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This is a short how to about how to sum sin(n q) and cos(n q).
This paper is inspired by this post. It asked how to count the number of solutions of the equation with non-negative integral solution:
x_1 + ... + x_k = n
without using the standard method of line and dot counting. I transform the problem into a partial difference equation, ...
I tried to find the n-th derivative of the function sec x + tan x in a specific form, and a partial difference equation arises from it. An attemp to solve the equation is stated in another article: Reducing a partial difference equation into a partial differential equation and solving for the generating function using method of characteristics
I am going to solve a partial difference equation, firstly to transform it into a partial differential equation by using generating function method. Then, the partial differential equation can be solved by using method of characteristics.
In this paper, I will derive an explicit formula for the Euler zigzag numbers (Up/down numbers). Euler zigzag number is the number of alternating permutation in a set. Therefore the explicit formula of Euler numbers(Secant numbers) and Bernoulli numbers are found as well. The formula involves two finite sum.
In this short how-to, I will show how to solve 2nd-order differential equation of the form:
y''(t) = f(y)
I will give an example at the end.
In this how-to, I am going to solve a recurrence differential equation using Binomial Expansion Theorem for non-commutative elements.
Given [A, B] = lambda, we would like to find out the expansion of (A+B)^n. This result is useful for quantum mechanics since [x,p] = i hbar. And it can be useful when solving mixed differential recurrence relation equation as well. I will give an example of it.
I am going to solve a partial difference equation in this paper, which arises from binomial expansion of non-commutative matrix/operator.
In this paper, I am going to prove the identity:
e^(A+B) = e^(-lambda/2) e^A e^B
given [A, B] = lambda.
This formula is quite useful in quantum mechanics, and I will prove it using power series expansion ...
