2
votes
2answers
96 views

Is there a recursive formula for Euler's Totient function

I have been looking for a recursive formula for Euler's totient function or Möbius' mu function to use these relations and try to create a generating function for these arithmetic functions.
2
votes
0answers
60 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
2
votes
1answer
59 views

Problem solving recurrences with generating functions

I'm trying to solve this linear recurrence with generating functions, but I keep getting stuck on the last few steps. I found the generating function, but after splitting it into partial fractions and ...
3
votes
1answer
90 views

Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
5
votes
2answers
263 views

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of ...
0
votes
1answer
130 views

Computing functions from generating functions

I am new to generating functions but understand how to derive them from given discrete numeric functions. Is there a simple way to derive the discrete numeric function given a generating function. For ...
3
votes
1answer
184 views

Is there any way to solve recurrence equations with variable coefficients?

So far I have done some problems that are best solved using generating functions. These mostly contain variable coefficients. A simple one is $H(n) = (n+2)H(n-2)$. I have found solutions to these ...