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Asymptotic expansions for second-order linear difference equations. (English) Zbl 0758.39005
Using the method of successive approximations, asymptotic formal series solutions of the linear second-order difference equation $x\sb{n+2}+a\sb nx\sb{n+1}+b\sb nx\sb n=0$, where $a\sb n$ and $b\sb n$ have asymptotic expansions of the form $a\sb n\sim\Sigma((c\sb i/n\sp i)$; $i=0,\dots,\infty)$ and $b\sb n\sim\Sigma((d\sb i/n\sp i;i=0,\dots,\infty)$, for large values of $n$, and $d\sb 0\ne 0$, are obtained.
Reviewer: H.Länger (Wien)

39A10Additive difference equations
Full Text: DOI
[1] Adams, C. R.: On the irregular cases of linear ordinary difference equations. Trans. amer. Math. soc. 30, 507-541 (1928) · Zbl 54.0483.01
[2] Bender, E. A.: Asymptotic methods in enumeration. SIAM rev. 16, 485-515 (1974) · Zbl 0294.05002
[3] Bender, C. M.; Orszag, S. A.: Advanced mathematical methods for scientists and engineers. (1978) · Zbl 0417.34001
[4] Birkhoff, G. D.: General theory of linear difference equations. Trans. amer. Math. soc. 12, 243-284 (1911) · Zbl 42.0359.02
[5] Birkhoff, G. D.: Formal theory of irregular linear difference equations. Acta math. 54, 205-246 (1930) · Zbl 56.0402.01
[6] Birkhoff, G. D.; Trjitzinsky, W. J.: Analytic theory of singular difference equations. Acta math. 60, 1-89 (1932) · Zbl 0006.16802
[7] Erdélyi, A.: Asymptotic expansions. (1956) · Zbl 0070.29002
[8] Hunter, C.: Asymptotic solutions of certain linear difference equations, with applications to some eigenvalue problems. J. math. Anal. appl. 24, 279-289 (1968) · Zbl 0197.06601
[9] Immink, G. K.: Asymptotics of analytic difference equations. Lecture notes in math. 1085 (1984) · Zbl 0548.39001
[10] Ince, E. L.: Ordinary differential equations. (1927) · Zbl 53.0399.07
[11] Magnus, W.; Oberhettinger, F.; Soni, R. P.: Formulas and theorems for special functions of mathematical physics. (1966) · Zbl 0143.08502
[12] Olver, F. W. J.: Asymptotics and special functions. (1974) · Zbl 0303.41035
[13] Tricomi, F. G.: Equazioni differenziali. (1953)
[14] Wimp, J.: Computation with recurrence relations. (1983) · Zbl 0543.65084
[15] Wimp, J.: Review of R. Wong’s book ’asymptotic approximations of integrals’. Math. comp. 56, 388-394 (1991)
[16] Wimp, J.; Zeilberger, D.: Resurrecting the asymptotics of linear recurrences. J. math. Anal. appl. 111, 162-176 (1985) · Zbl 0579.05007
[17] R. Wong and H. Li, Asymptotic expansions for second-order linear difference equations, II, Stud. Appl. Math., to appear. · Zbl 0780.39005