09 8 9

3 Chebyshev 5................................. 5........................................ 5.3............................. 6.4....................................... 8.4................................... 8.4............................... 9.4.3.............................. 9.4.4.............................. 0.4.5...................................4.6...................................5..................................5..........................5.................6................................. 3.6. Chebyshev-Gauss........................... 4.6.. Gauss........................ 4.6.. Chebyshev-Gauss................... 5.6..3 Chebyshev..................... 6.6. Chebyshev-Gauss-Lobatto...................... 6.6.. Gauss-Lobatto.................... 6.6.. Chebyshev-Gauss-Lobatto.............. 9.6..3 Chebyshev..................... 9.7............ 0.7. [z b, z t ]..................... 0.7................. 0.7... 0.7.......................... 3.8........ 4

09 8 9.8. Chebyshev................... 4.8. Chebyshev-Gauss-Lobatto..................... 5.9...................................... 7 Gegenbauer 9. Rodrigues........................... 9. Gegenbauer................................ 30.3..................................... 3.4........................... 33.5 Laplace................... 35.6.................................... 38.7.................................... 39.8...................................... 4 3 Hermite 43 3. Hermite Rodrigues........................ 43 3........................................ 43 3.3 Hermite............................... 44 3.4....................................... 45 3.4................................... 45 3.5...................................... 46 4

5 Chebyshev * Fourier. x T m x cos[m arccosx]. cos mθ cos θ m T m x m T m. T m m.3 max T mx.4 x. π n m 0 T m xt n x π/ n m 0 x 0 n m.5 * Chebyshev Tschebyscheff Wikipedia Chebyshev

.3. 09 8 9 x cos θ π T m xt n x cosmθ cosnθdθ x 0 π 0 π n m 0 π n m 0 0 n m {cos[m + nθ] + cos[m nθ]} dθ.6.3 m x cos θ T m cosmθ.7 T 0 T cos θ x T cos θ Re iθ Rcos θ + i sin θ cos θ sin θ cos θ x T 3 cos 3θ Re 3iθ Rcos θ + i sin θ 3 cos 3 θ 3 cos θ sin θ 4 cos 3 θ 3 cos θ 4x 3 3x T 4 cos 4θ Re 4iθ Rcos θ + i sin θ 4 cos 4 θ 6 cos θ sin θ + sin 4 θ cos 4 θ 6 cos θ cos θ + cos θ 8 cos 4 θ 8 cos θ + 8x 4 8x + T 5 cos 5θ Re 5iθ Rcos θ + i sin θ 5 cos 5 θ 0 cos 3 θ sin θ + 5 cos θ sin 4 θ cos 5 θ 0 cos 3 θ cos θ + 5 cos θ cos θ 6 cos 5 θ 0 cos 3 θ + 5 cos θ 6x 5 0x 3 + 5x.8 6

.3. 09 8 9 Wikipedia Chebyshev polynomials T 0 x.9 T x x.0 T x x. T 3 x 4x 3 3x. T 4 x 8x 4 8x +.3 T 5 x 6x 5 0x 3 + 5x.4 m T m cos mθ Re imθ Rcos θ + i sin θ m m/ k0 m/ k0 m/ k0 m/ k0 m k m k m k k l0 k cos m k θ sin k θ k cos m k θ cos θ k k cos m k θ m k k l k l k l l0 k l cos m k l θ cos l θ.5 α α n k l l n T m m/ n0 m/ kn m k k n Gould, 00 m/ kn m k k n mm n m n n m/ m 0 * T m m m m/ n0 m/ n0 m/ n0 m m n m m n n n cos m n θ.6 m n n n cos m n θ n m n! n!m n! n cos θ m n n m n! n!m n! n x m n * m 0 m n 0.7.8 7

