Difference between revisions of "Meixner polynomial"

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The Meixner polynomials $M_n(x;\beta,c); c \in (0,1)$ are defined by
 
The Meixner polynomials $M_n(x;\beta,c); c \in (0,1)$ are defined by
 
$$M_n(x;\beta,c) = {}_2F_1 \left(-n,-x;\beta; 1 - \dfrac{1}{c} \right)$$
 
$$M_n(x;\beta,c) = {}_2F_1 \left(-n,-x;\beta; 1 - \dfrac{1}{c} \right)$$
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=Properties=
 
=Properties=
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<div class="toccolours mw-collapsible mw-collapsed">
<strong>Theorem:</strong> The Meixner polynomials are orthogonal with respect to the inner product
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<strong>Theorem:</strong> The Meixner polynomials are [[orthogonal]] with respect to the [[inner product]]
$$\langle p,q \rangle = \displaystyle\sum_{k=0}^{\infty} p(k)q(k) \dfrac{\beta^{\overline{k}}}{k!} c^k$$
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$$\langle p,q \rangle = \displaystyle\sum_{k=0}^{\infty} p(k)q(k) \dfrac{\beta^{\overline{k}}}{k!} c^k,$$
 
and $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn};\beta>0,0<c<1,$
 
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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</div>
 
</div>
 
</div>
 
</div>
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[[Rodrigues formula for Meixner polynomial]]<br />
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[[Relationship between Meixner polynomials and Charlier polynomials]]<br />
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=References=
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Classical and quantum orthogonal polynomials in one variable - Mourad Ismail
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[[Category:SpecialFunction]]

Latest revision as of 02:40, 21 December 2016

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:

Rodrigues formula for Meixner polynomial
Relationship between Meixner polynomials and Charlier polynomials

References

Classical and quantum orthogonal polynomials in one variable - Mourad Ismail