On Series Involving Cubed Catalan Numbers

This paper derives new families of series involving the cubes and fourth powers of Catalan numbers using generalized binomial identities and Dougall's results, ultimately leading to a generalization of the Bauer series for $1/\pi$ and the discovery of Ramanujan-like series for $1/\pi^2$ and $1/\pi^3$.

The Gist

Imagine you are a chef in a grand mathematical kitchen. For years, mathematicians have been cooking up delicious recipes (formulas) using a special ingredient called Catalan numbers. These numbers are like a secret spice that appears everywhere in nature, from the way leaves branch on a tree to the number of ways you can arrange parentheses in a sentence.

In this paper, the author, Kunle Adegoke, is like a master chef who decides to take that spice, crush it into a fine powder, and then cube it (multiply it by itself three times) or even raise it to the fourth power. He then mixes these "super-spiced" ingredients into new, complex soups (infinite series) to see what flavor they produce.

Here is a simple breakdown of what he discovered, using everyday analogies:

1. The Ingredients: Catalan Numbers

Think of the Catalan numbers as a specific sequence of numbers (1, 1, 2, 5, 14, 42...) that describe patterns.

• The "Standard" Dish: Mathematicians already knew how to make a soup using just the Catalan numbers.
• The "Cubed" Dish: Adegoke asked, "What happens if we take these numbers, cube them, and add them up forever?"
• The "Fourth Power" Dish: He also wondered, "What if we raise them to the fourth power?"

2. The Secret Sauce: Dougall's Identities

To cook these new dishes, Adegoke didn't just guess. He used a set of ancient, powerful tools found in a cookbook written by a mathematician named John Dougall.

• Imagine Dougall's formulas as a universal translator. They can take a complicated, messy expression involving cubes and translate it instantly into a clean, beautiful answer involving famous constants like $\pi$ (the ratio of a circle's circumference to its diameter) and the Gamma function (a complex mathematical tool that extends the idea of factorials).

3. The Main Discoveries: New Recipes

The paper presents several "recipes" (formulas) that sum up infinite lists of these cubed numbers.

• The Surprise: When you add up an infinite number of these cubed terms, the result isn't a messy, random number. Instead, it simplifies into a very elegant equation involving $\pi$ and $\Gamma(1/4)$ (a specific value related to the shape of a circle and the number 4).
• The "Ramanujan-like" Series: The author also found a family of recipes that look very similar to those discovered by the legendary Indian mathematician Srinivasa Ramanujan. Ramanujan was famous for finding incredibly fast ways to calculate $\pi$ (the number 3.14159...).
  • Adegoke found new ways to calculate $1/\pi$, $1/\pi^2$, and even $1/\pi^3$ using these cubed Catalan numbers. It's like finding a new, faster highway to get to the destination of "Pi."

4. The "Odd Harmonic" Twist

In some of his recipes, Adegoke added a garnish called Odd Harmonic numbers.

• Think of this as adding a pinch of salt to the soup. It changes the flavor slightly but allows for even more complex and interesting recipes. He showed that even with this extra ingredient, the soup still boils down to a beautiful, clean formula involving $\pi$.

5. Why Does This Matter?

You might ask, "Who cares about summing infinite cubes of numbers?"

• The Puzzle Solver: For mathematicians, this is like solving a giant jigsaw puzzle. It connects different areas of math (combinatorics, calculus, and number theory) in unexpected ways.
• The Calculator: These formulas are not just pretty; they are powerful tools. Because they converge (add up) very quickly, they can be used by computers to calculate the digits of $\pi$ much faster than older methods.
• The Pattern Hunter: It reveals that the universe of numbers is deeply interconnected. The fact that cubing a simple counting number (Catalan) leads to the circle constant ($\pi$) suggests a hidden harmony in mathematics.

Summary

In short, Kunle Adegoke took a familiar mathematical ingredient (Catalan numbers), processed it in new ways (cubing and fourth powers), and used old, powerful tools (Dougall's identities) to discover new, elegant recipes that connect these numbers to the fundamental constants of the universe ($\pi$). He didn't just find one recipe; he found a whole new cookbook of infinite series that helps us understand the deep, hidden structure of mathematics.

Technical Summary

1. Problem Statement

The paper addresses the challenge of deriving closed-form evaluations for infinite series involving Catalan numbers ($C_k$), specifically focusing on:

• Series containing cubed Catalan numbers ($C_k^3$).
• Series containing fourth powers of Catalan numbers ($C_k^4$).
• Series incorporating odd harmonic numbers ($O_k$) and alternating signs.
• The derivation of Ramanujan-like series for $1/\pi$, $1/\pi^2$, and $1/\pi^3$ using these structures.

While specific identities involving cubed Catalan numbers (such as the one by Tauraso) were known, the author seeks to establish general families of such series that encompass these known results as special cases, thereby unifying and extending the existing literature.

