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Turán's inequality for ultraspherical polynomials revisited
Mathematical Inequalities & Applications, 20236 ...
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Zeros of pseudo-ultraspherical polynomials
Analysis and Applications, 2014The pseudo-ultraspherical polynomial of degree n can be defined by [Formula: see text] where [Formula: see text] is the ultraspherical polynomial. It is known that when λ < -n, the finite set [Formula: see text] is orthogonal on (-∞, ∞) with respect to the weight function (1 + x2)λ-½ and when λ < 1 - n, the polynomial [Formula: see text] has ...
Driver, Kathy, Muldoon, Martin E.
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Inequalities for Ultraspherical and Laguerre Polynomials. II
SIAM Journal on Mathematical Analysis, 1980The main result proved here is the inequality $(n + 1)F_n^\alpha (x) nF_n^\beta (x) - nF_{n - 1}^\alpha (x)F_{n - 1}^\beta (x) > 0$ for $ - 1 < x < 1$ and $\frac{1}{2} \leqq \alpha \leqq \beta \leqq \alpha + 1$, where $F_n^\lambda (x) = {{P_n^\lambda (x)} / {P_n^\lambda (1)}}$ and $P_n^\lambda (x)$ is the ultraspherical polynomial.
Bustoz, J., Savage, N.
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A generalization of ultraspherical polynomials
1983Some old polynomials of L. J. Rogers are orthogonal. Their weight function is given. The connection coefficient problem, which Rogers solved by guessing the formula and proving it by induction, is derived in a natural way and some other formulas are obtained. These polynomials generalize zonal spherical harmonics on spheres and include as special cases
R. Askey, Mourad E.-H. Ismail
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A multiplier theorem for ultraspherical polynomials
Studia Mathematica, 2019In this paper a result regarding the characterization of radial \(L^{p}_{\mathrm{rad}}(R^{d})\) Fourier multipliers acting on radial functions for ...
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A Relation Between Ultraspherical and Jacobi Polynomial Sets
Canadian Journal of Mathematics, 1953The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .
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On Bernstein's inequality for ultraspherical polynomials
Archiv der Mathematik, 1997The author offers a proof for \[ (\sin t)^{s}| P_{n}^{(s)}(\cos t)| < \frac{ 2^{1-s}}{\Gamma(s)} \frac{ \Gamma(n+(3s/2))}{\Gamma(n+1+(s/2))}, \] where ...
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Application of Ultraspherical Polynomials to Nonlinear Forced Oscillations
Journal of Applied Mechanics, 1967Approximate solutions to the conservative free-oscillation problem were obtained recently [1–4] through the use of ultraspherical polynomials. The present paper extends the technique to forced oscillations governed by x¨+g(x)˙+f(x)=F0sinpt+F1 Very accurate results are obtained either by setting the ultraspherical polynomial index λ = 0 or, better yet ...
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Characterizations of ultraspherical polynomials and their $q$-analogues
Proceedings of the American Mathematical Society, 2015The author studies some properties that characterize the ultraspherical polynomials and two of their \(q\)-analogues, namely the symmetric big \(q\)-Jacobi polynomials and the continuous \(q\)-ultraspherical (Roger) polynomials. In fact he improved some previous results by \textit{R. Lasser} and \textit{J. Obermaier} [Proc. Am. Math. Soc. 136, No.
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