.4. 09 8 9 m 0.4.4. T m+ x cos[m + arccosx].9 T m x cos[m arccosx].0 T m+ x + T m x cos[m arccosx] cos[arccosx] xt m x. T m+ x xt m x T m x. T 0 x.3 T x x.4 m T 0 x.5 T x x.6 T x x.7 T 3 x 4x 3 3x.8 T 4 x 8x 4 8x +.9 T 5 x 6x 5 0x 3 + 5x.30. T n+m x cos[n + m arccosx].3 T n m x cos[n m arccosx].3 T n+m x + T n m x cos[n arccosx] cos[m arccosx] T n xt m x.33 T n+m x + T n m x T n xt m x.34 8

.4. 09 8 9.4.. dt m+ T m + x dt m dt m.35 dt 0 0.36 dt.37 d T m+ 4 dt m + T m xd d T m.38 d T 0 0.39 d T 0.40.4.3 T m x cos[m arccosx].4 x d T sin[m arccosx] mx m x.4 T m+ x cos[m + arccosx].43 T m x cos[m arccosx].44 T m+ x T m x sin[m arccosx] sin[arccosx] sin[m arccosx] x.45 x d T mx m [T m x T m+ x].46 9

.4. 09 8 9 m m 0 dt 0 0.47.46 x d T mx x d T mx m [ d T m x d ] T m+x.48.46 x d T mx m x x [T m x T m+ x] m 4 x [m {T m x T m x} m + {T m x T m+ x}].49 4 x d T mx m [m T m x + 4xT m x mt m x 4xT m+ x + m + T m+ x].50 m m 0, d T 0 0.5 d T 0.5.4.4 dt m+ m + dt m m T m+ cos[m + θ].53 T m cos[m θ].54 x cosθ.55 {sin[m + θ] sin[m θ]} sin θ cosmθ.56 T m 0

.4. 09 8 9 m 0, T 0 dt dt T T m m + dt m+ dt m m m.57 dt m mt m + m dt m m.58.4.5..46 [ T m x SmT m x x + m x d ] T m x.59 [ T m+ x S mt + m x x m x d ] T m x.60.4.6.57 T 0 x T x + C.6 T x 4 T x + C.6 T m x [ Tm+ x m + T m x m ] + C m.63 C. m T m x T m x m m T m+x x m T mx + C.64 x m + T mx m m T m x + C.65

.5. 09 8 9 m 0, T 0 x T T.66 T x 4 T T 0.67 m T m x [ Tm+ T m+ T ] m T m m + m [ m+ ] m m + m [ m+ ] m { m even m 0 odd m.68.5.5. x F x *3 F x f 0 T 0 x + f m T m x.69 m f m π F xt m x x.70 x cos θ f m π π 0 F cos θt m cos θdθ.7.5. *3

.6. 09 8 9 dt n x dθ dt n cos θ dθ sin θ d cosnθ dθ n sinnθ sin θ n einθ e inθ e iθ e iθ n [e in θ + e in 3θ + + e in 3θ + e in θ] { n [cosn θ + cosn 3θ + + cos θ] even n n [ ] cosn θ + cosn 3θ + + cos θ + odd n { n [Tn x + T n 3 x + + T x] even n n [ T n x + T n 3 x + + T x + T 0x ] odd n { n n j,odd [ T jx even n n ] n j,even T jx + T 0x odd n T 0 { d T n n n dt j x j,odd even n { n n j,even dt j x odd n n [ n j,odd j j k,even T kx + T 0x even n n n j,even j j k,odd T kx odd n [ n n k,even n k,odd { 4n jk+,odd jt kx + ] n j j,odd T 0x even n 4n n jk+,even jt kx odd n { [ n ] 4n k,even n k 4 T k x + n 8 T 0x even n 4n n k,odd n k 4 T k x odd n { [ n ] n k,even n k T k x + n T 0x even n n n k,odd n k T k x odd n ].7.73.6 Lagrange sampling points collocation points Gauss Gauss-Lobatto Gauss Gauss-Lobatto Gauss-Lobatto 3

