Question

Let a and d > 0 be real numbers. Find a closed form of the generating function of the arithmetic progression

(a, a + d, a + 2d, a + 3d, ...).
The closed form is an expression for the generating function that does not use a power series.

Answer 1

The generating function for an arithmetic progression with first term 'a' and common difference 'd' is G(x) = a / (1 - x) + (dx) / (1 - x)^2, assuming |x| < 1.

Step-by-step explanation:

The closed form of the generating function for an arithmetic progression where a and d are positive real numbers can be derived using the concept of power series and the properties of geometric series. Consider the arithmetic progression (a, a + d, a + 2d, a + 3d, ...). The generating function G(x) for this series is the sum a + (a + d)x + (a + 2d)x2 + (a + 3d)x3 + .... This can be written as:

• 
  
• G(x) = a(1 + x + x2 + x3 + ...) + d(x + 2x2 + 3x3 + ...)

The first part of the function is a geometric series with common ratio x, whereas the second part can be obtained by differentiating the first part with respect to x and then multiplying by x. The closed form for the geometric series is 1 / (1 - x), assuming |x| < 1, and thus the generating function becomes:

• 
  
• G(x) = + d([a / (1 - x)])x

By simplifying, the final closed form of the generating function is:

G(x) = a / (1 - x) + (dx) / (1 - x)2.

This function encodes the entire arithmetic series and can be used to find the sum of the series up to any number of terms.