Calculating Binomial Sums Using Complex Roots of Unity

Introduction

This article delves into the sophisticated technique of calculating sums of binomial coefficients that are multiples of a specific index using the properties of complex roots of unity. We will explore how these roots enable us to isolate and evaluate such sums elegantly, providing a deeper understanding of their mathematical beauty and utility.

Overview of Binomial Sums and Complex Roots of Unity

Binomial sums are fundamental in combinatorics and have wide applications in mathematics and computer science. Complex roots of unity are a specialized subset of complex numbers that have applications in various fields. This article aims to combine these two areas to derive general formulas for specific types of binomial sums.

Deriving the Formula for Binomial Sums

Consider the binomial sum:

General Formula

For any natural number n, we can calculate the following binomial sum:

$$sum_{i0}^{n} binom{3n}{3i} frac{2 - 1^n 8^n}{3}$$

Starting with the Binomial Theorem, we expand x3n as:

$$x^{3n} equiv sum_{i0}^{3n} binom{3n}{i} x^i$$

By choosing specific values of x, we can extract different components of the sum.

When x 1

This provides:

$$2^{3n} equiv sum_{i0}^{3n} binom{3n}{i}$$

To isolate the sum of binomials with indices that are multiples of 3, we use the complex cubic roots of unity, denoted by w, which satisfy the properties:

$$w^3 1 quad text{and} quad 1 w w^2 0$$

When x w

Substituting x w gives:

$$w^{3n} equiv sum_{i0}^{3n} binom{3n}{i} w^i$$

Similarly, for x w2 gives:

$$w^{2 cdot 3n} equiv sum_{i0}^{3n} binom{3n}{i} w^{2i}$$

Combining these, we get:

$$2^{3n} - w^{3n} - w^{2 cdot 3n} sum_{i0}^{3n} binom{3n}{3i}$$

Because w and w2 are roots of unity, their powers simplify according to their periodicity and multiplicative properties:

When 3 ? i, 1 wi w2i 0

When 3 | i, 1 wi w2i 3 because w3 1

Therefore, the sum of binomials with indices multiples of 3 can be simplified to:

$$sum_{i0}^{n} binom{3n}{3i} 8^n - 1^n - 1^n frac{2 - 1^n 8^n}{3}$$

Thus, we conclude that:

$$sum_{i0}^{n} binom{3n}{3i} frac{2 - 1^n 8^n}{3}$$

Application to a Specific Binomial Sum

Given the sum:

$$sum_{i0}^{673} binom{2019}{3i}$$

We apply the derived formula to evaluate it:

First, we recognize that n 673, so:

$$sum_{i0}^{673} binom{2019}{3i} frac{8^{673} - 2}{3}$$

This shows the elegance and utility of the developed formula in simplifying complex binomial sums.

Conclusion

The use of complex roots of unity provides a powerful tool for solving binomial sums that are multiples of specific indices. This technique not only simplifies existing problems but also opens up new avenues for solving more complex mathematical expressions.

By leveraging the properties of these roots, we can derive elegant solutions for binomial sums, contributing to the broader field of combinatorial mathematics.