- #1

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## Main Question or Discussion Point

Hello, I've been working on solving the equation y''-2xy'+2py=0. where p is a positive integer.

I've assumed y=[itex]\sum a_{n}x^{n}[/itex] for n=0 to inf

I'm getting two formulas for [itex]a_{n}[/itex]

One is for odd n, the other for even n, related to [itex]a_{0}[/itex] and [itex]a_{1}[/itex]

However, the relation involves something that looks like a factorial but it skips every other number, such as p(p-2)(p-4)(p-6)...

My question is whether this is valid:

[itex]n!=n(n-1)(n-2)(n-3)...[/itex]

[itex]2^{n}(n)!=2n(2n-2)(2n-4)(2n-6)...[/itex]

letting [itex]p=2n [/itex] then

[itex]2^{p-1}(\frac {p}{2})!=p(p-2)(p-4)(p-6)...[/itex]

I've assumed y=[itex]\sum a_{n}x^{n}[/itex] for n=0 to inf

I'm getting two formulas for [itex]a_{n}[/itex]

One is for odd n, the other for even n, related to [itex]a_{0}[/itex] and [itex]a_{1}[/itex]

However, the relation involves something that looks like a factorial but it skips every other number, such as p(p-2)(p-4)(p-6)...

My question is whether this is valid:

[itex]n!=n(n-1)(n-2)(n-3)...[/itex]

[itex]2^{n}(n)!=2n(2n-2)(2n-4)(2n-6)...[/itex]

letting [itex]p=2n [/itex] then

[itex]2^{p-1}(\frac {p}{2})!=p(p-2)(p-4)(p-6)...[/itex]