Hypergeometric functions | Factorial and binomial topics

Egorychev method

The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers. The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the origin). The sought-for identity can now be found using manipulations of integrals. Some of these manipulations are not clear from the generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero, yielding a new expression for the original sum. The residue at infinity is particularly important in these considerations.Some of the integrals employed by the Egorychev method are: * First binomial coefficient integral where * Second binomial coefficient integral where * Exponentiation integral where * Iverson bracket where * Stirling number of the first kind where * Stirling number of the second kind where (Wikipedia).

Video thumbnail

Computational Physics Lecture 11, LU Decomposition and Matrix Inversion

In this lecture, we discuss the LU decomposition method for systems of linear algebraic equations. We then describe how to calculate the inverse of a matrix using this method. We also introduce the concept of the matrix condition number. Finally, we describe a simple iterative method for i

From playlist Nazarbayev: PHYS 270 - Computational Physics with Ernazar Ab

Video thumbnail

Russian Multiplication Algorithm - Intro to Algorithms

This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.

From playlist Introduction to Algorithms

Video thumbnail

Complex Analysis: Integral Representations of Binomial Coefficients

Today, we derive some contour integral representations for the binomial coefficients. Residues at higher order poles: https://www.youtube.com/watch?v=9hdZDHkKoAM Egorychev method: https://en.wikipedia.org/wiki/Egorychev_method

From playlist Complex Analysis

Video thumbnail

Russian Multiplication Algorithm Solution - Intro to Algorithms

This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.

From playlist Introduction to Algorithms

Video thumbnail

Least squares method for simple linear regression

In this video I show you how to derive the equations for the coefficients of the simple linear regression line. The least squares method for the simple linear regression line, requires the calculation of the intercept and the slope, commonly written as beta-sub-zero and beta-sub-one. Deriv

From playlist Machine learning

Video thumbnail

Russian Peasants Algorithm - Intro to Algorithms

This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.

From playlist Introduction to Algorithms

Video thumbnail

The Stepanov Method - Avi Wigderson

Avi Wigderson Institute for Advanced Study May 25, 2010 The Stepanov method is an elementary method for proving bounds on the number of roots of polynomials. At its core is the following idea. To upper bound the number of roots of a polynomial f(x) in a field, one sets up an auxiliary pol

From playlist Mathematics

Video thumbnail

[Calculus] Newton's Method || Lecture 36

Visit my website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any que

From playlist Calculus 1

Video thumbnail

Dirichlet Eta Function - Integral Representation

Today, we use an integral to derive one of the integral representations for the Dirichlet eta function. This representation is very similar to the Riemann zeta function, which explains why their respective infinite series definition is quite similar (with the eta function being an alte rna

From playlist Integrals

Video thumbnail

RubyConf 2021 - Control methods like a pro: A guide to Ruby's awesomeness, ... by Masafumi Okura

Control methods like a pro: A guide to Ruby's awesomeness, a.k.a. metaprogramming by Masafumi Okura Do you know that methods are objects in Ruby? We can manipulate method objects just like other object, meaning that we can store them in variables, get information from them and wrap them i

From playlist RubyConf 2021

Video thumbnail

RubyConf 2015 - Messenger: The (Complete) Story of Method Lookup by Jay McGavren

Messenger: The (Complete) Story of Method Lookup by Jay McGavren You call a method on an object, and it invokes the instance method defined on the class. Simple. Except when the method isn't on the class itself, because it's inherited from a superclass. Or a singleton class, mixin, or ref

From playlist RubyConf 2015

Video thumbnail

RubyConf 2015 - Ruby 2 Methodology by Akira Matsuda

Ruby 2 Methodology by Akira Matsuda This talk focuses on "Method" in Ruby. Although Method is the key feature of an OOP language like Ruby, Ruby's Method is still drastically evolving. This session is a quick tour on new features and changes around Method in recent versions of the Ruby l

From playlist RubyConf 2015

Video thumbnail

RubyConf 2016 - Metaprogramming? Not good enough! by Justin Weiss

RubyConf 2016 - Metaprogramming? Not good enough! by Justin Weiss If you know how to metaprogram in Ruby, you can create methods and objects on the fly, build Domain Specific Languages, or just save yourself a lot of typing. But can you change how methods are dispatched? Can you decide th

From playlist RubyConf 2016

Video thumbnail

The Revenge of method_missing()

Convinced that nobody can bully method_missing() and get away with it, Nusco resolved to present a talk about it. When is method_missing() appropriate, and when should you pick an alternative metaprogramming magic spell instead? Is method_missing() really dangerous? What are the common met

From playlist Madison Ruby 2012

Video thumbnail

[Rust Programming] Crafting Interpreters: Day 37, Chapter 28 (Part 1)

In this video we continue to look at the Crafting Interpreters book, and learn how to port it to Rust. Since I'm a Rust beginner, the intent is that it will help me learn the language more in-depth than before. The book: https://craftinginterpreters.com/contents.html We're getting close

From playlist Rust Ports

Video thumbnail

Madison Ruby 2012 - The Revenge of method_missing()

The Revenge of method_missing() by: Paolo Perrotta Convinced that nobody can bully method_missing() and get away with it, Nusco resolved to present a talk about it. When is method_missing() appropriate, and when should you pick an alternative metaprogramming magic spell instead? Is metho

From playlist Madison Ruby 2012

Video thumbnail

The Fourier Transform and Derivatives

This video describes how the Fourier Transform can be used to accurately and efficiently compute derivatives, with implications for the numerical solution of differential equations. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow

From playlist Fourier

Video thumbnail

RubyConf 2015 - Extremely Defensive Coding by Sam Phippen

Extremely Defensive Coding by Sam Phippen Defensive programming is one of those abstract ideas that seems great but is often unclear in practice. In this talk we'll look at some of the extremely defensive patterns that have been driven out in RSpec through the years. We'll look at buildi

From playlist RubyConf 2015

Related pages

Exponential integral | Residue at infinity | Rational function | Binomial coefficient | Identity (mathematics) | Iverson bracket | Generating function