[Paper Review] Base-$b$ analogues of classic combinatorial objects

Tanay Wakhare, Christophe Vignat|arXiv (Cornell University)|Jul 9, 2016

Advanced Combinatorial Mathematics5 references5 citations

TL;DR

This paper introduces base-$b$ analogues of classic combinatorial objects—Stirling numbers of the second kind, Fibonacci numbers, and the exponential function—using a novel base-$b$ binomial coefficient defined via digit-wise products. It establishes a general summation formula over carry-free indices, enabling new identities and analogues that generalize classical results, including a base-$b$ exponential function with asymptotic and convolution properties.

ABSTRACT

We study the properties of the base-$b$ binomial coefficient defined by Jiu and the second author, introduced in the context of a digital binomial theorem. After introducing a general summation formula, we derive base-$b$ analogues of the Stirling numbers of the second kind, the Fibonacci numbers and the classical exponential function.

Motivation & Objective

• To develop base-$b$ analogues of classical combinatorial objects such as Stirling numbers, Fibonacci numbers, and the exponential function.
• To generalize the digital binomial theorem to arbitrary bases $b$ using a digit-wise binomial coefficient $\binom{n}{k}_b$.
• To establish a general summation formula over carry-free indices $k \leq_b n$ that unifies and extends prior results on sums involving $\binom{n}{k}_b$.
• To define and analyze a base-$b$ exponential function $e_b(x,w)$ with properties including a convolution identity and asymptotic approximation.

Proposed method

• Define the base-$b$ binomial coefficient as $\binom{n}{k}_b = \prod_{i=0}^{N-1} \binom{n_i}{k_i}$, where $n_i, k_i$ are digits of $n, k$ in base $b$.
• Introduce a general summation formula: $\prod_{i=0}^{N-1} S(n_i) = \sum_{0 \leq k \leq_b n} \prod_{i=0}^{N-1} f(n_i, k_i)$, valid for carry-free $k$.
• Apply the summation formula to derive base-$b$ analogues of Stirling numbers of the second kind via generating functions.
• Construct a base-$b$ Fibonacci analogue by applying the formula to a recurrence relation involving digit-wise operations.
• Define the base-$b$ exponential function as $e_b(x,w) = \prod_{i=0}^\infty \sum_{k=0}^{b-1} \frac{(x w^{b^i})^k}{k!}$, using the upper incomplete gamma function for exact form.
• Prove a convolution identity $e_b(x,w) \star e_b(y,w) = e_b(x+y,w)$, where $\star$ enforces digit-wise dominance $k \leq_b n$.

Experimental results

Research questions

• RQ1 How can classical combinatorial objects like Stirling numbers and Fibonacci numbers be generalized to base-$b$ digit-wise structures?
• RQ2 What is the structure of a base-$b$ binomial theorem, and how does it extend the classical and digital binomial theorems?
• RQ3 Can a base-$b$ analogue of the exponential function be defined, and what are its analytic and algebraic properties?
• RQ4 How does the summation formula over carry-free indices unify and generalize previous results in digit-wise combinatorics?
• RQ5 What is the asymptotic behavior of the base-$b$ exponential function for small $w$?

Key findings

• The base-$b$ binomial coefficient $\binom{n}{k}_b = \prod_{i=0}^{N-1} \binom{n_i}{k_i}$ satisfies a generalized binomial theorem: $(X+Y)^{s_b(n)} = \sum_{k=0}^n \binom{n}{k}_b X^{s_b(k)} Y^{s_b(n-k)}$.
• A base-$b$ analogue of the Stirling numbers of the second kind is derived using the general summation formula, extending classical generating function identities.
• A base-$b$ Fibonacci sequence is defined via the summation formula, preserving recurrence-like properties under digit-wise operations.
• The base-$b$ exponential function is given by $e_b(x,w) = \exp\left(x \sum_{i=0}^\infty w^{b^i}\right) \prod_{i=0}^\infty \frac{\Gamma(b, x w^{b^i})}{(b-1)!}$, with asymptotic approximation $e_b(x,w) \simeq e^{xw + xw^b}$ for small $w$.
• The convolution identity $e_b(x,w) \star e_b(y,w) = e_b(x+y,w)$ holds, where $\star$ enforces digit-wise dominance $k \leq_b n$, generalizing $e^x e^y = e^{x+y}$.
• The identity $\binom{n}{k}_b \neq \frac{(n!)_b}{(k!)_b (n-k)!_b}$ is confirmed, showing that the base-$b$ factorial does not satisfy the classical binomial identity due to carry-free constraints.