Antidiagonal Asymptotics of the Power Multiplication Table and a Catalan–Wallis Structure: An Integer–Half-Integer Duality Locked to Multiples of Pi

Key Points

• The aim is to explore the asymptotic behavior of sums in the power multiplication table and establish a duality tied to multiples of pi.
• Analyzed sums over an N by N power multiplication table for integer and half-integer exponents.
• Compared row-wise sums with sums along anti-diagonals.
• Utilized the Beta function and Gamma duplication formula to derive limiting ratios.
• Identified a simplification of limiting ratios involving the Beta function.
• Established a duality relating these ratios to integer multiples of pi.
• Showed the connection to central binomial coefficients and Catalan numbers.

Abstract