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Resurrecting the asymptotics of linear recurrences. (English) Zbl 0579.05007
The theory of linear recurrences is presented and a number of examples are given. ”In Section 1 we will introduce several examples of combinatorial families that are counted by solving a recurrence equation. Then we will go on, in Section 2, to give an account of the Birkhoff- Trjitzinski method and finally, in Section 3, we will give a few examples of combinatorial interest.”
Reviewer: P.Reichensperger

MSC:
05A15 Exact enumeration problems, generating functions
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
68R99 Discrete mathematics in relation to computer science
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[1] Adams, C.R, On the irregular case of the linear ordinary difference equation, Trans. amer. math. soc., 30, 507-541, (1928) · JFM 54.0483.01
[2] Batchelder, P.M, An introduction to linear difference equations, (1927), Harvard Univ. Press Cambridge, Mass., · JFM 53.0430.10
[3] Bender, E.A, Asymptotic methods in enumeration, SIAM rev., 16, 485-515, (1974) · Zbl 0294.05002
[4] Birkhoff, G.D, Formal theory of irregular difference equations, Acta math., 54, 205-246, (1930) · JFM 56.0402.01
[5] Birkhoff, G.D; Trjitzinsky, W.J, Analytic theory of singular difference equations, Acta math., 60, 1-89, (1932) · Zbl 0006.16802
[6] Cole, R.J; Pescatore, C, Evaluation of ∝_0∞\(t\^{}\{n\} exp(−t\^{}\{2\} −tx) dt\), J. comput. phys., 32, 280-287, (1979) · Zbl 0415.65011
[7] Erdélyi, A; Magnus, W; Oberhettinger, F; Tricomi, F.G, Higher transcendental functions, (1953), McGraw-Hill New York, 3 vols. · Zbl 0052.29502
[8] Fields, J; Luke, Y; Wimp, J, Recursion formulae for generalized hypergeometric functions, J. approx. theory, 1, 137-166, (1968) · Zbl 0177.32302
[9] Fields, J; Wimp, J, On the factorization of a class of difference operators, Bull. amer. math. soc., 79, 1068-1071, (1968) · Zbl 0167.34701
[10] Knuth, D.E, Fundamental algorithms, () · Zbl 0191.17903
[11] Knuth, D.E, Sorting and searching, () · Zbl 0777.68012
[12] Moser, L; Wyman, M, On the solutions of xd = 1, Canad. J. math, 7, 159-168, (1955) · Zbl 0064.02601
[13] Poincaré, H, Asymptotic series, Amer. J. math., 7, 203-258, (1885)
[14] Slater, L.J, Confluent hypergeometric functions, (1960), Cambridge Univ. Press London/New York · Zbl 0086.27502
[15] Wimp, J, Recursion formulae for hypergeometric functions, Math. comp., 21, 363-373, (1967) · Zbl 0186.10401
[16] Wimp, J, On recursive computation, () · Zbl 0294.65010
[17] Wimp, J, On the computation of Tricomi’s ψ-function, Computing, 13, 195-203, (1979) · Zbl 0294.65010
[18] Wimp, J, Computation with recurrence relations, (1983), Pitman London
[19] Zeilberger, D, Sister Celine’s technique and its generalizations, J. math. anal. appl., 85, 114-145, (1982) · Zbl 0485.05003
[20] de Bruijn, N.G, Asymptotic methods in analysis, (1961), North-Holland Amsterdam · Zbl 0109.03502
[21] Van der Poorten, A, A proof that Euler missed, apéry’s proof of the irrationality of ζ (3), Math. intelligencer, 1, 195-203, (1979) · Zbl 0409.10028
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