Pell-Padovan tetranacci numbers and their Hadamard product with classical sequences

This paper presents a unified method for efficiently computing the generating functions of the Hadamard products between Pell-Padovan tetranacci numbers and eight specific classical second-order recurrence sequences.

The Gist

Imagine you are a master chef in a very busy kitchen. You have two main ingredients:

1. The "Pell-Padovan Tetranacci" Soup: This is a special, complex broth made by mixing four previous batches together in a specific recipe. It's a bit like a "super-soup" that depends on its own history to grow.
2. The "Classical" Spices: These are eight different, well-known spice blends (like Fibonacci, Pell, Jacobsthal, etc.). Each spice blend follows its own simple, two-step recipe.

The Problem: The "Hadamard" Taste Test

In the world of math, there's a special operation called the Hadamard Product. Think of it not as mixing the soup and the spices together in a pot, but rather as a taste test where you take a spoonful of the soup and a spoonful of the spice blend at the exact same moment (the same step in time) and see how they interact.

Mathematicians usually have to do this taste test one by one. If they want to see how the soup tastes with the "Fibonacci spice," they do a long calculation. Then they do it again for the "Pell spice," and again for the "Jacobsthal spice," and so on. It's tedious, like tasting eight different soups eight different times using eight different methods.

The Chef's Shortcut (The Paper's Solution)

Helmut Prodinger, the author of this paper, says, "Wait a minute! Why are we doing this eight times?"

He discovered a universal master recipe. Instead of treating each spice blend as a unique mystery, he realized they all fit into a single, flexible template:
$$\frac{a + bz}{1 + cd + dz^2}$$
Think of this template as a universal spice shaker. By simply turning the knobs (changing the numbers $a, b, c, d$), you can turn this one shaker into any of the eight specific spice blends mentioned in the paper.

The Magic Trick

Prodinger's method works like this:

1. The Setup: He puts the "Super-Soup" and the "Universal Spice Shaker" into a machine.
2. The Calculation: Instead of solving the math for every single spice blend separately, he uses a computer program (called gfun in the software Maple) to taste just the first 30 spoonfuls of the mixture.
3. The Guess: Based on those 30 spoonfuls, the computer looks at the pattern and says, "Ah! I know exactly what the rest of the recipe is!" It predicts the entire infinite formula for the result.
4. The Proof: Because the math behind these sequences is predictable (they follow strict rules), once the computer guesses the pattern correctly for the first 30 steps, it is mathematically guaranteed to be correct for all future steps.

The Result

The paper provides a single, giant formula (a long fraction with a numerator $N$ and a denominator $D$) that acts as the "Master Recipe."

• If you want the result for the k-Fibonacci spice, you just plug in the specific numbers for $a, b, c, d$ from Table 1.
• If you want the Chebyshev spice, you just change the numbers.

Why This Matters

Before this paper, if you wanted to know how the Pell-Padovan soup interacted with eight different spices, you had to write eight different long proofs. Now, you just have one proof that covers all eight cases instantly.

It's like realizing that instead of building eight different bridges to cross a river, you can build one modular bridge that can be reconfigured to fit any of the eight banks. It saves time, reduces errors, and shows that these seemingly different mathematical worlds are actually connected by a single, elegant structure.

In short: The paper found a "one-size-fits-all" formula that lets mathematicians instantly calculate how a complex sequence interacts with eight different classic sequences, turning a tedious eight-step process into a single, elegant calculation.

Technical Summary

Based on the provided text, here is a detailed technical summary of the paper "Pell-Padovan Tetranacci Numbers and Their Hadamard Product with Classical Sequences" by Helmut Prodinger.

1. Problem Statement

The paper addresses the computational challenge of determining the generating function for the Hadamard product of two specific types of sequences:

1. The Pell-Padovan Tetranacci sequence ($P_n$): Defined by the fourth-order linear recurrence $P_{n+4} = P_{n+2} + 2P_{n+1} + P_n$ with initial conditions $P_0=0, P_1=1, P_2=1, P_3=1$. Its generating function is given as $\frac{z(1+z)}{1-z^2-2z^3-z^4}$.
2. A class of second-order linear recurrence sequences ($X_n$): Represented generally by the rational generating function $\frac{a+bz}{1+cz+dz^2}$.

