Im looking for a formula to sum over Laguerre polynomials. The sum has the form
sum_n n!/(n+k!) (L_n^k (x))^2 * t^n
I found some formulas for similar sums on https://functions.wolfram.com/Polynomials/LaguerreL3/23/02/, but its only close and not excactly what I need. Does anyone have an idea where I can either find a better formula or how to derive it myself (although I guess the latter can be pretty tricky)
I'm currently in quantum mechanics and we are studying they hydrogen atom right now. The solutions for both the phi equation and the radial equation are expressed in associated special polynomials (associated Legendre and associated Laguerre polynomials) and I was wondering how you go about solving one of these equations (lets just go with the radial equation) from its original equation without using the known Laguerre Polynomial identities.
That is to say, how do I find some [; w(x) ;] such that [; \int dx \; w(x)\; L_n(x)\; L_m(x)\; e^{i\;(n+m)\;(\frac{c_1}{\sqrt{x}}-1)}=f(n)\delta_{mn} ;] where [;L_n(x);] is the n-th Laguerre polynomial? I know this is oddly specific, but it's getting to be kind of a roadblock on an optics project I'm working on. Thanks in advance for the help!
Edit: to clarify based on some of the responses, I'm looking for an inner product under which [; L_n(x)\;e^{i\;n\;(\frac{c_1}{\sqrt{x}}-1)} ;] are orthogonal. Just using the inner product that makes the Laguerre polynomials orthogonal doesn't work, and I'm not sure how to tweek it to make it work. Left the f(n) on the right side just for generality sake. Sorry if that lead to any confusion; I don't care about f(n), just that for mismatched n and m the inner product goes to 0.
Do your worst!
It really does, I swear!
For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.
I said "hey look, an escaPEA"
No one near me but it didn't half make me laugh for a good hour or so!
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Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.
This sub is mostly just NSFW puns now.
If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.
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When I got home, they were still there.
What did 0 say to 8 ?
" Nice Belt "
So What did 3 say to 8 ?
" Hey, you two stop making out "
I won't be doing that today!
You take away their little brooms
This morning, my 4 year old daughter.
Daughter: I'm hungry
Me: nerves building, smile widening
Me: Hi hungry, I'm dad.
She had no idea what was going on but I finally did it.
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Why
Itβs pronounced βNoel.β
After all his first name is No-vac
What, then, is Chinese rap?
Edit:
Notable mentions from the comments:
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Spanish/Swedish/Swiss/Serbian hits
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There hasn't been a single post this year!
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Nothing, it just waved
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Just to clarify, 12345678
Me grabbing a soda from my (what I thought was) half full 12pk...
Notices there's only 2;
Me: "Awe man... This is a damn bird box!" Her: "What the hell does that mean?!" Me: (Pulls both cans out & shows them to her) "It's only got Toucans."
I'm not ashamed to admit the look on her face was glorious.
I was just sitting there doing nothing.
βBOOMβ?!
"That's what they're fighting about."
