A recurrence formula for an integral

Click For Summary

Discussion Overview

The discussion revolves around a recurrence formula related to an integral, specifically the expression $$A(p,q) = A(p-1 , q ) + A(p,q-1)$$. Participants explore the implications of different base cases on the solution of this recurrence, seeking a general approach or algorithm for solving it.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about solving the recurrence formula and questions whether a general formula exists or if it depends on the base cases.
  • Another participant asks for the conditions on $p$ and $q$, confirming they are positive integers.
  • It is noted that if $A(p,1)=A(1,q)=1$, then $A(p,q)$ corresponds to binomial coefficients due to Pascal's rule.
  • Further discussion indicates that the value of the base case significantly influences the outcome, with examples provided showing different results based on varying base cases.
  • A participant inquires about the existence of a procedure or algorithm for solving the recurrence, suggesting that it may depend on the chosen base case.
  • Another participant suggests using the recurrence to find a functional equation for the generating function of $A(p, q)$ as a potential method for determining $A(p, q)$.

Areas of Agreement / Disagreement

Participants generally agree that the solution to the recurrence depends on the base cases chosen. However, there is no consensus on a specific algorithm or general formula applicable to all scenarios.

Contextual Notes

The discussion highlights the complexity of the base cases and their impact on the recurrence solution, indicating that different assumptions lead to different forms of $A(p, q)$.

alyafey22
Gold Member
MHB
Messages
1,556
Reaction score
2
I was solving an integral and on the process of integration by parts it seems that I have a certain recurrence formula that I have to solve. Before I post it I want to understand how to solve a simpler case which is

$$A(p,q) = A(p-1 , q ) +A(p,q-1) $$

where the base case $A(p,1)$ and $A(1,q)$ is known to me. Is there a general formula to solve it ? or does that depend on the base case ?

I am completely clueless!
 
Physics news on Phys.org
ZaidAlyafey said:
I was solving an integral and on the process of integration by parts it seems that I have a certain recurrence formula that I have to solve. Before I post it I want to understand how to solve a simpler case which is

$$A(p,q) = A(p-1 , q ) +A(p,q-1) $$

where the base case $A(p,1)$ and $A(1,q)$ is known to me. Is there a general formula to solve it ? or does that depend on the base case ?

I am completely clueless!

Hi Zaid,

What are the conditions on $p$ and $q$? Are they positive integers?
 
Euge said:
Hi Zaid,

What are the conditions on $p$ and $q$? Are they positive integers?

Of course. I should have noted that.
 
ZaidAlyafey said:
$$A(p,q) = A(p-1 , q ) +A(p,q-1) $$

where the base case $A(p,1)$ and $A(1,q)$ is known to me.
If $A(p,1)=A(1,q)=1$, then $A(p,q)$ are binomial coefficients due to Pascal's rule.
 
Evgeny.Makarov said:
If $A(p,1)=A(1,q)=1$, then $A(p,q)$ are binomial coefficients due to Pascal's rule.

So that depends on the value of the base case ?
 
It will depend on the value of the base case. If you had $A(p,1) = p + 1$ and $A(1, q) = q + 1$, then $A(p, q) = \binom{p+q}{q}$. However, if $A(p, 1) = 1$ and $A(1, q) = q$, then $A(p, q) = \binom{p+q-1}{q-1}$.
 
Oh. My approach seems very complicated because the base case is very complex but I think it is worth the trail. I wanted to ask one more question : Is there any procedure or algorithm to solve the recurrence ? or that depends on the base case chosen ?
 
It will depend on the base cases, but you use the recurrence above to find a functional equation for the generating function of $A(p, q)$, which may help determine $A(p, q)$.
 

Similar threads

Graduate Does this computation satisfy LTL formulas?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Solve Recurrence Equation for Domino Chart of 1xn
  • · Replies 22 ·
Replies
22
Views
5K
MHB Finding an Exact Solution for the Recurrence Formula f(n) = 2f(sqrt(n)) + n
  • · Replies 8 ·
Replies
8
Views
5K
Undergrad How Does the Equation D x E mod Φ(N) = 1 Relate to RSA Encryption?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solve Recurrence: $s_n = 2_{s_n-1} - s_{n-2}$ | Discrete Math
  • · Replies 2 ·
Replies
2
Views
2K
MHB Recurrence Relations - Determining a solution of the recurrence relation
  • · Replies 5 ·
Replies
5
Views
2K
MHB Solution of the recurrence relation
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Chernoff Bounds using importance sampling identity
  • · Replies 0 ·
Replies
0
Views
2K
MHB Solve Recurrence $$T(n)=aT(n-1)+bn^c$$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Solve Recurrence Equation w/2 Base Cases Same Result
  • · Replies 1 ·
Replies
1
Views
2K