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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 ..."
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Cited by 376 (4 self)
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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) = ..."
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Cited by 38 (1 self)
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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 ..."
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Cited by 15 (1 self)
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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 ..."
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Cited by 14 (3 self)
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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.
Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, submitted
"... Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, clai ..."
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Cited by 5 (5 self)
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Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents
, the orthonormal Jacobi polynomials P(α,β)
"... k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a str ..."
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k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a stronger 4 form by giving very explicit upper bounds. We also show that δ2 − x2 2 α (1 − x) ( P (α,α)
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 ..."
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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