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Can someone help me with this

$$\frac{1}{1-t}e^{-\frac{xt}{1-t}}=\sum_{n=0}^{n=\infty}L_{n}(x)\frac{t^{n}}{n!}$$

The author said that we should just expand it but I don't understand how and what $L_{n}$ is equal to.

Since there is $\frac{t^{n}}{n!}$ I guess the exponential function should be expanded but I don't know what to do with $\frac{1}{1-t}$ neither what is the full expression for the $L_{n}$. Thank you for your help, :)

GNUSupporter 8964民主女神 地下教會
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Vuk Stojiljkovic
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  • The answer depends on your definition of the Laguerre polynomials. – halbaroth Oct 30 '20 at 10:04
  • I know that there is a weight-ed laguerre polynomial, $L^{m}{n}(x)$ so i guess this is when m=0? Or what do you mean? When we solve the differential equation we get $$L{n}=\sum_{k=0}^{k=n}\frac{(-1)^{k}}{k!} {n\choose k} x^{k}$$ this is what wolfram gives me, are there any more definitions? – Vuk Stojiljkovic Oct 30 '20 at 10:08
  • But i dont know how to get this from the thing i am given. – Vuk Stojiljkovic Oct 30 '20 at 10:18
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    Add the above equation into your question & state that this is your definition of the Laguerre polynomials. – Donald Splutterwit Oct 30 '20 at 10:19
  • The calculation is now obvious ? ... $ \times t^n/n!$ ... sum on $n$ ... invert the order of the plums ... etc ... – Donald Splutterwit Oct 30 '20 at 10:21
  • @Vuk Stojiljkovic: your question makes as much sense as if you ask how to go to New York without mentioning any starting point. As soon as you choose your definition of the Laguerre polynomials, you could find a strategy to prove the identity. For instance, if you define the polynomials by a recursive definition, you can try to prove that the left-hand side satisfies some differential equation and deduce from that a recursive definition of its Taylor coefficients. – halbaroth Oct 30 '20 at 10:23

2 Answers2

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Using the following definition of Laguerre polynomials \begin{eqnarray*} L_{n}=\sum_{k=0}^{k=n}\frac{(-1)^{k}}{k!} {n\choose k} x^{k} \end{eqnarray*} Now we should have ... (See https://en.wikipedia.org/wiki/Laguerre_polynomials#Recursive_definition,_closed_form,_and_generating_function) \begin{eqnarray*} \sum_{n=0}^{\infty}L_{n}(x)\frac{t^{n}}{\color{red}{1}} &= & \sum_{n=0}^{\infty} \sum_{k=0}^{n}\frac{(-1)^{k}}{k!} {n\choose k} x^{k} t^{n} \\ &= & \sum_{k=0}^{\infty} \sum_{n=k}^{\infty }\frac{(-1)^{k}}{k!} {n\choose k} x^{k} t^{n}\\ &= & \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} x^{k}\sum_{n=k}^{\infty } {n\choose k} t^{n}\\ &= & \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} x^{k} \frac{t^{k}}{(1-t)^{k+1}} \\ &= & \frac{1}{1-t} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} \left( \frac{xt}{1-t} \right)^{k} \\ &= & \frac{e^{-\frac{xt}{1-t}}}{1-t}. \end{eqnarray*}

Donald Splutterwit
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  • Can i ask, under which conditions can we manipulate double summation in such way and how did the third row , the sum going from n=k to inf turned into $$\frac{t^{k}}{(1-t)^{k+1}}$$ , thank you. – Vuk Stojiljkovic Oct 30 '20 at 11:24
  • Scroll to the $7^{th}$ formula here ... https://en.wikipedia.org/wiki/Generating_function#Examples_of_generating_functions_for_simple_sequences ... This formula is valid provided $ \mid t \mid < 1$ ... & is easily proved by induction. – Donald Splutterwit Oct 30 '20 at 11:29
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$L_n^{(0)}(x)$ which is also written $L_n(x)$ is $L_n(x) = \sum_{k=0}^n (-1)^k {n \choose k}\frac{x^k}{k!}$. The expression in the question is an exponential generating function. Try to show that coefficients of like powers of $t$ on the left and right sides of the equation are equal. You can multiply the right side by 1 - t.

The equation in the question is incorrect. the correct generating function is not an exponential one as noted in red. Please see the Digital Library of Mathematical Functions, equation 18.12.13. I suggest you try some other examples first.

DavidW
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  • I am really confused with how to do such thing as in the book i was not given what $L_{n}(x)$ equals to. I was able to do all the things with legendre polynomials but i dont know how to do such thing with laguerre polynomials when i dont even have the definition of $L_{n}(x)$ nor how to start. Sorrry if my question is stupid. Could you show left and right side are equal? Thank you very much. – Vuk Stojiljkovic Oct 30 '20 at 10:32
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    Don't worry about the book. An equation for $L_n(x)$ is given above and can be checked with the literature. As you have looked at Legendre polynomials, Laguerre polynomials would be similar. – DavidW Oct 30 '20 at 11:02
  • Thank you for your positive comment. Other people seem to be irritated by my question. Also, thank you for the link. – Vuk Stojiljkovic Oct 30 '20 at 11:07
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    Not at all. Thanks for the questions. No matter what is asked there is always something to learn for everyone. – DavidW Oct 30 '20 at 11:09