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index, the associated Laguerre equation is xLk ′′ j (x) + ( 1 − x+ k ) Lk ′ j (x) + j L k j (x) = 0, (14) which reduces to the Laguerre equation when k = 0 . Some associated Laguerre polynomials are listed in table 13–4. L00(x) = L0(x) L 2 0(x) = 2 L01(x) = L1(x) L 0 3(x) = L3(x) L11(x) = −2x+ 4 L13(x) = −4x3 + 48x2 − 144x+ 96 L10(x) = 1 L 3 2(x) = 60x 2 − 600x+ 1200 L02(x) = L2(x) L 3 3(x) = −120x3 + 2160x2 − 10800x+ 14400 L12(x) = 3x 2 − 18x+ 18 L23(x) = −20x3 + 300x2 − 1200x+ 1200 L22(x) = 12x 2 − 96x+ 144 L31(x) = −24x+ 96 L21(x) = −6x+ 18 L30(x) = 6 Table 13 − 4

References 1. Benedict, R. C., Strange, E. D., and Lakritz, L.: 23rd Tobacco Chemists' Research Conference, Philadelphia, Penn., 1969. 2. Benedict, R. C., Lakritz, L., Strange, E. D., and Stedman, R. L.: Chem. & Incl., Soo, 13 June 1970. 3. Bielski, B. H. J ., and Alien, A. 0.: J. Am. Chem.Soc. 92 (1970) 3793· 4. Bilimoria, M. H., and Nisbet, M. A.: Beitr. Tabakforsch. 6 (1971) ???. 5. Calder, J. H., Curtis, R. C., and Fore, H.: The Lancet 1963 (:r.) 556. ' 6. Edgar, J. A.: Experientia 25 (1969) 1214. 7. Edgar, J. A.: Nature 22.7 (1970) 24. 8. Hagopian, M

, “Small-Scale Local Phenomena Related to the Magnetic Reconnection and Turbulence in the Proximity of the Heliopause”, Astrophysical Journal Letters, Volume 773, L23 pp. 1-5. [13] Kunz, M. W., Schekochihihin, A. A., Chen, C. H. K., Abel, I., Cowley, S. C., 2015, “Inertialrange kinetic turbulence in pressure-anisotropic astrophysical plasmas”, J. Plasma Phys., Volume 81, Issue 5, pp. 1-61. [14] Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E., Schekochihin, A. A., 2006, “Astrophysical Gyrokinetics: Basic Equations and Linear Theory”, The

+ {\sin ^2}\left( {\frac{t}{2}} \right)\,\left[ {\,\left( {\frac{{2\alpha \,{l^2}}}{3} + \frac{{\beta {l^3}}}{4}} \right)\cos t + \frac{{\beta \,{l^3}}}{8}\cos \,2t + \frac{{4\alpha {l^2}}}{3} + \frac{{\beta \,{l^3}}}{8}} \right] \end{array}$$ (35) The velocity becomes x ˙ ( t ) = − l sin t + 2 sin t 2 cos t 2 − 2 α l 2 3 + β l 3 4 sin ⁡ t − β l 3 4 sin 2 t $$\begin{array}{} \displaystyle \dot x(t) = - l\sin \,t + 2\sin \left( {\frac{t}{2}} \right)\cos \left( {\frac{t}{2}} \right)\,\left[ { - \,\left( {\frac{{2\alpha \,{l^2}}}{3} + \frac{{\beta {l^3}}}{4}} \right)\sin t