Lectures related to Counting with Recurrence Relations: Summary

Advanced Counting: Linear Homogeneous Recurrence Relations and Generating Functions

Explores solving linear homogeneous recurrence relations and generating functions for sequence formulas.

Covers the definition of generating functions and their application in solving recurrence relations.

Covers advanced counting techniques, including linear recurrence relations and generating functions, with examples from the Fibonacci sequence and differences between dice and poker cards.

Advanced Counting: Generating Functions and Recurrence Relations

Explores generating functions, recurrence relations, and advanced counting techniques.

Linear Recurrence Relations

Explores linear recurrence relations, including examples like the Fibonacci numbers and the proof of related theorems.

Generating Functions for Correlation Functions

Explores generating functions for correlation functions in the context of the forced harmonic oscillator.

Sequences: Arithmetic and Geometric Progressions

Covers arithmetic and geometric progressions, strings, and recurrence relations.

Sequences: Relations, Summation, Strings

Covers arithmetic and geometric progressions, strings, and recurrence relations in sequences.

Fibonacci numbers and the golden ratio

Explores the Fibonacci sequence, identities, golden ratio, and explicit formulas.

Extended Binomial Theorem

Explores the Extended Binomial Theorem and counting problems using generating functions.

Linear Recurrence Relations: Solving Techniques and Examples

Explains linear homogeneous recurrence relations and provides solving techniques and examples.

Generating Functions: Properties and Applications

Explores generating functions, Laplace transform, and their role in probability distributions.

Ramanujan Graphs: Generating Functions and Expander Graphs

Explores Ramanujan graphs, generating functions, non-backtracking walks, and expander graphs in relation to NP-hard problems.

Extreme Value Models: Technical Derivation

Explores the technical derivation and properties of Multivariate Extreme Value models.

Generating Functions: Moments and Cumulants

Explores generating functions for moments and cumulants, showcasing their role in distribution analysis.

Linear Recurrence Relations: Solving Techniques

Explores linear homogeneous recurrence relations, solving techniques, characteristic equations, and examples like Fibonacci numbers.

Pentagonal Number Theorem and Jacobi Identity

Covers the pentagonal number theorem, Jacobi identity, and modularity of eta and theta functions.

Meromorphic Functions & Differentials

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Functions Generators

Covers moment and cumulant generating functions for random variables, enabling calculation of moments and cumulants.

Functions: Limits, Continuity, Differentiability

Explores the origin of functions, continuity, differentiability, and the physical representation of chemical bonds.