Discrete Maths | Generating Functions-Introduction and Prerequisites

Discrete Maths | Generating Functions-Introduction and Prerequisites

Prerequisite – Combinatorics Basics, Generalized PnC Set 1, Set 2 

Definition: Generating functions are used to represent sequences efficiently by coding the terms of a sequence as coefficients of powers of a variable (say) [Tex]\big x  [/Tex]in a formal power series. 

Now with the formal definition done, we can take a minute to discuss why should we learn this concept. 

This concept can be applied to solve many problems in mathematics. There is a huge chunk of mathematics dealing with just generating functions. 

• It can be used to solve various kinds of Counting problems easily.
• It can be used to solve recurrence relations by translating the relation in terms of sequence to a problem about functions. 
• It can be used to prove combinatorial identities.

In simple words generating functions can be used to translate problems about sequences to problems about functions which are comparatively easy to solve using maneuvers. 

Some basic prerequisites… 

Before we start let’s go through some basic Combinatorics formulas. 

[Tex]%\documentclass{article} %\usepackage{amsmath} \newcommand*{\Perm}[2]{{}^{#1}\!P_{#2}}% \newcommand*{\Comb}[2]{{}^{#1}C_{#2}}% \begin{document} \Huge $\Perm{n}{k}=\frac{n!}{(n-k)!}$ $\binom nk=\Comb{n}{k}=\frac{n!}{k!(n-k)!}$ \end{document}  [/Tex]


A generating function is a “formal” power series in the sense that we usually regard x as a placeholder rather than a number. Only in rare cases will we actually evaluate a generating function by letting x take a real number value, so we generally ignore the issue of convergence. 

[Tex]$g_0, g_1, g_2, g_3….  [/Tex]of real numbers is the infinite series: 
[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \huge \func G(x) = g_0+{g_1}x+{g_2}x^{2}….. = \sum_{k=0}^\infty g_k x^k. \end{document}  [/Tex]

[Tex][g_{0},g_{1},g_{2},g_{3}…]\leftrightarrow g_{0} + g_{1}x+g_{2}x^{2}+g_{3}x^{3}….  [/Tex]

Note: we’ll indicate the correspondence between a sequence and its generating function with a double-sided arrow. 

Some Important sequences and their results…

[Tex]\textup{Let }G_{n}=1+x+x^{2}+x^{3}+…..+x^{n-1}+x^{n}……….\hspace{1cm} eq^{n} (1)\\ \textup{Multiplying it with x } \\ xG_{n}=x+x^{2}+x^{3}+….\hspace{1cm} eq^{n} (2) \\ \\ \textup{subtracting equation (2) with (1) we will get.. } \\ G_{n}-xG_{n} =1 \\ \\G_{n}=\frac{1}{1-x}  [/Tex]

If you want to know more about the derivation of the given sequences you may refer here 


[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \Large \(\frac{1} {(1-x)}\) = 1 + x + x^2 + x^3…. \longleftrightarrow [1, 1, 1, 1…\normalsize\infty] \end{document}  [/Tex]

[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \Large \(\frac{1} {(1+x)}\)=1-x+x^2-x^3…. \longleftrightarrow [1, -1, 1, -1….\normalsize\infty] \end{document}  [/Tex]


[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \Large \(\frac{1} {(1-ax)}\) = 1 + ax + {a^2}x^2 + {a^3}x^3…. \longleftrightarrow [1, a, a^2, a^3….\normalsize\infty] \end{document}  [/Tex]

[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \Large %\(\frac{1} {(1-ax)}\) = 1 – ax + {a^2}x^2 – {a^3}x^3…. \longleftrightarrow [1, a, a^2, a^…\infty] \(\frac{1} {(1-x^{2})}\) = 1 + x^2 + x^4…. \longleftrightarrow [1, 0, 1, 0…\normalsize\infty] \end{document}  [/Tex]

Note that all the above series is infinite. There is one peculiar result of finite series which will be very handy so it’s recommended to remember these results to solve problems quickly. 

An important result: 



[Tex]%\documentclass{article} %\usepackage{amsmath} \begin{document} \LARGE %\(\frac{1} {(1-ax)}\) = 1 – ax + {a^2}x^2 – {a^3}x^3…. \longleftrightarrow [1, a, a^2, a^…..] 1+x+x^2+x^3+….+x^n = \(\frac{1 + x^{n+1}} {(1-x)}\) \end{document}  [/Tex]

 

Solved Examples

Example 1: Generating Function for a Simple Sequence

Problem: Find the generating function for the sequence {1,1,1,1,…}.

Solution: G(x)=1+x+x2+x3+⋯Using the formula for an infinite geometric series: G(x)=11−x​1

Example 2: Generating Function for an Arithmetic Sequence

Problem: Find the generating function for the sequence {0,1,2,3,…}.

Solution: G(x)=0+1x+2×2+3×3+⋯ Using the formula: G(x)=x(1−x)2x​

Example 3: Generating Function for a Geometric Sequence

Problem: Find the generating function for the sequence {1,2,4,8,…}.

Solution: G(x)=1+2x+4×2+8×3+⋯ Using the formula for an infinite geometric series with ratio 2x2x: G(x)=11−2×1​

Example 4: Sum of Binomial Coefficients

Problem: Find the generating function for the sequence {(n0),(n1),(n2),…}.

Solution: G(x)=∑k=0n(nk)xk=(1+x)n

Example 5: Generating Function for a Shifted Sequence

Problem: Find the generating function for the sequence {0,1,1,1,…}.

Solution: G(x)=x+x2+x3+⋯⋯ Using the formula: G(x)=x1−x​

Practice Problems on Generating Functions

• Problem 1: Find the generating function for the sequence {1,2,3,4,…}
• Problem 2: Determine the generating function for the sequence {1,−1,1,−1,…}.
• Problem 3: Find the generating function for the sequence {1,0,1,0,1,0,…}.
• Problem 4: Derive the generating function for the sequence {1,3,5,7,…}.
• Problem 5: Determine the generating function for the sequence {1,4,9,16,…}.
• Problem 6: Find the generating function for the sequence {0,1,0,1,0,1,…}.
• Problem 7: Derive the generating function for the sequence {1,1/2,1/4,1/8,…}.
• Problem 8: Find the generating function for the sequence {1,1,0,0,1,1,0,0,…}.
• Problem 9: Determine the generating function for the sequence {1,2,1,2,1,2,….}.
• Problem 10: Derive the generating function for the sequence {1,3,6,10,15,…}.

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