.6. 09 8 9 minimax Lagrange Runge Gibbs 0 0 0 FFT.6. Chebyshev-Gauss.6.. Gauss N T N cosnθ.74 x cos θ.75 Gauss Chebyshev-Gauss grid points k,..., N N k + θ k π.76 N N k + x k cos π.77 N N θ [π, 0] m 0 m N T m cosmθ.78 x cos θ.79 Gauss N k + T m x k cos m π N.80 0 n, m N N N n m 0 T m x k T n x k N/ n m 0 k 0 n m.8 4

.6. 09 8 9 N T m x k T n x k k N k cos m N k + cos N k + π cos n N [ cos m + n m n N k + N ] N k + π N N k + π N π.8 cosine l < l < N N k N k + cos l π N R N cos l k N π N R exp il k N π N exp il k N π k k k Re il/nπ e ilπ e il/nπ l R e il/nπ e il/nπ [ l ]R i sinl/nπ 0.83 cosine l 0 n m 0 N T m x k T n x k k N k + cos [ cos m + n m n N k + N ] N k + π N N n m 0 N/ n m 0 0 n m π.84.6.. Chebyshev-Gauss x F x F N x f 0T 0 x + 5 N m f m T m x.85

.6. 09 8 9 Gauss x k k,..., N F N x k F x k.86 F x k f 0T 0 x k + f m N m f m T m x k.87 f m N N F x k T m x k.88 k FFT.6..3 Chebyshev f m π π Gauss 0 F cos θt m cos θdθ.89 f m N F cos θ k T m cos θ k θ.90 π k Gauss θ θ π N.9 f m N N F x k T m x k.9 k.6. Chebyshev-Gauss-Lobatto.6.. Gauss-Lobatto N T N cos[n θ].93 x cos θ.94 x dt N x dθ dt N dθ N sin[n θ].95 6

.6. 09 8 9 Gauss-Lobatto Chebyshev-Gauss-Lobatto grid points k,..., N θ k N k N π.96 N k x k cos N π.97 N θ [π, 0] T N x k cos[n kπ] N k.98 m 0 m N T m cosmθ.99 x cos θ.00 Gauss-Lobatto T m x k cos m N k N π 0 n, m N.0 T mx T n x + N k T m x k T n x k + T mx N T n x N N n m 0, N N / n m 0, N 0 n m N T mx T n x + T m x k T n x k + T mx N T n x N k T m T n + N k cos m N k N π cos n N k N π + T mt n.0.03 m + n m + n T m T n T m T n N T mx T n x + T m x k T n x k + T mx N T n x N 0 N k N k k cos m N k N π cos n N k N π [ cos m + n N k N π + cos m n N k ] N π.04 7

.6. 09 8 9 0 l k N k + cos l N k N π N N k + + cos l π N cos lπ l k N π + cos l k N π [coslπ + ] cos l k N π [ l + ] cos l k N π 0.05 N k N + / N N + / l cos l π cos N π 0.06 l m + n T m T n T m T n N T mx T n x + T m x k T n x k + T mx N T n x N N k k cos m N k N π N k cos n N k N π [ cos m + n N k N π + cos m n N k ] N π N n m 0, N N / n m 0, N 0 n m cosine 0 l < l < N N cos l N k N N π k cos l N π k k0 N k R exp il N π k0 N k R exp il N π k0 e ilπ R e i[l/n ]π 0.07.08 k k N / l e ilθ e ilθ N / 8

.6. 09 8 9 k k N m + n m + n k k N /.6.. Chebyshev-Gauss-Lobatto x F x F N x f 0T 0 x + N m Gauss-Lobatto x k k,..., N F x k f 0T 0 x k + f m T m x + f N T N x.09 F N x k F x k.0 N m f m T m x k + f N T N x k. f m [ ] f m N N F x T m x + F x k T m x k + F x N T m x N k FFT..6..3 Chebyshev f m π π 0 F cos θt m cos θdθ.3 Gauss-Lobatto Gauss Gauss-Lobatto f m π N F cos θ k T m cos θ k θ.4 k Gauss-Lobatto θ θ π N [ ] f m N N F T m + F x k T m x k + F T m k 9.5.6

.7. 09 8 9 F x f 0 T 0 x + N m f m T m x.7 m N /.7.7. [z b, z t ] F z [z b, z t ] z z b + z t z b + x.8 x z z b z t z b.9 z x x [, ] F x.7..7...5. F x F x f 0 T 0 x + m f m T m x.0.7 dt n x { n n j,odd [ T jx even n n ] n j,even T jx + T 0x odd n. 0

.7. 09 8 9 F x T 0 df m n f m dt m x dt n x dt n x f n + f n n n n f n n T j x + T 0x + f n 4n j n f n T 0 x + n 4n f n T j x + j nj n f n T 0 x + n j nj+ n n T j x j n f n T j x j + n f j+n T j x j n. df x df 0 T 0 x + m df m T m x.3 df m m + n f m+n.4 n df m m + n f m+n + m + f m+ n m + + n f m++n + m + f m+ n.5 df m+ + m + f m+ m.7...73 d T n { [ n ] n k,even n k T k x + n T 0x even n n n k,odd n k T k x odd n.6

.7. 09 8 9 F x T 0 d F m n n + f m d T m x f n d T n x + n f n dt nx n f n n {n j }T j x n j n f n n {n j }T j x + n T 0 x j 4n 3 fn T 0 x + + n j j nj+ nj+ 4n 3 fn T 0 x + n 4n 3 fn T 0 x + n 4 n {n j } f n T j x n{n j } f n T j x n 3 fn T 0 x + n j nj+,n j even nn j f n T j x j + n[j + n j ] f j+n T j x j j n 4 nj + nj + n f j+n T j x n.7 d F x ddf 0 T 0 x + m ddf m T m x.8 ddf m 4 nm + nm + n f m+n.9 n

.7. 09 8 9.7...57 T 0 dt dt T T m m + dt m+ dt m m m.30 F x x F x f 0 T 0 x + m df x df 0 T 0 x + f m T m x.3 m T 0.3 df m T m x.3 df x m f m dt m.33.3 [ df x dt df 0 + [ dt df 0 + m df dt + m df dt + m3 [ dfm m df ] dtm m+.33.34 { dt m+ df m m + m dt df m m m df m+ m m } ] dt m ] dt m.34 f m [ dfm m df m+].35 m df m df m+ + m + f m+.36 3

.8. 09 8 9.8 x [, ] u t u x.37 u, t u b.38 u, t u t.39.8. Chebyshev u.37.9 dũ m dt ddu m.40 4 nm + nm + nũ m+n n n ũ n u b.4 n0 ũ n u t.4 n0 n N 0 n N 0 m N dũ m dt N m/ 4 nm + nm + nũ m+n.43 n 4

.8. 09 8 9 α α ] N ũ N ũ N [u N b n ũ n ũ N + ũ N u t N n0 n0.44 ũ n.45 N N ũ N u t u b ũ N u t + u b + N 3 n,odd N n0,even ũ n.46 ũ n.47 N ũ N u t + u b ũ N u t u b + N 3 n,even N n0,odd ũ n.48 ũ n.49.8. Chebyshev-Gauss-Lobatto Gauss-Lobatto θ k N k N π.50 N k x k cos N π.5 k,..., N T m;k T m x k.5 T m;k d T m x k.53 5

.8. 09 8 9 u N u N x k, t T 0;ku 0 t + N m T m;k u m t + T N ;ku N t.54 d u N x k, t N T 0;ku 0 t + T m;ku m t + T N ;ku N t.55 m.37 T du 0 0;k N dt t + m T m;k du m dt t + T N ;k du N t dt N T 0;ku 0 t + T m;ku m t + T N ;ku N t m.56 k,..., N k, N N T 0;u 0 t + T m; u m t + T N ;u N t u b.57 m N T 0;N u 0 t + T m;n u m t + T N ;N u N t u t.58 m u m t + t Euler T 0; T ;... T N ; T N ; u 0 t + t T 0; T ;... T N ; T N ; u t + t....... T 0;N T ;N... T N ;N T. N ;N u N t + t T 0;N T ;N... T N ;N T N ;N u N t + t u b rhs. rhs N u t.59 rhs m t u m t T 0; T 0;... T 0;N T 0;N T ; T ;... T ;N T ;N N....... T N ; T N ;... T N ;N T N ;N T N ; T N ;... T N ;N T N ;N 6.60

.9. 09 8 9 u 0 t + t u t + t. u N t + t u N t + t T 0; T 0;... T 0;N T 0;N T ; T ;... T ;N T ;N. N...... T N ; T N ;... T N ;N T N ;N T N ; T N ;... T N ;N T N ;N u b rhs. rhs N u t.6.9 Doman, Brian George Spencer 06 The Classical Orthogonal Polynomials, World Scientific 5 Glatzmaier, Gary A. 04 Introduction to Modeling Convection in Planets and Stars, Prenceton University Press 9.4 Gould, H.W. 00 Combinatorial Identities: Table III: Binomial Identities Derived from Trigonometric and Exponential Series ed., Jocelyn Quaintance, https://www.math.wvu.edu/~gould/vol.6.pdf 004, 5 986 I Protas, Bartosz 004 Topics in Numerical Analysis Spectral Methods III Chebyshev Spectral Methods, http://ms.mcmaster.ca/~bprotas/math745a/spectr_03.pdf 7

9 Gegenbauer Gegenbauer Legendre Chebyshev Gegenbauer Chandrasekhar 98 6. Rodrigues Xµ µ µ + µ. ρ α µ µ α. n Fn α µ d n ρ α µ dµ n [ρ αµxµ n ].3 Rodrigues n 0 F α 0 µ.4 F α n µ [, ] n Π n µ F α n, Π n ρ α µf α n µπ n µdµ Π n µ dn dµ n [ρ αµxµ n ]dµ ] [Π n µ dn dµ n [ρ αµxµ n ] Π n µ dn dµ n [ρ αµxµ n ]dµ.5

.. Gegenbauer 09 8 9 µ + µ 0 0 n F α n, Π n n Π n n µ[ρ αµxµ n ]dµ.6 n Π n µ n 0 0 F α n, Π n 0.7 F α 0, F α,..., F α n n F α n, F α l 0 l,,..., n.8 Fl α, Fk α ρ α µfl α µfk α µdµ 0 l k.9 Fn α [, ] F 0 α F α F 0 α ρ α Fn α F 0 α, F α,..., Fn α ρ α n F α n F α n n Fn α d n lim µ µ α dµ n [ µn+α + µ n+α ].0 µ µ n+α n + µ n+α F α n n n + αn + α α + n n n α + n. Pochhammer β n β n ββ + β + n Γβ + n Γβ.. Gegenbauer Gegenbauer F α n µ C α+ n µ n Γα + Γn + α + n n! Γα + Γn + α + F n α µ.3 Cn α µ n Γα + /Γn + α n n! ΓαΓn + α + / F α n µ.4 30

.3. 09 8 9 α + /.6 n 0 µ C α n n! C α 0 µ.5 Γn + α Γα m + / C m+ n µ n n n! m!n + m! m!n + m! m 0 Legendre α n n!.6 d n µ m dµ n [ µ n+m ].7 P n x C n µ n d n n n! dµ n [ µ n ].8 Chandrasekhar 98 6 m C 3 n µ n n + n+ n! µ d n dµ n [ µ n+ ].9 Chebyshev 0 Gegenbauer Γ0 T n µ Cnµ 0 n Γ/Γn + n n! ΓΓn + / F n µ n n n! n n n! π Γn + / F n µ π Γn + / F n µ n n F n µ / n.0 Chebyshev Gegenbauer U n µ C nµ..3 Gegenbauer I α+ n [ ρ α µ C α+ n µ] dµ n Γα + Γn + α + n n! Γα + Γn + α + [ ] C α+ d n n µ dµ n [ρ αµxµ n ]dµ. 3