The specific goal is to compute the generating function for the sequence formed by the term-wise product $P_n X_n$ (denoted as $\sum P_n X_n z^n$) for eight specific classical sequences (including $k$-Fibonacci, $k$-Pell, $k$-Jacobsthal, $k$-Mersenne, and four types of Chebyshev polynomials) simultaneously, rather than treating them individually.

2. Methodology

Prodinger employs a computer-assisted experimental mathematics approach, specifically utilizing the Maple package gfun. The methodology proceeds as follows:

• Coefficient Generation: Instead of deriving the result algebraically for each case, the author computes a list of initial coefficients (e.g., the first 30 terms) for the Hadamard product of the general Pell-Padovan generating function and the general second-order generating function.
• Rational Function Guessing: Using the guessgf command within gfun, the program attempts to fit these coefficients to a rational generating function.
• Theoretical Justification: The author relies on a known theoretical bound: the Hadamard product of a sequence with a recurrence of order $k$ and a sequence with a recurrence of order $m$ results in a sequence satisfying a recurrence of order $k \times m$.
  • Here, $k=4$ (Pell-Padovan) and $m=2$ (Second-order sequences).
  • Therefore, the resulting generating function must be a rational function with a denominator of degree at most $8$.
  • Since the number of initial coefficients computed (30) significantly exceeds the degrees of freedom required to define a rational function with an 8th-degree denominator, the "guessed" result is mathematically guaranteed to be correct.
• Generalization: The method treats the second-order sequence parameters ($a, b, c, d$) as symbolic variables, allowing the derivation of a single general formula that covers all specific instances.

3. Key Contributions

• Unified General Formula: The paper derives a single, closed-form rational generating function for the Hadamard product of the Pell-Padovan sequence and any second-order sequence defined by parameters $a, b, c, d$.
  • The result is expressed as $N/D$, where $N$ and $D$ are polynomials in $z$ with coefficients dependent on $a, b, c, d$.
  • The denominator $D$ is an 8th-degree polynomial, confirming the theoretical order of the recurrence.
• Efficiency: This approach replaces the need for eight separate, complex algebraic derivations (as done in previous literature, specifically reference [2]) with a single computational procedure.
• Parameter Specialization: The paper provides a clear mapping (Table 1) showing how to substitute specific values for $a, b, c, d$ to recover the generating functions for the eight specific classical sequences.

4. Results

The paper presents the explicit rational generating function:
$$\sum_{n \ge 0} P_n X_n z^n = \frac{N}{D}$$
Where:

• Numerator ($N$): A polynomial of degree 5 involving terms like $b-ac$, $(-ad-cb+ac^2)z$, etc.
• Denominator ($D$): A polynomial of degree 8:
  $$D = 1 + (-c^2+2d)z^2 + 2c(c^2-3d)z^3 + (-c^4+4c^2d-d^2)z^4 - 2cd^2z^5 + d^2(c^2+2d)z^6 - 2cd^3z^7 + d^4z^8$$

The paper explicitly lists the parameter sets $(a, b, c, d)$ for the eight target sequences:

1. $k$-Fibonacci
2. $k$-Pell
3. $k$-Jacobsthal
4. $k$-Mersenne
5. Chebyshev (1st kind)
6. Chebyshev (2nd kind)
7. Chebyshev (3rd kind)
8. Chebyshev (4th kind)

5. Significance

• Methodological Shift: The paper demonstrates the power of "guessing" rational generating functions from initial coefficients when the theoretical order of the recurrence is known. This validates a technique popularized by Doron Zeilberger, shifting the burden from manual algebraic manipulation to computational verification.
• Scalability: By solving the problem for a general class of second-order sequences, the result is immediately applicable to any sequence within that class, not just the eight listed.
• Unification: It consolidates eight distinct results found in previous literature into one compact mathematical expression, simplifying future research involving these specific number sequences and their interactions.
• Accessibility: The use of standard tools like Maple's gfun makes the derivation accessible to researchers who may not wish to perform complex symbolic algebra by hand.