[Paper Review] Several generalizations and variations of Chu-Vandermonde identity

Romeo Meštrović|arXiv (Cornell University)|Jul 23, 2018

Advanced Combinatorial Mathematics18 references3 citations

TL;DR

This paper generalizes the classical Chu-Vandermonde identity using a probabilistic method based on the k-th moments (k=1,2,3) of complex-valued discrete random variables. It derives new combinatorial identities involving sums over multisets of complex numbers, establishes two new combinatorial congruences, and extends results to matrix forms, including quadratic and Hermitian matrix identities, all grounded in moment calculations of random subset sums with fixed size m.

ABSTRACT

In this paper we prove some combinatorial identities which can be considered as generalizations and variations of remarkable Chu-Vandermonde identity. These identities are proved by using an elementary combinatorial-probabilistic approach to the expressions for the $k$-th moments ($k=1,2,3$) of some particular cases of recently investigated discrete random variables. Using one of these Chu-Vandermonde-type identities, two combinatorial congruences are established.

Motivation & Objective

• To extend the Chu-Vandermonde identity to new combinatorial identities using probabilistic moment analysis.
• To investigate the statistical properties of subset sums of complex numbers via discrete random variables with fixed size m.
• To derive and prove new identities for the first, second, and third moments of such random variables.
• To establish combinatorial congruences using one of the derived identities.
• To generalize results to matrix-valued settings, including real and complex matrices with Hermitian transposes.

Proposed method

• Uses the probabilistic method to compute the k-th moments (k=1,2,3) of a complex-valued discrete random variable X(m,Φ_N), defined as the sum of m randomly selected elements from a multiset Φ_N of complex numbers.
• Applies the definition of expectation to derive closed-form expressions for E[X(m,Φ_N)], E[|X(m,Φ_N)|²], and E[(X(m,Φ_N))³] under uniform sampling without replacement.
• Derives combinatorial identities by equating moment expressions to symmetric sums over subsets, leading to identities involving multinomial coefficients and power sums of z_i.
• Extends results to matrix forms by replacing scalars with matrices, proving identities for (k₁A₁ + ... + kₛAₛ)³ and related quadratic forms.
• Applies linearity and combinatorial counting to generalize identities to matrix algebras and Hermitian transpose settings.
• Verifies matrix identities for small cases and conjectures their general validity using a counting method from prior work.

Experimental results

Research questions

• RQ1 How can the Chu-Vandermonde identity be generalized using the first, second, and third moments of a discrete complex-valued random variable?
• RQ2 What new combinatorial identities emerge from equating moment expressions to symmetric sums over subsets of fixed size m?
• RQ3 Can the derived identities be extended to matrix-valued settings, particularly involving Hermitian transposes and matrix powers?
• RQ4 What combinatorial congruences can be deduced from the derived identities involving scalar and matrix coefficients?
• RQ5 How do the identities behave when the variables z_i are complex versus real numbers, and can they be extended to polynomial rings?

Key findings

• The first moment of X(m,Φ_N) is E[X(m,Φ_N)] = (m/N)∑z_i, generalizing the expected sum of m randomly selected elements from a multiset.
• The second moment is E[|X(m,Φ_N)|²] = (m/(N(N-1)))((N−m)∑|z_i|² + (m−1)|∑z_i|²), valid for complex z_i.
• For real z_i, the third moment is E[(X(m,Φ_N))³] = (m/(N(N−1)))((N+2−3m)∑z_i³ + 3(m−1)(∑z_i²)(∑z_i)).
• A new identity (27) generalizes the third moment to weighted sums: ∑(k₁z₁+…+kₛzₛ)³ over multinomial coefficients, yielding a linear combination of ∑n_i z_i³ and (∑n_i z_i²)(∑n_i z_i).
• Matrix identities (28), (29), and (30) extend the results to M×N and N×N matrices, expressing weighted sums of matrix powers and products in terms of binomial coefficients and total sums.
• The identities hold over complex numbers by linearity and polynomial extension, and matrix versions are conjectured to hold generally based on verification for small cases.