.3. 09 8 9 n I α+ n Γα + Γn + α + n n! Γα + Γn + α + d n [ ] dµ n C α+ n µ [ρ α µxµ n ]dµ.3 C α+ n µ n n µ n k n k n Γα + Γn + α + dn n µ n+α µ n+α n! Γα + Γn + α + dµ n Γα + Γn + α + Γn + α + n n! Γα + Γn + α + Γn + α + Γα + Γn + α + n n! Γα + Γn + α +.4 I α+ n k n Γα + Γn + α + n n! Γα + Γn + α + k n n Γα + Γn + α + Γα + Γn + α + k n n Γα + Γn + α + Γα + Γn + α + d n dµ n xn [ρ α µxµ n ]dµ [ρ α µxµ n ]dµ µ n+α dµ.5 ξ + µ.6 µ n+α dµ n+α+ ξ n+α ξ n+α dξ 0 n+α+ Bn + α +, n + α +.7 n+α+ [Γn + α + ] Γn + α + I α+ n Γα + Γn + α + n n! Γα + Γn + α + Γα + Γn + α + n Γα + Γn + α + n+α+ [Γn + α + ] Γn + α + α+ [Γα + ] Γn + α + n!n + α + [Γα + ].8 α 0 Legendre I n [P n µ] dµ n +.9 3

.4. 09 8 9 α I 3 n [ ] µ C 3 3 n +! n µ dµ n!n + 3 n + n + n + 3.30.4 F α n µ n Gegenbauer C α+ n µ d F α n dµ α + µdf α n dµ + n n + α + F α n 0.3 F α n Cα n µ d C α n dµ α + µdcα n dµ + n n + α Cα n 0.3 α / Legendre µ d P n dµ µdp n dµ + nn + P nµ 0.33 α 3/ Gegenbauer µ d C 3 n 3 n dµ 4µ dc dµ + nn + 3C 3 n µ 0.34 33

.4. 09 8 9.3 Xµ d n+ [ dµ n+ Xµ d ] dµ [ρ αµxµ n ] Xµ dn+ dµ n+ [ρ αµxµ n ] + n + X µ dn+ dµ n+ [ρ αµxµ n ] nn + + X µ dn dµ n [ρ αµxµ n ] Xµ d d n dµ dµ n [ρ αµxµ n ] + n + X µ d d n dµ dµ n [ρ αµxµ n ] nn + + X µ dn dµ n [ρ αµxµ n ] Xµ d dµ [ρ αµfn α µ] + n + X µ d dµ [ρ αµfn α µ] nn + + X µ [ρ α µfn α µ] µ d [ µ dµ α Fn α µ ] n + µ d [ µ α Fn α µ ] dµ nn + [ µ α Fn α µ ] { µ α µ d Fn α α dµ n + α + µdf n dµ [ ] } 4αα + nµ + µ nn + α Fn α µ.35 34

.5. Laplace 09 8 9 d n+ [ dµ n+ Xµ d dn+ dµ n+ dn+ dµ n+ ] dµ [ρ αµxµ n ] [ Xµ d dµ [ ρα µxµ Xµ n ]] [ Xµ n d dµ [ρ αµxµ] + n Xµ n ρ α µx µ dn+ dµ n+ [Xµn ρ α µf α µ + n Xµ n ρ α µx µ] dn+ dµ n+ [{F α µ + n X µ} ρ α µxµ n ] {F α µ + n X µ} dn+ dµ n+ [ρ αµxµ n ] + n + {F α µ + n X µ} dn dµ n [ρ αµxµ n ] {F α µ + n X µ} d d n dµ dµ n [ρ αµxµ n ] + n + {F α µ + n X µ} dn dµ n [ρ αµxµ n ] {F α µ + n X µ} d dµ [ρ αµfn α µ] + n + {F α µ + n X µ} [ρ α µfn α µ] µα + n d dµ α + n µ α { µ df α n dµ + [ µ α F α n µ ] n + α + n [ µ α F α n µ ] ] } [n + αµ µ Fn α ].36.35.36 µ d F α n dµ α + µdf α n dµ + nn + α + F α n µ 0.37.5 Laplace Gegenbauer C N/ n N N zonal N N x i i.38 35

.5. Laplace 09 8 9 N x r cos θ.39 x r sin θ cos θ.40 x 3 r sin θ sin θ cos θ 3.4..4 x N r sin θ sin θ sin N cos θ N.43 x N r sin θ sin θ sin N sin θ N.44 0 θ i π for i,,..., N 0 θ N π 0 N r + N N r r + r N r N r r i + ρ i N i θ i + N i tan θ i ρ i sin θ i N i θ i [ sin θ i N i ].45 θ i θ i { r i ρ i r i l sin θ l i.46 N H 0 H r θ i i zonal Zr, θ { r N r N + r r r sin θ N [ sin θ N θ θ ]} Z 0.47 Zr, θ RrΘθ.48 d Rr N 3 r dr N dr d dr Θsin θ N dθ [ sin θ N dθ dθ ].49 r θ λ Θ Θ [ ] d N dθ sin θ N sin θ + λθ 0.50 dθ dθ µ cos θ.5 36

.5. Laplace 09 8 9 Mµ Θθ.5 M µ d M N µdm dµ dµ + λm 0.53 N 5 Chandrasekhar 6 α N.54 µ d M α + µdm dµ dµ + λm 0.55 n λ nn + α M a n µ n.56 n0 µ nn a n µ n α + µ na n µ n + λ a n µ n 0.57 n0 n0 n0 µ n 0 n + n + a n+ nn a n α + na n + λa n 0.58 a n+ a n nn + α λ n + n +.59 µ lim n a n+ /a n µ < µ Gauss Bressoud 006 u m a m u m+ u m mm + α λ m + m + m + αm λ/4 m + 3/m + /.60 Gauss m m α < / α / Chandrasekhar 6 α 3/ λ nn + α n n n 37

.6. 09 8 9.4 Gegenbauer C α n µ N Gegenbauer Cn N/ µ Gegenbauer.6 Gegenbauer α 0 wµ, h µ wµ, h µ wµ, h µh + h α.6 αh µh + h α+.6 4αα + h µh + h α+.63 wµ, h αh µ h µh + h α+.64 h α { } α+ wµ, h h αh [ α + µh µ α + α + h + α + µ ] h h µh + h α+.65 µ wµ, h wµ, h µ α + µ µ h α { α+ wµ, h h h h }.66 wµ, h µh + h α h n ϕ n µ.67 n0 ϕ n µ µ n.66 n0 { h n µ d ϕ n µ dµ α + µ dϕ } nµ dµ h n nn + αh n ϕ n x.68 µ d ϕ n µ dµ α + µ dϕ nµ + nn + αϕ n x 0.69 dµ Gegenbauer.3 ϕ n Gegenbauer C α n.67 n0 38

.7. 09 8 9 µ h n ϕ n h α α n n0.6 n0 n0 n0 h n αα + α + n h n n! α n h n n! ϕ n α n n!.70.7 Cn α µ ϕ n µ.7 µh + h α h n Cn α µ.73 Gegenbauer Gegenbauer n0.7 Goursat fz n f n z n! πi C t z C ft dt.74 t z n+ Gegenbauer C α+ n µ n Γα + Γn + α + d n n n! Γα + Γn + α + µ α dµ n [ µ α+n ].75 C α+ n µ n Γα + Γn + α + n πi Γα + Γn + α + Γα + Γn + α + πi Γα + Γn + α + C 39 µ α [ t t µ C ] n [ t µ t α+n t µ ] α dt t µ n+ dt.76