2. Methodology

The author employs a rigorous analytical approach based on the following mathematical tools:

• Generalized Binomial Coefficients: The paper utilizes the extension of binomial coefficients to complex arguments via the Gamma function: $\binom{r}{s} = \frac{\Gamma(r+1)}{\Gamma(s+1)\Gamma(r-s+1)}$.
• Dougall's Identities: The core of the methodology relies on three specific hypergeometric series identities derived by John Dougall (Lemma 6), which involve sums of the form $\sum (-1)^k \binom{x}{k+1}^3$ and their variations with linear factors $(2k+2-x)$.
• Differentiation Techniques: To introduce harmonic numbers ($H_n$) and odd harmonic numbers ($O_n$) into the series, the author differentiates the base Dougall identities with respect to the parameter $x$.
• Gamma Function Properties: Extensive use is made of:
  • Euler's reflection formula.
  • Recurrence relations ($\Gamma(z+1) = z\Gamma(z)$).
  • Specific values of $\Gamma(1/4)$ and $\Gamma(3/4)$.
  • Trigonometric evaluations of $\sin(\pi x)$ and $\cos(\pi x)$ at half-integer arguments.
• Transformation of Terms: The Catalan number is expressed in terms of central binomial coefficients and Gamma functions to map the series into the form required by Dougall's theorems.

3. Key Contributions and Results

A. Families of Series with Cubed Catalan Numbers

The paper derives general families of series where the $k$-th term involves $\left(\frac{C_k}{2^{2k}} \prod \frac{1}{2k-2j+1}\right)^3$.

• Corollary 2: Establishes a family for even product lengths ($2m$ terms), recovering Tauraso's identity (Eq. 1) as the $m=0$ case and deriving a new identity (Eq. 2) for $m=1$.
• Theorem 4 & Corollary 5: Derive families involving linear factors $(4k - 4m + 3)$ and $(4k - 4m + 5)$, recovering identities (3)–(6) from the introduction.
• Theorem 1: Provides a general formula for the sum without the alternating sign, expressed in terms of $\Gamma(1/4)$ and trigonometric factors.

B. Series Involving Odd Harmonic Numbers

By differentiating the base identities, the author derives series containing odd harmonic numbers ($O_k = \sum_{j=1}^k \frac{1}{2j-1}$).

• Theorem 6: Derives a family for $\sum \left(\frac{C_k}{2^{2k}} \dots\right)^3 O_{k-m}$. This yields specific results like Eq. (7) and Eq. (8).
• Theorem 7: Derives a family involving the term $(4k - 2m + 3)O_{k-m} + 1/3$, recovering identities (9) and (10).

C. Series Involving Fourth Powers of Catalan Numbers

Using a variation of Dougall's identity for fourth powers (Eq. 32), the paper establishes:

• Theorem 8: A family for $\sum \left(\frac{C_k}{2^{2k}} \dots\right)^4 (4k - 2m + 3)$, recovering identities (11) and (12).
• Theorem 9: A family involving fourth powers and odd harmonic numbers, recovering identities (13) and (14).

D. Ramanujan-like Series for $1/\pi$, $1/\pi^2$, and $1/\pi^3$

A significant portion of the paper is dedicated to series involving central binomial coefficients $\binom{2k}{k}$ that converge to powers of $\pi$.

• Theorem 10: Derives a family of Ramanujan-like series for $1/\pi$:
  $$\sum_{k=0}^\infty \left( \frac{(-1)^k}{2^{2k}} \binom{2k}{k} \prod_{j=1}^m \frac{1}{2k-2j+1} \right)^3 (4k - 2m + 1) = \left( \frac{m!}{2^m} \binom{2m}{m} \right)^{-3} \frac{2}{\pi}$$
  This generalizes the Bauer series (1859) for $m=0$.
• Theorem 12: Derives a family for $1/\pi^2$ involving fourth powers:
  $$\sum_{k=0}^\infty \left( \frac{1}{2^{2k}} \binom{2k}{k} \prod_{j=1}^m \frac{1}{2k-2j+1} \right)^4 (4k - 2m + 1) = \frac{(-1)^m 2^{8m}}{(m!)^4 \pi^2} \binom{2m}{m}^{-5}$$
• Corollary 14: Derives a family for $1/\pi^3$ involving cubed central binomial coefficients, recovering the identity $\sum \left( \frac{1}{2^{2k}} \binom{2k}{k} \right)^3 = \frac{\Gamma^4(1/4)}{4\pi^3}$.

4. Significance

• Unification: The paper successfully unifies disparate known identities (such as those by Tauraso and Bauer) into single, parameterized families (Theorems 1–14). This reveals the underlying structural connections between series of different powers and harmonic weights.
• New Identities: It provides numerous new closed-form evaluations (e.g., Eqs. 2, 4, 6, 8, 10, 12, 14) that were not previously documented in the literature.
• Extension to Harmonic Numbers: The systematic derivation of series involving odd harmonic numbers ($O_k$) extends the scope of Ramanujan-like series, which traditionally focus on rational or algebraic coefficients.
• Generalization of Bauer's Series: The discovery of the family in Theorem 10 generalizes the classical Bauer series, offering a parameter $m$ that generates an infinite sequence of $1/\pi$ series.
• Methodological Clarity: The paper demonstrates a powerful, reproducible method using Dougall's identities and differentiation, which can likely be applied to other hypergeometric series problems.

In conclusion, this work significantly advances the field of combinatorial series by providing a comprehensive framework for evaluating series involving powers of Catalan numbers and central binomial coefficients, linking them directly to fundamental constants like $\pi$ and the Gamma function.