Results 1  10
of
13
The Askeyscheme of hypergeometric orthogonal polynomials and its qanalogue
, 1998
"... We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erent ..."
Abstract

Cited by 580 (6 self)
 Add to MetaCart
We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erential or di#erence equation, the forward and backward shift operator, the Rodriguestype formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between di#erent classes of orthogonal polynomials listed in the Askeyscheme. In chapter 3 we list the qanalogues of the polynomials in the Askeyscheme. We give their definition, orthogonality relation, three term recurrence relation, second order di#erence equation, forward and backward shift operator, Rodriguestype formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally, in chapter 5 we...
On some inequalities for the gamma and psi functions
 MATH. COMP
, 1997
"... We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) = ..."
Abstract

Cited by 78 (3 self)
 Add to MetaCart
(Show Context)
We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) =
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Generalized Jacobi Weights, Christoffel Functions, and Zeros of Orthogonal Polynomials
 JOURNAL OF APPROXIMATION THEORY 69. 111132
, 1992
"... We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sharp lower and upper bounds for the corresponding Christoffel functions, and for the distance of the consecutive zeros of the corresponding orthogonal polynomials. The novelty of our results is that our c ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
(Show Context)
We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sharp lower and upper bounds for the corresponding Christoffel functions, and for the distance of the consecutive zeros of the corresponding orthogonal polynomials. The novelty of our results is that our constants depend only on the degree of the weight function but not on the weight itself.
Monotonicity and convexity for the gamma function
 J. Inequal. Pure Appl. Math
"... ABSTRACT. Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)] 1/(b−a) − x is strictly convex and decreasing on (−a, ∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function. Key words and phrases: Gamma fun ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)] 1/(b−a) − x is strictly convex and decreasing on (−a, ∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function. Key words and phrases: Gamma function; monotonicity; convexity.
HOW SHARP IS BERNSTEIN’S INEQUALITY FOR JACOBI POLYNOMIALS?∗
"... Abstract. Bernstein’s inequality for Jacobi polynomials P (α,β)n, established in 1987 by P. Baratella for the region R1/2 = {α  ≤ 1/2, β  ≤ 1/2}, and subsequently supplied with an improved constant by Y. Chow, L. Gatteschi, and R. Wong, is analyzed here analytically and, above all, computation ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Bernstein’s inequality for Jacobi polynomials P (α,β)n, established in 1987 by P. Baratella for the region R1/2 = {α  ≤ 1/2, β  ≤ 1/2}, and subsequently supplied with an improved constant by Y. Chow, L. Gatteschi, and R. Wong, is analyzed here analytically and, above all, computationally with regard to validity and sharpness, not only in the original regionR1/2, but also in larger regionsRs = {−1/2 ≤ α ≤ s,−1/2 ≤ β ≤ s}, s> 1/2. Computation suggests that the inequality holds with new, somewhat larger, constants in any region Rs. Best constants are provided for s = 1:.5: 4 and s = 5: 1: 10. Our work also sheds new light on the socalled Erdélyi–Magnus–Nevai conjecture for orthonormal Jacobi polynomials, adding further support for its validity and suggesting.66198126... as the best constant implied in the conjecture. Key words. Bernstein’s inequality, Jacobi polynomials, sharpness, Erdélyi–Magnus–Nevai conjecture
On Some Inequalities for the Gamma Function
"... We present some elementary proofs of wellknown inequalities for the gamma function and for the ratio of two gamma functions. The paper is purely expository and it is based on the talk that the first author gave during the memorial conference in Patras, 2012. AMS Subject Classifications: 33B15, 26D0 ..."
Abstract
 Add to MetaCart
(Show Context)
We present some elementary proofs of wellknown inequalities for the gamma function and for the ratio of two gamma functions. The paper is purely expository and it is based on the talk that the first author gave during the memorial conference in Patras, 2012. AMS Subject Classifications: 33B15, 26D07.
Reconstruction of sparse Legendre and Gegenbauer
"... We present a new deterministic algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony–like methods. Furthermore we show that the suggested method can be extended t ..."
Abstract
 Add to MetaCart
(Show Context)
We present a new deterministic algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony–like methods. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order. Key words and phrases: Legendre polynomials, sparse Legendre expansions, Gegenbauer polynomials, ultraspherical polynomials, sparse Gegenbauer expansions, sparse recovering, sparse Legendre interpolation, sparse