Orthogonal Polynomials and Special Functions

By Francisco Marcellán, Walter Van Assche. 2003

583 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/77822dc34499b1ea000c206a53461864edf8d419

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In Computational Theory and Mathematics

Citations in the corpus

Jacobi interpolation approximations and their applications to singular differential equations (2001, background): “One of them is as follows (see Askey [3]),”
Generalized Jacobi polynomials/functions and their applications (2009, background): “The classical Jacobi polynomials with indexes α,β > −1 satisfy the following recurrence relations (see Szegö [35], Askey [2] and Rainville [31]): Jα,βn (−x) = (−1)nJ β,αn (x); (A.3) J α,β n−1(x) = Jα,β−1n (x) − Jα−1,βn (x), α,β > 0, n 1; (A.4) Jα,βn (x) = 1 n + α + β [ (n + β)J α,β−1n (x) + (n + α)J”
Some connection and linearization problems for polynomials in and beyond the Askey scheme (2001, unknown): “The literature on these topics is extremely vast (see, e.g., [1] [2] [4] [21] [23] [24]), and a wide variety of methods have been devised for computing the linearization and connection coeecients, either in closed form or by means of recurrence relations (usually in k).”
Graphical functions and single-valued multiple polylogarithms (2013, unknown): “Most conveniently Gegenbauer polynomials are defined by the following generating series [1, 22, 46]:”
A positive radial product formula for the Dunkl kernel (2002, methodology): “We now employ a well-known degenerate version of the addition theorem for Gegenbauer polynomials (see [2], (4.”
A necessary and sufficient condition for strictly positive definite functions on spheres (2003, methodology): “we have the following formula (see [ A , p. 30]):”
Product of random projections, Jacobi ensembles and universality problems arising from free probability (2004, result): “√ u) √ vJb+1( √ v)+Jb( √ v) √ uJb+1( √ u) 2(u−v) For future use, note that the coefﬁcient in ub/2+kvb/2+l of −Jb( √ u) √ vJb+1( √ v)+Jb( √ v) √ uJb+1( √ u) is (38) (k−l) (b+k)!k!(b+l)!l! According to [Ask75], p. 7 Formula (2.2) Pa,b n (−1+ 2u n2 )= (b+1)n n! Xn k=0 (−n)k(n+a+b+1)k k!(b +1)k u n2 k w”
Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball (2007, unknown): “A function â ∈ C∞[0,∞) is called admissible of type (a) if supp â ⊂ [0, 2] and â(t) = 1 on [0, 1], and of type (b) if supp â ⊂ [1/2, 2].”
Universally optimal distribution of points on spheres (2006, background): “That follows from Theorem 3.1, Lemma 8.4, and the fact that the set of nonnegative linear combinations of q0, q1, q2, . . . is closed under taking products (see [Gas] for the most general case or Theorem 5.2 in [ As , p. 44] for a simpler proof that applies to these orthogonal polynomials).”