[Paper Review] The Binomial Coefficient for Negative Arguments

M. J. Kronenburg|arXiv (Cornell University)|May 18, 2011

Advanced Combinatorial Mathematics4 references20 citations

TL;DR

This paper extends the binomial coefficient to negative integer arguments using the gamma function and a symmetry identity, ensuring consistency with fundamental identities like symmetry, trinomial revision, and the binomial theorem. The key contribution is a unified definition that preserves continuity and symmetry across all integer arguments, resolving inconsistencies in prior treatments.

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.

Motivation & Objective

• To define the binomial coefficient for negative integer arguments using the gamma function and analytic continuation.
• To ensure the symmetry identity $\binom{n}{k} = \binom{n}{n-k}$ holds for all integer $n$ and $k$.
• To verify consistency with core identities such as trinomial revision, absorption, and addition, despite exceptions at singular points.
• To demonstrate compatibility with the binomial theorem for negative integer powers through convergent series expansions.
• To establish continuity in specific complex directions, validating the definition's analytic robustness.

Proposed method

• Uses the gamma function representation $\binom{x}{y} = \frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$ for complex $x$ and $y$.
• Applies the reflection and symmetry formula for the gamma function, $\Gamma(s-a+1)/\Gamma(s-b+1) = (-1)^{b-a}\Gamma(b-s)/\Gamma(a-s)$, to handle poles at nonpositive integers.
• Derives two equivalent expressions for $\binom{n}{k}$ when $n < 0$, based on whether $k \geq 0$ or $k \leq n$, using gamma function substitutions.
• Establishes continuity by taking limits of the binomial coefficient as arguments approach integers from complex directions, using asymptotic expansions of the gamma function.
• Validates identities by substituting gamma function forms and checking consistency, especially for the absorption and addition identities.
• Compares series expansions of $(x+y)^n$ for negative $n$, showing convergence in different regions depending on whether $|x| < |y|$ or $|x| > |y|$.

Experimental results

Research questions

• RQ1 How can the binomial coefficient be consistently defined for negative integer values of $n$ while preserving symmetry?
• RQ2 To what extent do standard binomial identities—such as symmetry, trinomial revision, absorption, and addition—remain valid under this extended definition?
• RQ3 Does the extended definition maintain compatibility with the binomial theorem for negative integer powers?
• RQ4 Are the binomial coefficients continuous in complex neighborhoods of integer arguments, and in which directions?
• RQ5 What are the exceptional cases in the absorption and addition identities, and how do they arise from gamma function singularities?

Key findings

• The binomial coefficient for negative integer $n$ and integer $k$ is defined as $\binom{n}{k} = (-1)^k \binom{-n+k-1}{k}$ for $k \geq 0$, and $\binom{n}{k} = (-1)^{n-k} \binom{-k-1}{n-k}$ for $k \leq n$, with zero otherwise.
• The symmetry identity $\binom{n}{k} = \binom{n}{n-k}$ holds for all integer $n$ and $k$ under this definition.
• The absorption identity $\binom{x}{y} = \frac{x}{y}\binom{x-1}{y-1}$ holds for all complex $x,y$ except when $y = 0$, due to a singularity in the gamma function.
• The addition identity $\binom{x}{y} = \binom{x-1}{y} + \binom{x-1}{y-1}$ holds for all complex $x,y$ except when $x = y = 0$, due to non-associative behavior of limits involving $0/0$.
• The binomial theorem for negative integer $n$ yields two different convergent series: one for $|x| < |y|$ and one for $|x| > |y|$, corresponding to the two cases in the extended definition.
• The definition ensures continuity in complex directions approaching integer arguments, as shown by the limits $\binom{n+\delta}{k} \to (-1)^k \binom{-n+k-1}{k}$ as $\delta \to 0$.