@MISC{Pitman99alattice, author = {Jim Pitman and Gamma Z}, title = {A lattice path model for the Bessel polynomials}, year = {1999} }

Bookmark

OpenURL

Abstract

The (n \Gamma 1)th Bessel polynomial is represented by an exponential generating function derived from the number of returns to 0 of a sequence with 2n increments of \Sigma1 which starts and ends at 0. AMS 1991 subject classification. Primary: 05A15. Secondary: 33C10, 33C45. It is well known [21, x3.71 (12)],[6, (7.2(40)] that the McDonald function or Bessel function of imaginary argument K (x) := 1 2 ` x 2 ' \Gamma Z 1 0 t \Gamma1 e \Gammat\Gamma(x=2) 2 =t dt (1) admits the evaluation K n+1=2 (x) = r ß 2x e \Gammax ` n (x)x \Gamman (n = 0; 1; 2; : : :) (2) Research supported in part by N.S.F. Grant 97-03961 where ` n (x) := n X m=0 fi n;n\Gammam x m with fi n;k := (n + k)! 2 k (n \Gamma k)!k! : (3) The Bessel polynomials ` n (x) and y n (x) := n X k=0 fi n;k x k = x n ` n (x \Gamma1 ) (4) have been extensively studied and applied: see the book of Grosswald [9] for a review. Dulucq and Favreau [4, 5] gave a combinatorial model for the Bes...