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Search: All articles in the CJM digital archive with keyword hypergeometric functions

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1. CJM 2011 (vol 64 pp. 961)

Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
Densities of Short Uniform Random Walks
We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

Keywords:random walks, hypergeometric functions, Mahler measure
Categories:60G50, 33C20, 34M25, 44A10

2. CJM 2010 (vol 62 pp. 261)

Chiang, Yik-Man; Ismail, Mourad E. H.
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
No abstract.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Problem
Categories:34M10, 33C15, 33C47

3. CJM 2006 (vol 58 pp. 726)

Chiang, Yik-Man; Ismail, Mourad E. H.
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above ``special function approach" can be described by a classical Heine problem for differential equations that admit polynomial solutions.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble
Categories:34M10, 33C15, 33C47

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