Orthogonal polynomial 3 term recurrence.

[crowlogic]

Implement Empirical Calculation of Coefficients and Recurrence Solver Integration

Task

• Calculate Coefficients: Implement empirical calculation of sequences (A_n), (B_n), and (C_n) for orthogonal polynomials.
• Integrate Solver: Use Maple or Mathematica to solve the recurrence relation.

Coefficients Formulas

$$A_n = \frac{\int_0^{\infty} x P_n(x)^2 \, dx}{\int_0^{\infty} P_n(x)^2 \, dx}$$

$$B_n = \frac{\int_0^{\infty} x Pn(x) P{n-1}(x) \, dx}{\int0^{\infty} P{n-1}(x)^2 \, dx}$$

$$C_n = \frac{\int0^{\infty} P{n+1}(x)^2 \, dx}{\int_0^{\infty} P_n(x)^2dx}$$

Detailed Proof of Favard's Theorem

Theorem Statement

A sequence of polynomials ${P_n(x)}$ is orthogonal with respect to some positive measure if and only if it satisfies:

P_n(x) = (A_n x + B_n)P_{n-1}(x) - C_n P_{n-2}(x) \quad \text{for} \; n \geq 2

where $A_n > 0$, $C_n > 0$, and $B_n$ is any real number.

Proof

Part 1: Recurrence Relation Implies Orthogonality

1. Polynomial Construction: Define $P_n(x)$ using the given recurrence relation.
2. Demonstrating Orthogonality:
   • Prove $\int P_n(x) P_m(x) d\mu(x) = 0$ for $m \neq n$ under measure $\mu$.
   • Apply linear operators in Hilbert spaces and orthogonal projections.

Application of Mathematical Principles

• Measure Theory:
  • Integral with respect to measure $\mu$ for inner product definition in square-integrable function spaces.
  • Define orthogonality via this inner product.
• Linear Algebra and Functional Analysis:
  • Analyze polynomial space as a Hilbert space.
  • Utilize linear operators for polynomial manipulation.
  • Employ orthogonal projections for function decomposition into polynomial bases.

Part 2: Orthogonality Implies Recurrence Relation

1. Starting with Orthogonality: Assume an orthogonal sequence under measure $\mu$.
2. Deriving Recurrence Relation:
   • Represent $xP_n(x)$ as a linear combination of orthogonal polynomials.
   • Use orthogonality to find coefficients, leading to the recurrence relation.

Conclusion

This proof demonstrates the relationship between the algebraic structure of polynomial sequences and their analytical properties of orthogonality under a measure, using principles from measure theory, linear algebra, and functional analysis.

[crowlogic]

Yes, in principle, it is possible to find recurrences or closed-form expressions for the coefficients $A_n$, $B_n$, and $C_n$ in terms of $n$ or in terms of each other, depending on the specific family of orthogonal polynomials and their properties. These expressions would relate the coefficients to the polynomial degree $n$ or establish a relationship between successive coefficients (e.g., $Bn$ in terms of $B{n-1}$, etc.), reflecting the underlying structure and symmetries of the polynomial sequence.

For many well-known families of orthogonal polynomials (like Legendre, Chebyshev, Hermite, and Laguerre polynomials), such relationships are well-established. These polynomials have specific weight functions and are defined on specific intervals, which dictate the form of their three-term recurrence relations, including explicit formulas for $A_n$, $B_n$, and $C_n$ as functions of $n$.

However, deriving such expressions for a general or newly identified sequence of orthogonal polynomials requires analyzing the polynomials' orthogonality conditions, the weight function, and the interval of definition. This process often involves:

1. Identifying Orthogonality Conditions: Understanding the weight function $w(x)$ and the interval over which the polynomials are orthogonal.
2. Analyzing Polynomial Properties: Investigating any symmetries, asymptotic behavior, or other properties that the polynomials exhibit.
3. Computing Integrals: Calculating the integrals that define $A_n$, $B_n$, and $C_n$ and attempting to express these in terms of $n$ or find patterns in the coefficients.

If a pattern or direct relationship can be identified for $A_n$, $B_n$, and $C_n$, either as functions of $n$ or in terms of their predecessors, this can greatly facilitate the analysis of the polynomials, enabling the derivation of closed-form expressions for the polynomials themselves or simplifying the computation of the polynomials for various degrees $n$.

For arbitrary sequences of orthogonal polynomials, especially those derived from specific applications or those with complex weight functions, finding such closed-form expressions or recurrence relations for the coefficients may be challenging and require detailed mathematical investigation.