General Formula for Multiplying Polynomials?

[kpoltorak]

Homework Statement

Does a general formula exist?
$$\sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k$$

Homework Equations

The Attempt at a Solution

I am having trouble understanding the relation between the c coefficients in the product and the respective a and b coefficients in the factors. Could somebody shed some light on this? I am fairly certain there is a relation between the two which is based on combinatorics but cannot find the answer anywhere.

 

[deluks917]

See if you can solve an "easier" problem. Try assuming m1=m2 (if you get this the general case isn't hard to handle).

Look at a specific case m1 = m2 = 2.

(a0 +a1x +a2x^2) * (b0 +b1x + b2x^2) =

(a0b0)x^0 + (a1b0 + b1a0)x + (a2b0 +a1b1 + a0b2)x^2 + (a1b2 + a2b1)x^3 + (a2b2)x^4

Do you see how to write this as a summation?

 

[Bman900]

I know am digging this thread out of the bottom of the universe but Google got me here, and am sure more people could use this answer.

I am writing a program for multiplying polynomials of equal length and need a general formula for the coefficients in the resulting polynomial. I literary stumped as I don't see much of a pattern here. Any one have a clue?

 

[LCKurtz]

Look up the Cauchy Product. One place is here:
http://www.astarmathsandphysics.com/university_maths_notes/analysis/university_maths_notes_analysis_cauchy_products_of_series.html

 

[Bman900]

Thank you :)

 

[Ray Vickson]

Homework Statement

Does a general formula exist?
$$\sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k$$

Homework Equations

The Attempt at a Solution

I am having trouble understanding the relation between the c coefficients in the product and the respective a and b coefficients in the factors. Could somebody shed some light on this? I am fairly certain there is a relation between the two which is based on combinatorics but cannot find the answer anywhere.

The sequence $\{ c_k \}$ is the (discrete) convolution of the two sequences $\{ a_k \} \text{ and } \{ b_k \}.$ See, eg., http://www.cg.tu-berlin.de/fileadmin/fg144/Courses/07WS/compPhoto/Convolution_charts.pdf .

RGV