Proof for recurrence relation of laguerre polynomials

Question

How can I prove the following recurrence relation for Laguerre polynomials. $$ (n+1)L_{n+1}(x)=(2n+1-x)L_n(x)-nL_{n-1}(x) $$ please help me. thanks

Answer 1

Before speaking about Rodrigues formula, you can use the closed for of Laguerre polynomials which is
L[n] = Sum[ Binomial[n,k] (-x)^k / k! , {k,0,n} ]
If you start with L[n+1] and expand it, you will arrive to the recurrence relation.

You can also prove it by induction starting with the first terms.

More serious help will be found at http://aw.twi.tudelft.nl/~koekoek/documents/thesis.pdf. This also can help you : https://www.math.lsu.edu/system/files/MD1%20Paper.pdf

I wonder if proper manipulation of Rodrigues formula would not be easier.