Solve Generating Function to Find Linear Recurrence Relation

In summary, a generating function is a mathematical tool used to represent a sequence of numbers or objects as a power series. It can be used to find patterns and relationships within a sequence and can be used to find a linear recurrence relation by manipulating the power series. Generating functions can be used for any type of sequence, but may not always be the most efficient method and may not work for sequences with complex or non-linear patterns. They are specifically designed for finding linear recurrence relations and may not be effective for finding non-linear relationships.
  • #1
saubbie
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Homework Statement



Find a linear homogeneous recurrence relation satisfied by an=2^n+n!

Homework Equations





The Attempt at a Solution



The teacher gave us a hint using generating functions. The generating function for f(x) is
f(x)=1+2x+4x^2+8x^3+...+1+x+2x^2+6x^3+24x^4+... The first part is a geometric series which equals 1/(1-2x), so f(x)=1/(1-2x)+1+x+2x^2+6x^3+24x^4+... Then he said to multiply both sides by x, take the derivative, and relate f'(x) to f(x). I found f'(x) after multiplying by x to be f(x)+xf'(x)=1/((1-2x)^2)+1+2x+6x^2+24x^3+... Now I don't know how to proceed. I need to relate the two, but am lost as to how to do that. Any suggestions? Thanks.
 
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  • #2


To find a linear homogeneous recurrence relation satisfied by the given sequence, we can use the following steps:

1. Construct the generating function for the sequence an=2^n+n!. In this case, the generating function is f(x)=1+2x+4x^2+8x^3+...+1+x+2x^2+6x^3+24x^4+...

2. Use the fact that the generating function for a geometric series is 1/(1-x) to rewrite the first part of the generating function as 1/(1-2x).

3. Multiply both sides of the equation by x and take the derivative. This will give us f(x)+xf'(x)=1/((1-2x)^2)+1+2x+6x^2+24x^3+...

4. Now, we can use the fact that f'(x) is equal to the generating function for the sequence of derivatives of the original sequence an=2^n+n!. In this case, the generating function for the sequence of derivatives is f'(x)=2+4x+12x^2+24x^3+...

5. Substitute this into the equation from step 3 to get f(x)+x(2+4x+12x^2+24x^3+...)=1/((1-2x)^2)+1+2x+6x^2+24x^3+...

6. Simplify the equation to get f(x)+2x+4x^2+12x^3+...=1/((1-2x)^2)+1+2x+6x^2+24x^3+...

7. Finally, we can equate the coefficients of each power of x on both sides to get the linear homogeneous recurrence relation. In this case, we get the following recurrence relation: an=2an-1+4(n-1)an-2+12(n-1)(n-2)an-3+...

I hope this helps! Let me know if you have any further questions.
 

1. What is a generating function?

A generating function is a mathematical tool used to represent a sequence of numbers or objects as a power series. It is typically denoted by G(x) and can be used to find patterns and relationships within a sequence.

2. How is a generating function used to find a linear recurrence relation?

A generating function can be used to find a linear recurrence relation by representing the given sequence as a power series and then manipulating the series to find a closed form expression. This expression can then be used to determine the recurrence relation for the sequence.

3. Can a generating function be used for any type of sequence?

Yes, a generating function can be used for any type of sequence, including numerical sequences, sequences of polynomials, and even sequences of objects.

4. Are there any limitations to using a generating function to find a linear recurrence relation?

While generating functions can be a useful tool for finding linear recurrence relations, they may not always provide the most efficient or straightforward method. Additionally, they may not work for sequences with complex or non-linear patterns.

5. Can a generating function be used to find a non-linear recurrence relation?

No, a generating function is specifically designed to find linear recurrence relations and may not be effective for finding non-linear relationships. Other mathematical tools, such as difference equations, may be more suitable for finding non-linear recurrence relations.

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