scholarly journals Zeros of a binomial combination of Chebyshev polynomials

2021 ◽  
pp. 1-13
Author(s):  
Summer Al Hamdani ◽  
Khang Tran
Keyword(s):  
Power Series ◽  
Linear Combination ◽  
Chebyshev Polynomials ◽  
Constant Independent ◽  
Number Of Zeros

For [Formula: see text], we study the zeros of the sequence of polynomials [Formula: see text] generated by the reciprocal of [Formula: see text], expanded as a power series in [Formula: see text]. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of [Formula: see text] outside the interval [Formula: see text] is bounded by a constant independent of [Formula: see text].

On the Zeros of Power Series with Exponential Logarithmic Coefficients

1981 ◽  
Vol 24 (3) ◽  
pp. 257-271 ◽  
Author(s):  
W. Gawronski ◽  
U. Stadtmüller
Keyword(s):  
Power Series ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Number Of Zeros ◽  
Logarithmic Coefficients ◽  
Image Position

In this paper we investigate the zeros of power series1for some functions of coefficients A. In particular, we derive upper and lower bounds for the number of zeros of f in its domain of analyticity.

scholarly journals Linear combinations of polynomials with three-term recurrence

2021 ◽  
Vol 110 (124) ◽  
pp. 29-40
Author(s):  
Khang Tran ◽  
Maverick Zhang
Keyword(s):  
Linear Combination ◽  
Chebyshev Polynomials ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Zero Distribution ◽  
Necessary And Sufficient ◽  
Linear Polynomials

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

scholarly journals On the number of zeros of exponential polynomials and related questions

1992 ◽  
Vol 46 (3) ◽  
pp. 401-412 ◽  
Author(s):  
A.J. van der Poorten ◽  
I.E. Shparlinski
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Zeros Of Polynomials ◽  
Number Of Solutions ◽  
Exponential Polynomials ◽  
Number Of Zeros ◽  
Polynomial Coefficients ◽  
Congruence Modulo ◽  
Linear Differential ◽  
Zeros Of Exponential Polynomials

We apply Straßmann's theorem to p–adic power series satisfying linear differential equations with polynomial coefficients and note that our approach leads to our estimating the number of integer zeros of polynomials on a given interval and thence to an investigation of the number of p–adic small values of a function on such an interval, that is, of the number of solutions of a congruence modulo pr.

scholarly journals The cyclicity of a class of quadratic reversible centers defining elliptic curves

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guilin Ji ◽  
Changjian Liu
Keyword(s):  
Elliptic Curves ◽  
Linear Combination ◽  
Asymptotic Expansions ◽  
Parameter Family ◽  
Reversible System ◽  
Abelian Integrals ◽  
Number Of Zeros ◽  
Period Annulus ◽  
New Criteria

<p style='text-indent:20px;'>In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [<xref ref-type="bibr" rid="b28">28</xref>] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in <inline-formula><tex-math id="M1">\begin{document}$ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $\end{document}</tex-math></inline-formula>. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to <inline-formula><tex-math id="M2">\begin{document}$ (-2,-\frac{8}{5}) $\end{document}</tex-math></inline-formula>.</p>

XXII.—Two Numerical Applications of Chebyshev Polynomials

1946 ◽  
Vol 62 (2) ◽  
pp. 204-210
Author(s):  
J. C. P. Miller
Keyword(s):  
Power Series ◽  
Chebyshev Polynomials ◽  
Preceding Paper ◽  
Mathieu Functions ◽  
Fourier Synthesis ◽  
Computational Processes

In this paper two computational processes are outlined in which the table of Chebyshev Polynomials Cn(x) = 2 cos (n cos−1 ½x) given in the preceding paper may be used with effect; these processes are (a) interpolation and (b) Fourier synthesis. A brief outline is also given of the idea behind the process of “Economization of Power Series” developed in Lanczos, 1938; this is related to (a). Finally the application of (b) to the calculation of Mathieu functions is considered.

Asymmetries of the Voigt line shape

2001 ◽  
Vol 34 (2) ◽  
pp. 136-143 ◽  
Author(s):  
H. Flores-Llamas
Keyword(s):  
Power Series ◽  
Linear Combination ◽  
Line Shape ◽  
Series Expansions ◽  
Second Line ◽  
X Ray ◽  
Intermediate Case ◽  
Line Shapes ◽  
First Case ◽  
Power Series Expansions

Two cases of asymmetric Voigt line shape are analysed, from which two power series expansions are obtained. In the first case, the line shape is built as a linear combination of the Voigt line shape (VLS) and its derivatives. This is valid for small asymmetries. The second line shape is built as the sum of the VLS and an odd-parity line shape; this second case is valid for intermediate asymmetries. An improved line shape, defined as a linear combination of the traditional VLS with the odd-parity line shape, is also presented. The improved procedure and intermediate case are particularly useful for dealing with X-ray line shapes at lower diffraction angles, as is demonstrated by a practical example.

X-ray microanalysis of thin specimens in the transmission electron microscope at voltages up to 1000 Kv.

1978 ◽  
Vol 36 (1) ◽  
pp. 540-541
Author(s):  
G. Cliff ◽  
M.J. Nasir ◽  
G.W. Lorimer ◽  
N. Ridley
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Weight Fraction ◽  
Present Series ◽  
Constant Independent ◽  
X Ray ◽  
Transmission Electron ◽  
Series Of Experiments ◽  
Image Position ◽  
X Ray Absorption

In a specimen which is transmission thin to 100 kV electrons - a sample in which X-ray absorption is so insignificant that it can be neglected and where fluorescence effects can generally be ignored (1,2) - a ratio of characteristic X-ray intensities, I1/I2 can be converted into a weight fraction ratio, C1/C2, using the equationwhere k12 is, at a given voltage, a constant independent of composition or thickness, k12 values can be determined experimentally from thin standards (3) or calculated (4,6). Both experimental and calculated k12 values have been obtained for K(11<Z>19),kα(Z>19) and some Lα radiation (3,6) at 100 kV. The object of the present series of experiments was to experimentally determine k12 values at voltages between 200 and 1000 kV and to compare these with calculated values.The experiments were carried out on an AEI-EM7 HVEM fitted with an energy dispersive X-ray detector.

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