I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 A. Gram-Schmidt orthogonalization.
 B. Definition and existence of orthogonal polynomials.
 C. Three-term recurrence relation for orthogonal polynomials.
 D. Orthogonal polynomials and quadrature rules.
 E. Extremal properties of orthogonal polynomials.
 F. Chebyshev polynomials.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Three-term recurrence relation for orthogonal polynomials.

roposition

(Three-term recurrence relation) Let be the polynomials related to measure as in the definition ( Orthogonal polynomials ) and the inner product is positive definite. We have

Proof

By the definition ( Orthogonal polynomials )-1 ) we have Hence, by the proposition ( Basic property of orthogonal polynomials ), for some numbers .

Next, we apply the operation to both sides and use consequences of orthogonality to calculate and .

For we get For we get Note that where is -th degree polynomial with leading coefficient equal to one. Thus, by the proposition ( Basic property of orthogonal polynomials ), for some numbers . We continue transformation of using orthogonality: We combine the last result with the relationship and obtain For we use the property and orthogonality to find

Definition

(Jacobi matrix) We introduce the following notation

1.

2. The numbers are zeros of : for each .

Proposition

(Zeros of orthogonal polynomials) Let be the polynomials related to the measure as in the definition ( Orthogonal polynomials )-2 and the inner product is positive definite. The zeros are eigenvalues of the matrix for each and are the corresponding eigenvectors.

Proof

The proposition's ( Three-term recurrence relation ) main statement is It may be rewritten as We substitute then We divide the last relationship by then or We restate the last result in matrix form as where and . The statement is apparent after the substitution .

 Notation. Index. Contents.