User:Egm6321.f10.team4.Yoon/Mtg46

 Mtg 46: Mon, 6 Dec 10

[[media: 2010_12_02_14_28_37.djvu | Page 46-1]] Other applications: Dislocation theory, Quantum mechanics, Spectral Methematics. Discrete var. : Coding theory, Probability(Queing Theory, Birth and Death) Ref: Nikitorrv et al.(1991)

Rodrigues's formula: Another way to generate $$\displaystyle \{ P_n \}$$

Other orthogonal functions: (in particular, polynomials)

[[media: 2010_12_02_14_28_37.djvu | Page 46-2]] Unified general theory of orthogonal functions Note: Also include Bessel Functions.

Def: (After some transformations) Fundermental equations of Hypergeometric type(FEHT)

Ex: Legendre differntial equation(LDE)

[[media: 2010_12_02_14_28_37.djvu | Page 46-3]]

Theorem 1: All derivatives of a FHT, which is a solution of a FEHT, is again a FHT. Def: y is a FHT, i.e., a solution of a FEHT

Theorem 2: "Self-adjoint" form

[[media: 2010_12_02_14_28_37.djvu | Page 46-4]]

Theorem 3:

[[media: 2010_12_02_14_28_37.djvu | Page 46-5]] By Theorem 2:

note: [[media: 2010_12_02_14_28_37.djvu | (1)p.46-5]] is general Rodrigues's formula with [[media: 2010_12_02_14_28_37.djvu | (1)p.46-1]] as particular case.

Canonical form of classical orthogonal polynomials