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The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

The paper deals with a new fuzzy version of the Lee-Carter (LC) mortality model, in which mortality rates as well as parameters of the LC model are treated as triangular fuzzy numbers. As a starting point, the fuzzy Koissi-Shapiro (KS) approach is recalled. Based on this approach, a new fuzzy mortality model -- CNMM -- is formulated using orthonormal expansions of the inverse exponential membership functions of the model components. The paper includes numerical findings based on a case study with the use of the new mortality model compared to the results obtained with the standard LC model. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [ABSTRACT FROM AUTHOR]

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]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number a is defined by Da p(z) = np(z)+(a -z)p'(z). If p(z) has all its zeros in |z| ≥ k, k ≤ 1, then for |a| ≥ k, Aziz and Rather [Math. Inequal. Appl., 1, (1998), 231-238] proved. In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta Math. Hungar., 104, (2004), 115-126] and also prove an inequality for a subclass of polynomials having all its zeros in |z| ≥ k, k ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper,we study the growth of solutions of a k-th order linear differential equation and that of a k+1-th order linear differential equation. From this we affirmatively answer a uniqueness question concerning a conjecture given by Brück in 1996 under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM. The results in this paper also improve some known results. Some examples are provided to show that the results in this paper are best possible. [ABSTRACT FROM AUTHOR]

We study the complexity of approximating the value of the independent set polynomial ZG(\lambda) of a graph G with maximum degree \Delta when the activity \lambda is a complex number. When \lambda is real, the complexity picture is well understood, and is captured by two real-valued thresholds \lambda \ast and \lambda c, which depend on \Delta and satisfy 0 < \lambda \ast < \lambda c. It is known that if \lambda is a real number in the interval (\lambda \ast, \lambda c) then there is a fully polynomial time approximation scheme (FPTAS) for approximating ZG(\lambda) on graphs G with maximum degree at most \Delta. On the other hand, if \lambda is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds \lambda \ast and \lambda c on the \Delta -regular tree. The ''occupation ratio"" of a \Delta -regular tree T is the contribution to ZT (\lambda) from independent sets containing the root of the tree, divided by ZT (\lambda) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if \lambda \in [ \lambda \ast, \lambda c]. Unsurprisingly, the case where \lambda is complex is more challenging. It is known that there is an FPTAS when \lambda is a complex number with norm at most \lambda \ast and also when \lambda is in a small strip surrounding the real interval [0, \lambda c). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the complex values of \lambda for which the occupation ratio of the \Delta -regular tree converges. These values carve a cardioid-shaped region \Lambda \Delta in the complex plane, whose boundary includes the critical points \lambda \ast and \lambda c. Motivated by the picture in the real case, they asked whether \Lambda \Delta marks the true approximability threshold for general complex values \lambda. Our main result shows that for every \lambda outside of \Lambda \Delta, the problem of approximating ZG(\lambda) on graphs G with maximum degree at most \Delta is indeed NP-hard. In fact, when \lambda is outside of \Lambda \Delta and is not a positive real number, we give the stronger result that approximating ZG(\lambda) is actually \#P-hard. Further, on the negative real axis, when \lambda < \lambda \ast, we show that it is \#P-hard to even decide whether ZG(\lambda) > 0, resolving in the affirmative a conjecture of Harvey, Srivastava, and Vondrak. Our proof techniques are based around tools from complex analysis--specifically the study of iterative multivariate rational maps. [ABSTRACT FROM AUTHOR]

Let P(z) = a0 + Σn ν=µ aνzν, 1 ≤ µ ≤ n, be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1. We obtain an improvement and a generalization of an inequality in polar derivative proved by Somsuwan and Nakprasit [1]. Further, we also extend a result proved by Chanam and Dewan [2] to its polar version. Besides, our results are also found to generalize and improve some known inequalities [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

We establish some -inequalities for the polar derivative of a polynomial. Our results generalize some known Bernstein-type polynomial inequalities and include as special cases several interesting generalizations of some Zygmund-type inequalities for polynomials as well. [ABSTRACT FROM AUTHOR]

We prove that if P( z) is a polynomial of degree n with zeros z , that satisfy $${|z_m| \geq K_m \geq 1}$$ , $${1 \leq m \leq n}$$ , then for any p > 0, and for every complex number α, with $${|\alpha| \geq 1}$$ , we have where $${G_{p}=\big\{\frac{2\pi}{\int_0^{2\pi}|t_0+e^{i\theta}|^{p}\,d\theta}\big\}^{{1}/{p}}}$$ , and $${t_0=\big\{1+\frac{n}{\sum_{m=1}^n\frac{1}{K_m-1}}\big\}}$$ if $${K_{m}>1}$$ $${(1\leq m \leq n)}$$ , and t = 1 if K = 1 for some m, $${1\leq m\leq n}$$ . Our results generalize and sharpen several of the known results. [ABSTRACT FROM AUTHOR]

