]

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form Q C (p) = a p n + m p + c , where n ≥ 2 . It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals. [ABSTRACT FROM AUTHOR]

This paper aims to study the uniqueness of certain types of differential polynomials generated by two meromorphic functions sharing one value with finite weight. We obtain some results which extend and generalize some recent results due to Rajeshwari S, Husna V, and Nagarjun V. [1]. [ABSTRACT FROM AUTHOR]

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

The aim of this research is to present a new generalization of d-orthogonal $ (d\geq 2) $ (d ≥ 2) polynomials of Laguerre type by utilizing a suitable generating function from Sheffer class and employing operational rules associated with the lowering and raising operators that satisfy d-orthogonality. We derive several properties of these polynomials and establish the recurrence relation. Moreover, we provide explicit, connection and inversion formulas, the $ (d+1) $ (d + 1) -order differential equation, and the canonical d-dimensional functional vector. [ABSTRACT FROM AUTHOR]

In this article, we investigate some properties of meromorphic solution of the following differentialdifference equation ∑j=0n cjf (z + j) + ∑j=0n ljf(j) (z + j) + ∑j=0m djf(j) (z) -- c = (f (z + η) -- f (z) -- b)eQ, where n,m ≥ 1 are two integers, ηis a constant with |η| ≠ n, Q is a polynomial. We study the growth of solutions of a more general differential-difference equation given by Lü et al [Rest Math 74 (2019):1-18]. Meantime we obtain the relation between deg Q and p(f). [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the uniqueness problem of derivatives of a meromorphic function and its difference polynomial in view of a partially sharing. As a consequence of the main result, we improve the recent result of W. J. Chen and Z. G. Huang with the weaker hypotheses and also supplement several results in particular cases. [ABSTRACT FROM AUTHOR]

. We consider a minimal realization of a rational matrix. We perturb all the coefficients of matrix polynomial and some coefficients from the realization part present in the realization form of rational matrix. We derive explicit computable formulae for backward error of approximate eigenvalues and eigenpairs of regular rational matrix. We also determine minimal perturbations for all the coefficients of matrix polynomial and some coefficients from the realization part for which approximate eigenvalues are exact eigenvalues of the perturbed rational matrix. [ABSTRACT FROM AUTHOR]

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour. [ABSTRACT FROM AUTHOR]

The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented. [ABSTRACT FROM AUTHOR]

In this research article, we have studied the results of P. Sahoo and H. Karmakar, intending to determine, in any manner, whether it is possible to relax the nature of sharing by replacing the shift of non-constant transcendental entire functions of finite order with the product of shift. In this direction, we have investigated the uniqueness of shift difference polynomials of two entire functions when they share a non-zero polynomial with a finite weight and one being the Mobius transformation of the other satisfying n = 2d-s +3, and also when they share a small function with a finite weight satisfying n = m + s + 5. We also investigate the same situation when the original functions f and g share the value zero counting multiplicities (CM) satisfying n > 2(G1 + d) - s. Our results in this paper extend and generalizes the previous results of P. Sahoo and H. Karmakar [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2017, 52 (2), pp. 102-110]. [ABSTRACT FROM AUTHOR]

We investigate the uniqueness problem of entire functions that share two polynomials with their kth derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of zero set of reverse indexed polynomial, first introduced by us, we have investigated the uniqueness of meromorphic functions under weighted sharing in wider sense criteria [4], which in turn extend some earlier results in different directions. We have succeeded to identify a subclass of meromorphic functions for which uniqueness property exists for higher reverse indexed polynomial in literature. In the last section, we have presented the application of our results in case of derivatives of the concerned functions accompanied by series of examples. [ABSTRACT FROM AUTHOR]