TY - JOUR
T1 - Uniform asymptotics of some q-orthogonal polynomials
AU - Wang, X. S.
AU - Wong, R.
PY - 2010/4/1
Y1 - 2010/4/1
N2 - Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q-1-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials. © 2009 Elsevier Inc. All rights reserved.
AB - Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q-1-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials. © 2009 Elsevier Inc. All rights reserved.
KW - q-Airy function
KW - q-Laguerre polynomial
KW - q-Theta function
KW - q-1-Hermite polynomial
KW - Stieltjes-Wigert polynomial
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U2 - 10.1016/j.jmaa.2009.10.038
DO - 10.1016/j.jmaa.2009.10.038
M3 - RGC 21 - Publication in refereed journal
VL - 364
SP - 79
EP - 87
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -