Skip to main content
Chemistry LibreTexts

M3: Laguerre Polynomials

  • Page ID
    13500
  • The Laguerre polynomials are solutions of Laguerre's differential equation:

    \[x\,y'' + (1 - x)\,y' + n\,y = 0\,\]

    These are the first few Laguerre polynomials:

    \(n\) \(L_n(x)\,\)
    0 1
    1 \(-x+1\,\)
    2 \({\frac{1}{2}} (x^2-4x+2) \,\)
    3 \({\frac{1}{6}} (-x^3+9x^2-18x+6) \,\)
    4 \({\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \,\)
    5 \({\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \,\)
    6 \({\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,\)