Difference between revisions of "Meixner polynomial"
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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 $\langle M_n(\cdot;\beta,c),M_m(\cdot;\beta,c) \rangle = \dfrac{n! (1-c)^{-\beta}}{c^n \beta^{\overline{n}}} \delta_{mn};\beta>0,0<c<1,$ | 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]]. | ||
Revision as of 07:51, 10 June 2015
The Meixner polynomials $M_n(x;\beta,c); c \in (0,1)$ are defined by $$M_n(x;\beta,c) = {}_2F_1 \left(-n,-x;\beta; 1 - \dfrac{1}{c} \right)$$
Contents
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: █
Theorem
The following formula holds: $$\dfrac{\beta^{\overline{x}}c^x}{x!} M_n(x;\beta,c)= \nabla^n \left[ \dfrac{(\beta+n)^{\overline{x}}}{x!}c^x \right],$$ where $\nabla$ denotes the backwards difference operator $\nabla f = f(x)-f(x-1)$, $\beta^{\overline{x}}$ denotes a rising factorial and $M_n$ is a Meixner polynomial.
Proof
References
References
Classical and quantum orthogonal polynomials in one variable - Mourad Ismail
Theorem
The following formula holds: $$\displaystyle\lim_{\beta \rightarrow \infty} M_n \left(x;\beta,\dfrac{a}{\beta+a} \right) = C_n(x;a),$$ where $M_n$ denotes a Meixner polynomial and $C_n$ denotes a Charlier polynomial.
