# Linear recurrence with polynomial coefficients

• Pere Callahan
In summary, a linear recurrence with polynomial coefficients was discussed, and the usual methods for solving such equations did not work. A particular substitution was suggested to simplify the equation, but the question about a general theory or algorithm for solving these types of recursion remains.
Pere Callahan
Hi all,

I came across a linear recurrence with polynomial coefficients and realized that I don't have a clue as to how to solve it. The usual methods like generating functions or guessing seem not to work in that case.

Here is the equation:

$$i (i - 1) (i - 2) b = 1/3 (i + 1) i (1 - i) b[i - 3] + (1 + i) i (i - 2) b[i - 2] + (i + 1) (i - 1) (i - 2) b[i - 1]$$

Is there any general theory on recursions of that type or maybe even a general algorithm to compute the solution (in terms of some initial valeus b[0], b[1], b[2])?

Thanks a lot!

Pere

So, for this particular recursion, the substitution $b\mapsto(i+1)c$ reduces the equation to a very easy form. The question about the general theory remains.

Last edited:

## 1. What is a linear recurrence with polynomial coefficients?

A linear recurrence with polynomial coefficients is a mathematical sequence in which each term is a polynomial function of the previous terms. It can be represented by a recursive formula, where the next term is expressed in terms of a linear combination of the previous terms.

## 2. How is a linear recurrence with polynomial coefficients different from a regular linear recurrence?

In a regular linear recurrence, each term is a linear function of the previous terms, while in a linear recurrence with polynomial coefficients, each term is a polynomial function of the previous terms. This means that the terms in a linear recurrence with polynomial coefficients can have higher degrees and more complex relationships with the previous terms.

## 3. What is the degree of a linear recurrence with polynomial coefficients?

The degree of a linear recurrence with polynomial coefficients is the highest degree of the polynomials involved in the recurrence. It determines the complexity and behavior of the sequence, and can be used to predict the long-term behavior of the sequence.

## 4. How is a linear recurrence with polynomial coefficients used in scientific research?

Linear recurrences with polynomial coefficients are often used in mathematical modeling and data analysis in various scientific fields. They can be used to describe and predict complex systems, such as population growth, economic trends, and chemical reactions.

## 5. Are there any practical applications of linear recurrence with polynomial coefficients?

Yes, linear recurrences with polynomial coefficients have many practical applications in fields such as computer science, engineering, and physics. They are used to design algorithms, analyze data, and model physical systems. For example, they are used in signal processing to filter and analyze signals, and in cryptography to generate pseudorandom numbers.

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