Asymptotics of linear normal lambda terms: WKB structure, Airy turning point and a positive recurrence
Abstract
Let a n be the number of closed linear λ-terms in β-normal form with n variable occurrences. We prove
is the large-order constant of the Riccati-Airy sequence K m attached to the turning point of the WKB expansion. We show that the K m are exactly rescaled coefficients of the formal logarithmic derivative of the Airy asymptotic series, and deduce κ ∞ = 1/(2π) from the explicit Airy coefficients. Hence
The proof combines four further ingredients: (i) the WKB coefficients are polynomials with nonnegative coefficients in v = (1 -4w) -1/2 and satisfy an explicit positive recurrence; (ii) the linearisation of the WKB equation has a combinatorial meaning (λ-chain terms) and is a translation in the action coordinate s, whose singularity sits at the instanton action A 0 = ln 2 -1 2 ; (iii) an exact identity shows that, after conjugation by the transported WKB prefactor µ = r/(1+r) 2 , the remaining non-linear terms are either negative or of relative size O(m -2 ); (iv) under the conjugated linear dynamics the depth profile is dominated by its top coefficient. A finite bound on an explicit ratio sequence is obtained by exact computer arithmetic (the program is given in an appendix); it also yields a n ≤ 1.14736 f ′′ n (1) for all n, with f ′′ n (1) ∼ 5 144 (n -1)!λ n .
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BibTeX
@misc{bodini2026asymptotics,
title = {{Asymptotics of linear normal lambda terms: WKB structure, Airy turning point and a positive recurrence}},
author = {Olivier Bodini},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05841200v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotics of linear normal lambda terms: WKB structure, Airy turning point and a positive recurrence
AU - Olivier Bodini
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05841200v1
AB - Let a n be the number of closed linear λ-terms in β-normal form with n variable occurrences. We prove is the large-order constant of the Riccati-Airy sequence K m attached to the turning point of the WKB expansion. We show that the K m are exactly rescaled coefficients of the formal logarithmic derivative of the Airy asymptotic series, and deduce κ ∞ = 1/(2π) from the explicit Airy coefficients. Hence The proof combines four further ingredients: (i) the WKB coefficients are polynomials with nonnegative coefficients in v = (1 -4w) -1/2 and satisfy an explicit positive recurrence; (ii) the linearisation of the WKB equation has a combinatorial meaning (λ-chain terms) and is a translation in the action coordinate s, whose singularity sits at the instanton action A 0 = ln 2 -1 2 ; (iii) an exact identity shows that, after conjugation by the transported WKB prefactor µ = r/(1+r) 2 , the remaining non-linear terms are either negative or of relative size O(m -2 ); (iv) under the conjugated linear dynamics the depth profile is dominated by its top coefficient. A finite bound on an explicit ratio sequence is obtained by exact computer arithmetic (the program is given in an appendix); it also yields a n ≤ 1.14736 f ′′ n (1) for all n, with f ′′ n (1) ∼ 5 144 (n -1)!λ n .
ER -