(n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)
From specialfunctionswiki
Theorem
The following formula holds: $$(n+1)L_{n+1}(x)=(2n+1-x)L_n(x)-nL_{n-1}(x),$$ where $L_{n+1}$ denotes Laguerre L.
Proof
References
- 1968: {{ #if: |{{{2}}}|W.W. Bell}}{{#if: |{{#if: |, [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]}}|}}: [[Book:W.W. Bell/Special Functions for Scientists and Engineers{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Special Functions for Scientists and Engineers{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}}|{{#if: | ({{{ed}}} ed.)}}}}]]{{#if: | (translated by [[Mathematician:{{{translated}}}|{{ #if: |{{{2}}}|{{{translated}}}}}]])}}{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Orthogonality of Laguerre L | ... (previous)|}}{{#if: xL n'(x)=nL n(x)-n L (n-1)(x) | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: Theorem 6.5 (i)