.7. 09 8 9 t h h t t µ.77 ht t + µ h 0.78 t h µh + h.79 dt dh h µh + h + µ h h dh µh + h µh µh + h h dh µh + h.80 t µ h µh µh + h.8 dt t µ h dh.8 µh + h t µ µh + µh + h.83 Goursat C α+ n µ Γα + Γn + α + πi Γα + Γn + α + α µh + dh µh + h α µh + h h n+ around h0 Γα + Γn + α + Γα + Γn + α + [ ] d n α n! dh n µh + µh + h α µh + h h0.84 α µh + µh + h α µh + h Γα + Γn + α + Γα + Γn + α + Cα+ n µh n α + n C α+ n µh n α + n n0 n0.85 40

.8. 09 8 9 α / µh + µh + h α / µh + h n0 n0 ΓαΓn + α + / Γα + /Γn + α Cα n µh n α + / n α n Cn α µh n.86 / Legendre C n µh n P n µh n.87 µh + h n0 n0.8 Bressoud, David M. 006 Gauss s Test, Appendix to A Radical Approach to Real Analysis, nd ed., https://www.macalester.edu/aratra/edition/chapter4/chapt4d.pdf Chandrasekhar, S. 98, Dover edition; 96, original Hydrodynamics and Hydromagnetic Stability, Dover. Doman, Brian George Spencer 06 The Classical Orthogonal Polynomials, World Scientific 9 96, 5 0 n, http://wasan.hatenablog.com/entry/00605/307389 4

43 3 Hermite 3. Hermite Rodrigues Hermite Rodrigues H m x m e x dm m e x 3. m 0 m 5 H 0 x e x e x 3.a H x e x H x e x H 3 x e x H 4 x e x d e x d e x d3 3 e x d4 4 e x e x xe x x 3.b e x [4x e x ] 4x 3.c e x [ 8x 3 + xe x ] 8x 3 x 3.d e x [6x 4 48x + e x ] 6x 4 48x + 3.e H 5 x e x d5 5 e x e x [ 3x 5 + 60x 3 0xe x ] 3x 5 60x 3 + 0x 3.f 3. Rodrigues e x H m xh l x 0 m l 3.3

3.3. Hermite 09 8 9 m > l m < l e x H m xh l x m d m m e x H l x { [ d m m e x H m l x m+ m 0 ] d m e x d m H lx e x dm m H lx } d m e x d m H lx 3.4 H l l m> l 0 3.3 Hermite Rodrigues Hermite Hermite d H m x dh m + mh m 0 3.5 m dm m e x e x H m x 3.6 m dm+ e x e x xh m+ m x + dh mx 3.7 e x H m x d e x H m x d [e x xh m x + dh ] mx 4x e x H m x 4xe x dh mx e x [ 4x H m x 4x dh mx + e x d H m x ] + d H m x 3.8 44

3.4. 09 8 9 3.6 3.7 d e x H m x m dm+ m+ e x m dm+ m+ xe x m [ m + dm m e x x dm+ m+ e x] m + m dm m e x x m dm+ m+ e x m + e x H m x xe x xh m x + dh mx [ e x 4x m H m x x dh ] mx 3.9 4x H m x 4x dh mx + d H m x 4x m H m x x dh mx 3.0 d H m x dh m + mh m 0 3. 3.4 3.4. Φ m x e x / H m/ m x m / ex dm m/ m e x 3. A ± d + x A + Φ m Φ m+ A Φ m Φ m 3.3 3.4a 3.4b 45

3.5. 09 8 9 ] A + Φ m m+ d [e x / dm m+/ m e x + m / xex dm m+/ m e x m+ / ex dm+ e x + m+ / xex dm m+/ m+ m+/ m e x + m / xex dm m+/ m e x / ex dm+ m+/ m+ Φ m+ e x m+ ] A Φ m m d [e x / dm m+/ m e x + m / xex dm m+/ m e x m / ex dm+ e x + m / xex dm m+/ m+ m+/ m e x + m / xex dm m+/ m e x m / ex dm m+/ m xe x / ex dm m / m Φ m e x m + m / xex dm m / m e x 3.5 3.6 3.5 05 SGC 6, 46