All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox-Wright function 1Ψ1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another 1Ψ1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the 1Ψ1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the 1Ψ1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree nn and for any complex number aa, let DαP(z)=nP(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials. [ABSTRACT FROM AUTHOR]

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1; then for any real r ≥ 1, Aziz [J. Approx. Theory, 55(1988), 232-239] proved the integral mean inequality ... In this paper, we obtain an improved extension of the above inequality by using the polar derivative instead of the ordinary derivative and involving the leading coefficient and the constant term of the polynomial. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number α is defined by D α p (z) = n p (z) + (α - z) p ′ (z). If p (z) = z s ∑ j = 0 n - s c j z j , 0 ≤ s ≤ n , has all its zeros in | z | ≤ k , k ≥ 1 , then for | α | ≥ k , Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved max | z | = 1 | D α p (z) | ≥ n (| α | - k) 1 + k n - s 1 + (| c n - s | k n - | c 0 | k s) (k - 1) 2 (| c n - s | k n + | c 0 | k s + 1) max | z | = 1 | p (z) |. In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in | z | < k , k ≤ 1 . [ABSTRACT FROM AUTHOR]

The purpose of this article is to generalize the ring of q-Appell polynomials to the complex case. The formulas for q-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex q-Appell polynomials are also q-complex analytic functions, we are able to give a first example of the q-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define q-complex Bernoulli and Euler polynomials. Previously, in order to obtain the q-Appell polynomial, we would make a q-addition of the corresponding q-Appell number with x. This is now replaced by a q-addition of the corresponding q-Appell number with two infinite function sequences C;q(x; y) and S;q(x; y) for the real and imaginary part of a new so-called q-complex number appearing in the generating function. Finally, we can prove q-analogues of the Cauchy-Riemann equations. [ABSTRACT FROM AUTHOR]

Let $$P(z)=a_{0}+\sum _{\nu =\mu }^{n}a_{\nu }z^{\nu },~1\le \mu \le n,$$ be a polynomial of degree $$n$$ such that $$P(z)\ne 0$$ in $$|z|< k,~k> 0,$$ then for $$01$$ by proving and In this paper, we obtain improvements of the above inequalities by involving some of the coefficients of the polynomial $$P(z).$$ Besides this, our results also generalize and refines the results of Barchand and Dewan (J Math Anal Appl 336:171-179, ). [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

The article discusses the fundamental theorem of algebra. It mentions that every non-constant single-variable polynomial with complex coefficients has at least one complex root. It includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. It adds that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots.

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]

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The polar derivative of a polynomial P(z) of degree n with respect to a complex number γ is a polynomial nP(z) + (γ-z)P′(z) of degree at most n-1 and is denoted by DγP(z). We consider the class of polynomials P(z) = a0 + Σv=μN avzv, μ ≥ 1, of degree n such that P(z) ≠0 in |z| < k, k ≥ 1 and establish some upper bound estimates for the maximum modulus of DγP(z) on the unit disk by involving some of the coefficients of P(z). The obtained results refine and generalize some well known polynomial inequalities. [ABSTRACT FROM AUTHOR]

The well-known Erdös-Lax and Turán-type inequalities, which relate the uniform norm of a univariate complex coefficient polynomial to its derivative on the unit circle in the plane, are discussed in this paper. We create some new inequalities here when there is a restriction on its zeros. The obtained results strengthen some recently proved ErdösLax and Turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in a very rich literature on this subject. [ABSTRACT FROM AUTHOR]

Let $\mathscr{F}$ be a family of solutions of Laplace equations in a domain D and for each $f\in\mathscr{F}$, f has only zeros of multiplicity at least k. Let n be a positive integer and such that $n\geq\frac{1+\sqrt{1+4k(k+1)^{2}}}{2k}$. Let a be a complex number such that $a\neq0$. If for each pair of functions f and g in $\mathscr{F}$, $f^{n}f^{(k)}$ and $g^{n}g^{(k)}$ share a value in D, then $\mathscr{F}$ is normal in D. [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]

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]

Let p(z) be a polynomial of degree n and for a complex number α, let Dαp(z) np(z) + (α - z)p'(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros in /z/ ≤ K. Our results generalize and sharpen a famous inequality of Turán and some other known results in this direction. [ABSTRACT FROM AUTHOR]