Asymptotic enumeration via singularity analysis

Asymptotic formulae for two-dimensional arrays (fr,s)r,s≥0 where the associated generating function F (z, w) := ∑ r,s≥0 fr,sz w is meromorphic are provided. Our approach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients fr,s along a compact set of directions specified by smooth points of the singular variety of the denominator of F (z, w). The scheme we develop can lead to a high level of complexity. However, it provides the leading asymptotic order of fr,s if some unusual and pathological behavior is ruled out. It relies on the asymptotic analysis of a certain type of stationary phase integral of the form ∫ e−s·P A(d, θ)dθ, which describes up to an exponential factor the asymptotic behavior of the coefficients fr,s along the direction d = r s in the (r, s)-lattice. The cases of interest are when either the phase term P (d, θ) or the amplitude term A(d, θ) exhibits a change of degree as d approaches a degenerate direction. These are handled by a generalized version of the stationary phase and the coalescing saddle point method which we propose as part of this dissertation. The occurrence of two special functions related to the Airy function is established when two simple saddles of the phase term coalesce. A scheme to study the asymptotic behavior of big powers of generating functions is proposed as an additional application of these generalized methods.

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