TY - JOUR
T1 - Asymptotic expansion of the modified lommel polynomials hn,ν(x) and their zeroszeros
AU - LEE, Kei Fung
AU - Wong, R.
PY - 2014/7
Y1 - 2014/7
N2 - The modified Lommel polynomials satisfy the second-order linear difference equation h(n +1),(v)(x) - 2(n + v) x h(n,v)(x)+ h(n-1,v)(x) = 0, n = 0, with initial values h(-1,v)(x) = 0 and h(0,v)(x) = 1, where x is a real variable and v is a fixed positive parameter. An asymptotic expansion, as n - infinity, is derived for these polynomials by using a turning-point theory for three-term recurrence relations developed by Wang and Wong Numer. Math. 91 (2002) and 94 (2003)]. The result holds uniformly in the infinite interval 0 = x infinity, containing the critical value x = 1/N, where N = n + v. Behavior of the zeros of these polynomials is also studied.
AB - The modified Lommel polynomials satisfy the second-order linear difference equation h(n +1),(v)(x) - 2(n + v) x h(n,v)(x)+ h(n-1,v)(x) = 0, n = 0, with initial values h(-1,v)(x) = 0 and h(0,v)(x) = 1, where x is a real variable and v is a fixed positive parameter. An asymptotic expansion, as n - infinity, is derived for these polynomials by using a turning-point theory for three-term recurrence relations developed by Wang and Wong Numer. Math. 91 (2002) and 94 (2003)]. The result holds uniformly in the infinite interval 0 = x infinity, containing the critical value x = 1/N, where N = n + v. Behavior of the zeros of these polynomials is also studied.
KW - Modified Lommel polynomials
KW - second-order linear difference equations
KW - uniform asymptotic expansions
KW - Airy function
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U2 - 10.1090/S0002-9939-2014-12134-4
DO - 10.1090/S0002-9939-2014-12134-4
M3 - RGC 21 - Publication in refereed journal
VL - 142
SP - 3953
EP - 3964
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 11
ER -