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Approximation formulas related to Somos' quadratic recurrence constant.

Bo Zhang¹, Chao-Ping Chen¹

¹School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, China.

Journal of Inequalities and Applications

|October 27, 2018

Summary

This study introduces two asymptotic expansion classes for Somos' quadratic recurrence constant. Recursive relations for coefficients are derived using Bell polynomials, alongside new continued fraction approximations.

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Area of Science:

Mathematics
Number Theory
Computational Analysis

Background:

Somos' quadratic recurrence constant is a significant mathematical entity.
Understanding its asymptotic behavior is crucial for theoretical and computational applications.

Purpose of the Study:

To develop and present two novel classes of asymptotic expansions for Somos' quadratic recurrence constant.
To establish recursive relations for the coefficients of these expansions.
To introduce new continued fraction approximations for the constant.

Main Methods:

Utilizing Bell polynomials and other advanced mathematical techniques.
Deriving recursive formulas for asymptotic expansion coefficients.
Developing continued fraction representations.

Keywords:

Asymptotic formula Bell polynomials Ordered Bell numbers Somos’ quadratic recurrence constant

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Main Results:

Two distinct classes of asymptotic expansions for Somos' quadratic recurrence constant have been successfully derived.
Recursive relations for determining the coefficients of each expansion class have been established.
Novel continued fraction approximations related to the constant have been presented.

Conclusions:

The presented methods offer a systematic approach to analyzing Somos' quadratic recurrence constant.
These findings contribute to a deeper understanding of the constant's properties and approximations.
The derived expansions and approximations have potential applications in numerical analysis and theoretical mathematics.