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On Certain Properties of Perturbed Freud-Type Weight: A Revisit

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Abstract

In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fulfill linear differential equations with some polynomial coefficients in their holonomic form. The aim of this work is to explore certain characterizing properties of perturbed Freud-type polynomials such as nonlinear recursion relations, finite moments, differential-recurrence, and differential relations satisfied by the recurrence coefficients as well as the corresponding semiclassical orthogonal polynomials. We note that the obtained differential equation fulfilled by the considered semiclassical polynomials are used to study an electrostatic interpretation for the distribution of zeros based on the original ideas of Stieltjes.KeywordsOrthogonal polynomialFreud-typeThree-term recurrenceDifferential-recurrence equationElectrostatic zeros2010 Mathematics Subject Classification:33C45

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In this study, the free axisymmetric vibrations of functionally graded circular plate of linearly varying thickness controlled by a taper parameter in radial direction and non-linear temperature rise along the thickness have been investigated on the basis of classical plate theory. The plate material is graded in transverse direction and its mechanical properties are temperature-dependent. The temperature over the top and bottom surfaces is assumed to be uniformly distributed. Hamilton’s principle has been used to derive the governing equations for thermo-elastic equilibrium and axisymmetric motion of such a plate model. The generalized differential quadrature method has been employed to obtain the thermal displacements and characteristic equations, for clamped and simply supported plates. The lowest three roots of these equations have been computed and reported as the values of frequency parameter for the first three modes of vibration. Effect of thickness parameter, volume fraction index and temperature difference has been analyzed on the vibration characteristics of the plate. A study with temperature-independent material properties has also been performed. Results have been compared with the published work.
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In this paper our aim is to establish several new properties for some class of functions related to the Fox-Wright functions. In particular, the monotonicity, convexity, sub-additivity and super-additivity properties involving the Fox-Wright functions are proved, this results is also closely connected with some functional inequalities (like Tur´an type inequalities). Moreover, we investigate certain criteria for the univalence and starlikeness for a certain class of functions associated with the Fox-Wright functions.
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This paper investigates the nonlinear functional-integral equation comprising of Hadamard fractional operator. Using the concept of measure of non compactness, existence of solutions in Banach algebra has been studied under certain relevant assumptions in conjunction with fixed point theory. Finally an example has been considered to substantiate the validity of the result.