Difference between revisions of "Meixner polynomial"
(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...") |
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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 $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn},$ | + | 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]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following three-term recurrence holds for Meixner polynomials: | ||
| + | $$xM_n(x;\beta,c)=c(\beta+n)(1-c)^{-1}M_{n+1}(x;\beta,c)-[n+c(\beta+n)](1-c)^{-1}M_n(x;\beta,c)+n(1-c)^{-1}M_{n-1}(x;\beta,c).$$ | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
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: █
