TY - JOUR

T1 - Linear difference equations with a transition point at the origin

AU - Cao, Li Hua

AU - Li, Yu Tian

N1 - Funding Information:
The work of Y. Li is supported in part by the HKBU Strategic Development Fund, a start-up grant from Hong Kong Baptist University, and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU 201513). The authors thank two anonymous referees for many valuable suggestions and comments.

PY - 2014/1

Y1 - 2014/1

N2 - A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation Pn+1(x) ? (Anx + Bn)Pn(x) + Pn?1(x) = 0, where An and Bn have asymptotic expansions of the form An ∼ n-o ∞ θ s=o αs/ns, Bn ∼ ∼ n-o ∞ θ s=o βs/ns with θ ≠ 0 and A0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight x Aexp(-qmxm), x > 0, where m is a positive integer, A > -1 and qm > 0.

AB - A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation Pn+1(x) ? (Anx + Bn)Pn(x) + Pn?1(x) = 0, where An and Bn have asymptotic expansions of the form An ∼ n-o ∞ θ s=o αs/ns, Bn ∼ ∼ n-o ∞ θ s=o βs/ns with θ ≠ 0 and A0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight x Aexp(-qmxm), x > 0, where m is a positive integer, A > -1 and qm > 0.

KW - Bessel functions

KW - Difference equation

KW - Orthogonal polynomials

KW - Three-term recurrence relations

KW - Transition point

KW - Uniform asymptotic expansions

UR - http://www.scopus.com/inward/record.url?scp=84890443686&partnerID=8YFLogxK

U2 - 10.1142/S0219530513500371

DO - 10.1142/S0219530513500371

M3 - Journal article

AN - SCOPUS:84890443686

SN - 0219-5305

VL - 12

SP - 75

EP - 106

JO - Analysis and Applications

JF - Analysis and Applications

IS - 1

ER -