Section 3. The Operant and Modes of Perception

     

The proposition, section 2, asserts that there may be defined an infinite sequence of orthogonal polynomial functions of a single independent variable and of the classical kind, satisfying the boundary conditions that the functions vanish at the endpoints of the interval of their domain of the real line. The general expression describing any one polynomial of the sequence ascertained from the knowledge of the first five functions which satisfy their linear second-order differential equation in x of the Sturm-Liouville type. These, the principal generating functions, posited as deriving a priori in the mind from its operant and modes of perception.

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For simplicity, let the domain of the functions y_n(x), n=1, 2, 3, 4, 5, correspond to that of the unit interval [0,1]. The first function, n=1, being a polynomial of degree two with two real zeros (one at each endpoint of the domain at zero and one) and one relative extremum with either the downward or upward concavity of a local maximum or minimum (Fig. 4):
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y_1(x)=a_11X^2+a_12x+a_13.
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The second function, n=2, being of degree three with three real zeros (one at each endpoint of the domain and one at one-half at the center ) and two alternating relative extrema beginning in the same way as the previous function (from left to right) with either the downward or upward concavity of a local maximum or minimum:
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y_2(x)=a_21x^3+a_22x^2+a_23x+a_24.
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The third function, n=3, being of degree four with four real zeros (one at each endpoint of the domain and one at one-third and two-thirds) and three alternating relative extrema beginning in the same way as the previous functions:
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y_3(x)=a_31x^4+a_32x^3+a_33x^2+a_34x+a_35.
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The symmetry pattern thus repeating for the remaining two functions, n=4, 5.
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Aesthetically, the graphs of the functions which exhibit the following two properties:
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(i) The zeros divide the unit interval into equal lengths.
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(ii) The shapes of the graphs are symmetric with respect to the unit interval.
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The boundary conditions, y_n(0)=y_n(1)=0, requiring that the constant terms and the sums of the coefficients of the polynomials vanish:
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a_11+a_12=0,

a_21+a_22+a_23=0,
a_31+a_32+a_33+a_34=0,
a_41+a_42+a_43+a_44+a_45=0,
a_51+a_52+a_53+a_54+a_55+a_56=0.
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The problem being to solve for the coefficients so that the polynomials are forced to assume the symmetry properties (i) and (ii). The characteristics of which manifest in analogy to the physical constraints of an object bound to the unit interval. The coefficients which then characterize the polynomials as representations of the object, their values encoded, through the modes of the object’s perception, in the operant set of two elements identified as the boundary points of zero and one. The modes which are applied under seven lines of reasoning to yield the first five orthogonal polynomials as the principal generating functions.
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