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Turán Inequalities for Ultraspherical and Continuous q-Ultraspherical Polynomials
SIAM Journal on Mathematical Analysis, 1983Paul Turan discovered that Legendre polynomials satisfy the inequality \[ P_n^2 - P_{n + 1} P_{n - 1} > 0\quad {\text{for }} - 1 0,\quad 0 < x < 1,\quad \frac{1}{2} < a \leq \beta \leq \alpha + 1.\]
Bustoz, Joaquin, Ismail, Mourad E. H.
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Some Characterizations of the Ultraspherical Polynomials
Canadian Mathematical Bulletin, 1968Let be the nth ultraspherical polynomial. Also let . The following generating relation is well known (3, p.98).It can also be written as1.1This suggests the consideration of the class of polynomial sets {Qn(x), n = 0, 1, 2,…}, Qn(x) is of exact degree n and1 ...
Al-Salam, N. A., Al-Salam, W. A.
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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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Low-Pass Filters Using Ultraspherical Polynomials
IEEE Transactions on Circuit Theory, 1966The problem of approximating the ideal normalized amplitude response of a low-pass filter by the use of a set of ultraspherical polynomials is considered. The amplitude response obtained is more general than the analogous response of the Chebyshev filter because of an additional parameter available with the ultraspherical polynomials.
D. Johnson, J. Johnson
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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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