Exponential Generating Functions

In summary, the conversation is asking for help understanding the step of a solution involving a summation in a binomial expansion. The second step is a binomial expansion of (e^x-1)^k and the solution involves multiplication of exponential generating functions. The person asking for help realizes they were overcomplicating the solution and thanks the other person for their assistance.
  • #1
kensaurus
9
0
I got a question here, and I am stuck at understanding the step of the solution. Any help will be appreciated.

http://img841.imageshack.us/img841/6589/40155869.jpg

I would like to know how to get from the second to the third step, where the summation comes in.

It looks like multiplication of 2 exponential generating function, but the second step is not one. Thanks for any help.
 
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  • #2
the second step is the binomial expansion of [itex](e^x-1)^k[/itex].

cheers
 
  • #3
oh my, I am slapping my head... i went to derive it through complex, exponential functions and other means...

thanks alot!
 
  • #4
cheers.
 

1. What is an exponential generating function?

An exponential generating function is a mathematical tool used to represent a sequence of numbers by associating each number with a specific power of x. It is commonly used in combinatorics and other areas of mathematics to study the growth and behavior of sequences.

2. How is an exponential generating function different from an ordinary generating function?

An ordinary generating function represents a sequence of numbers by associating each number with a specific coefficient in a power series. In contrast, an exponential generating function associates each number with a specific power of x, which allows for easier manipulation and analysis of sequences that involve factorials and other exponential terms.

3. What are some applications of exponential generating functions?

Exponential generating functions are commonly used in combinatorics to solve problems involving permutations, combinations, and other counting problems. They are also used in probability to analyze the behavior of random variables and in differential equations to solve problems involving exponential growth or decay.

4. How do you find the exponential generating function for a given sequence?

To find the exponential generating function for a sequence, you can use the definition: G(x) = ∑(k=0 to ∞) ak (xk)/k!, where ak is the kth term in the sequence. You can also use known properties of exponential generating functions, such as linearity and differentiation, to manipulate and find the generating function for a given sequence.

5. Can an exponential generating function be used to find the terms in a sequence?

Yes, an exponential generating function can be used to find the terms in a sequence by using the formula: ak = G(k)(0)/k!, where G(k)(x) is the kth derivative of the exponential generating function, evaluated at x=0. This method is commonly used in combinatorics to find the terms in a sequence involving combinations or permutations.

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