Sign In

Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Comprehensive Analysis

Core Concepts
The paper provides a detailed analysis of binomial sums and Mellin asymptotics, offering explicit error bounds for a comprehensive understanding.
The content delves into the application of the Mellin transform in analyzing binomial sums, focusing on 132-avoiding permutations. It introduces a new package in SageMath for asymptotic expansions with explicit error bounds. The analysis involves approximating binomial coefficients, handling tails of the sum, and evaluating specific integrals using the Mellin transform. The paper concludes with an asymptotic formula for F(n) and detailed error bound calculations. Structure: Introduction to Binomial Sums and Mellin Asymptotics Importance of binomial coefficients in combinatorics. Tools like Laplace method, Stirling approximation, and Mellin transform. Reducing the Problem Combinatorial interpretations using lattice paths. Decomposition of Dyck paths. B-terms and Asymptotics with Explicit Error Bounds Introduction to B-terms for precise computations. Extension of capabilities in SageMath for computations involving dependent variables. Asymptotic Analysis Approximating binomial coefficients. Handling tails of the sum. Evaluating integrals using the Mellin transform.
1/n+1n + knnnk yields large Schröder numbers. Cn = 1/n+12nn is the n-th Catalan number.
"Making use of a newly developed package in the computer algebra system SageMath." "While these methods are well known and in some sense mechanical, it is still not straightforward to implement them."

Key Insights Distilled From

by Benjamin Hac... at 03-15-2024
Binomial sums and Mellin asymptotics with explicit error bounds

Deeper Inquiries

How can explicit error bounds enhance computational mathematics beyond this specific case


What alternative approaches could be explored to simplify complex summations like those discussed


How does the integration of software tools impact traditional mathematical analysis methodologies