Khinchin Families, Set Constructions, Partitions and Exponentials.

Area of Science:

• Mathematics
• Combinatorics
• Analysis

Background:

• Analytic combinatorics frequently utilizes generating functions.
• The Hayman class is important for understanding the asymptotic behavior of coefficients in these functions.
• Previous criteria for verifying membership in the Hayman class were complex.

Purpose of the Study:

• To develop a simplified criterion for determining if functions belong to the Hayman class.
• To apply this criterion to obtain new asymptotic formulas for generating functions.
• To facilitate the analysis of set constructions in analytic combinatorics.

Main Methods:

• A new, simple criterion is presented for functions of the form $$ where g is a power series with nonnegative coefficients.
• The Hayman and Báez-Duarte formulas are employed.
• Asymptotic analysis of generating function coefficients is performed.

Main Results:

• A straightforward criterion for verifying Hayman class membership is established.
• Simplified derivation of asymptotic formulas for coefficients of generating functions is achieved.
• The new criterion significantly streamlines previous methods.

Conclusions:

• The developed criterion offers a more accessible method for classifying functions within the Hayman class.
• This simplification aids in the study of set constructions via generating functions in analytic combinatorics.
• The findings contribute to more efficient asymptotic analysis in the field.