Difference between revisions of "Meixner polynomial"

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(Created page with "The Meixner polynomials $M_n(x;\beta,c); c \in (0,1)$ are defined by $$M_n(x;\beta,c) = {}_2F_1 \left(-n,-x;\beta; 1 - \dfrac{1}{c} \right)$$ =Properties= <div class="toccolo...")
 
(Properties)
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<strong>Theorem:</strong> The Meixner polynomials are orthogonal with respect to the inner product
 
<strong>Theorem:</strong> The Meixner polynomials are orthogonal with respect to the inner product
 
$$\langle p,q \rangle = \displaystyle\sum_{k=0}^{\infty} p(k)q(k) \dfrac{\beta^{\overline{k}}}{k!} c^k$$
 
$$\langle p,q \rangle = \displaystyle\sum_{k=0}^{\infty} p(k)q(k) \dfrac{\beta^{\overline{k}}}{k!} c^k$$
and $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn},$
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and $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn};\beta>0,0<c<1,$
 
where $\delta_{mn}$ denotes the [[Dirac delta]] and $\beta^{\overline{k}}$ denotes a [[rising factorial]].
 
where $\delta_{mn}$ denotes the [[Dirac delta]] and $\beta^{\overline{k}}$ denotes a [[rising factorial]].
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<strong>Proof:</strong> █
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<strong>Theorem:</strong> The following three-term recurrence holds for Meixner polynomials:
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$$xM_n(x;\beta,c)=c(\beta+n)(1-c)^{-1}M_{n+1}(x;\beta,c)-[n+c(\beta+n)](1-c)^{-1}M_n(x;\beta,c)+n(1-c)^{-1}M_{n-1}(x;\beta,c).$$
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
 
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</div>
 
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Revision as of 09:44, 20 May 2015

The Meixner polynomials $M_n(x;\beta,c); c \in (0,1)$ are defined by $$M_n(x;\beta,c) = {}_2F_1 \left(-n,-x;\beta; 1 - \dfrac{1}{c} \right)$$

Properties

Theorem: The Meixner polynomials are orthogonal with respect to the inner product $$\langle p,q \rangle = \displaystyle\sum_{k=0}^{\infty} p(k)q(k) \dfrac{\beta^{\overline{k}}}{k!} c^k$$ and $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn};\beta>0,0<c<1,$ where $\delta_{mn}$ denotes the Dirac delta and $\beta^{\overline{k}}$ denotes a rising factorial.

Proof:

Theorem: The following three-term recurrence holds for Meixner polynomials: $$xM_n(x;\beta,c)=c(\beta+n)(1-c)^{-1}M_{n+1}(x;\beta,c)-[n+c(\beta+n)](1-c)^{-1}M_n(x;\beta,c)+n(1-c)^{-1}M_{n-1}(x;\beta,c).$$

Proof: