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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 MATH be the polynomials related to measure $d\lambda$ as in the definition ( Orthogonal polynomials ) and the inner product MATH is positive definite. We have MATH MATH

Proof

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

Next, we apply the operation MATH to both sides and use consequences of orthogonality to calculate MATH and $\gamma_{k,i}$ .

For $s=k$ we get MATH For $s=k-1$ we get MATH Note that MATH where $t\pi_{k-1}$ is $k$ -th degree polynomial with leading coefficient equal to one. Thus, by the proposition ( Basic property of orthogonal polynomials ), MATH for some numbers MATH . We continue transformation of MATH using orthogonality: MATH We combine the last result with the relationship MATH and obtain MATH For $s<k-1$ we use the property MATH and orthogonality to find MATH

Definition

(Jacobi matrix) We introduce the following notation

1. MATH

2. The numbers MATH are zeros of $\pi_{k}$ : MATH for each $k$ .

Proposition

(Zeros of orthogonal polynomials) Let MATH be the polynomials related to the measure $d\lambda$ as in the definition ( Orthogonal polynomials )-2 and the inner product MATH is positive definite. The zeros MATH are eigenvalues of the matrix $J_{k}$ for each $k$ and MATH are the corresponding eigenvectors.

Proof

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





Notation. Index. Contents.


















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