Section 2. The Proposition

by Emilio Charles Mazzie in Section 2. The Proposition
     
Let the transformation of my soul lie within its infinite sequence of orthogonal polynomial functions. Those satisfying the boundary conditions that the functions vanish at the endpoints of the unit interval of their domain, and in being of the class of Legendre, Laguerre and Hermite, satisfy their second-order, ordinary differential equation as the characteristic functions of their boundary value problem:
A(x)y”+B(x)y’+k_ny=0.
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Where A(x) is at most quadratic (e.g., ax^2+bx+c), B(x) is at most linear (e.g., mx+t) and k_n is the nth term of the sequence of characteristic values corresponding to the characteristic functions y=y_n(x), n=1,2, etc. The second and first derivatives of the functions denoted by y” and y’, respectively.

 

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Knowing the first five polynomials of the sequence, y_1(x), …, y_5(x), these and their derivatives may be substituted into the differential equation in order to solve for the unknowns A(x), B(x) and k_n. The results of which give the exact form of the differential equation which may be solved to obtain the general expression describing any one polynomial of the infinite sequence.

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The exact form of the differential equation and its solution which provide the key for determing all further analytical and metaphysical properties of the proposed functions as the constituents of the metakinetics of my soul. The first five polynomials posited as deriving a priori in the mind from its operant and modes of perception as the principal generating functions of the boundary value problem.
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The operant being that which is given to the mind as the seed and catalyst of the transformation; the modes of perception being the faculties of the mind which instantiate the transformation.

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