Generating functions are powerful tools for solving counting problems and working with sequences. They represent sequences as coefficients in formal power series, allowing for easier manipulation and analysis of complex mathematical patterns.

In this section, we'll explore ordinary generating functions, their properties, and applications. We'll learn how to use them to solve recurrence relations, extract coefficients, and simplify complex sequences into manageable closed forms.

Generating Functions and Power Series

Understanding Generating Functions and Power Series

• Generating function represents a sequence as coefficients of a formal power series
• Formal power series consists of an infinite sum of terms with increasing powers of a variable
• Taylor series expansion approximates functions using polynomial terms
• Closed form expresses a generating function as a finite algebraic expression

Applications and Properties of Generating Functions

• Generating functions solve counting problems by encoding sequence information
• Formal power series manipulations include addition, multiplication, and composition
• Taylor series expansion centers around a specific point, often x = 0
• Closed form simplifies complex generating functions for easier analysis

Examples and Techniques

• Generating function for arithmetic sequence: $\frac{1}{(1-x)^2}$ represents 1, 2, 3, 4, ...
• Formal power series for exponential function: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
• Taylor series for sine function: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$
• Closed form for geometric series: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$

Coefficient Extraction Techniques

Fundamentals of Coefficient Extraction

• Coefficient extraction retrieves specific terms from generating functions
• Partial fraction decomposition breaks complex rational functions into simpler forms
• Convolution combines two sequences to produce a third sequence

Methods for Coefficient Extraction

• Use of square brackets notation [x^n] to denote the coefficient of x^n
• Partial fraction decomposition splits function into sum of simpler fractions
• Convolution applies multiplication of generating functions

Practical Applications and Examples

• Extracting coefficients from $\frac{1}{1-x-x^2}$ yields Fibonacci sequence
• Partial fraction decomposition of $\frac{x}{(1-x)(1-2x)}$ into $\frac{1}{1-x} - \frac{1}{1-2x}$
• Convolution of sequences (1,1,1,...) and (1,2,3,...) produces (1,3,6,10,...)

Recurrence Relations

Fundamentals of Recurrence Relations

• Recurrence relation defines sequence terms using previous terms
• Linear recurrence relations involve linear combinations of previous terms
• Homogeneous recurrence relations have no constant term
• Inhomogeneous recurrence relations include a constant or function of n

Solving Recurrence Relations with Generating Functions

• Convert recurrence relation to an equation involving generating function
• Solve the resulting equation for the generating function
• Extract coefficients to obtain closed form of sequence terms

Examples and Applications

• Fibonacci recurrence: F_n = F_(n-1) + F_(n-2) with F_0 = 0, F_1 = 1
• Generating function for Fibonacci: $F(x) = \frac{x}{1-x-x^2}$
• Closed form solution: $F_n = \frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^n - (\frac{1-\sqrt{5}}{2})^n]$
• Catalan number recurrence: C_n = C_0C_(n-1) + C_1C_(n-2) + ... + C_(n-1)C_0