The Binomial Coefficient for Negative Arguments

TL;DR

This paper extends the definition of binomial coefficients to negative integers using gamma functions, ensuring the symmetry identity holds universally and exploring its consistency with known identities and the binomial theorem.

Contribution

It introduces a gamma function-based extension of binomial coefficients to negative arguments, maintaining symmetry and compatibility with classical identities.

Findings

The gamma function definition aligns with binomial identities for negative integers.

Symmetry of binomial coefficients is preserved for all integer arguments.

The extended definition agrees with the binomial theorem and other known identities.

Abstract

The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a symmetry formula for the gamma function, this definition is extended to negative integer arguments, making the symmetry identity for binomial coefficients valid for all integer arguments. The agreement of this definition with some other identities and with the binomial theorem is investigated.