LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN
LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN
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DOI：
10.1142/s0219530513500371
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发表时间：
2013-03
影响因子：
2.2
通讯作者：
L. Cao;Yutian Li
中科院分区：
文献类型：
--
作者：
L. Cao;Yutian Li
A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation $$ P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0,$$ where An and Bn have asymptotic expansions of the form $$ A_n \sim n^{-\theta} \sum_{s=0}^\infty \frac{\alpha_s}{n^s}, \quad B_n \sim \sum_{s=0}^\infty \frac{\beta_s}{n^s},$$ with θ ≠ 0 and α0 ≠ 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 xαexp(-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.
A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation $$ P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0,$$ where An and Bn have asymptotic expansions of the form $$ A_n \sim n^{-\theta} \sum_{s=0}^\infty \frac{\alpha_s}{n^s}, \quad B_n \sim \sum_{s=0}^\infty \frac{\beta_s}{n^s},$$ with θ ≠ 0 and α0 ≠ 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 xαexp(-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.