Charlier Polynomials: Generating Function, Orthogonality

Here we study Charlier polynomials along with its generating function and orthogonality relation. Before doing this, let us recall the series notation of exponential and binomial functions.

e^ax = $\displaystyle \sum_{n=0}^\infty \dfrac{(ax)^n}{n!}$ …(∗)

Also, recall how the Cauchy product of two infinite series.

$\displaystyle \sum_{n=0}^\infty a_n \times \displaystyle \sum_{n=0}^\infty b_n$ $=\displaystyle \sum_{n=0}^\infty c_n$ where c_n = $\displaystyle \sum_{k=0}^n a_k b_{n-k}$.

Generating Function of Charlier Polynomials

Theorem: The generating function for the Charlier polynomials P_n(x) is given by $G(x, w) = e^{-aw}(1 + w)^x$. That is,

$e^{-aw}(1 + w)^x = \displaystyle\sum_{n=0}^{\infty} P_n(x) w^n$

Proof:

We have to show that we can get the Charlier polynomials P_n(x), for each n ∈ ℕ, out of this function G(x, w).

We have

$G(x, w) = e^{-aw}(1 + w)^x$

⇒ $G(x, w) = \displaystyle\sum_{m=0}^{\infty} \dfrac{(-a)^m w^m}{m!}
\displaystyle \sum_{n=0}^{\infty} \binom{x}{n} w^n$

⇒ $G(x, w) = \displaystyle\sum_{n=0}^{\infty} P_n(x) w^n$ …(∗∗) [using the Cauchy product of infinite series.] Here,

$P_n(x) = \displaystyle \sum_{k=0}^{n} \binom{x}{k} \dfrac{(-a)^{n-k}}{(n – k)!}$ …(1)

Note that $\binom{x}{k}=\dfrac{x!}{k!(x-k)!}$ $=\dfrac{1}{k!}x(x-1)\cdots (x-k+1)$. Therefore, from Equation (1) we see that P_n(x) is a polynomial of degree n, called the Charlier Polynomial of degree n (n= 1, 2, 3, …).

As the polynomials P_n(x) are generating from G(x, w), this function G(x, w) is called the generating function of the Charlier polynomials.

Orthogonal Relation of Charlier Polynomials

The orthogonality relation of Charlier polynomials is obtained as follows.

Note that

$a^x G(x, v) G(x, w) = e^{-a(v + w)} [a(1 + v)(1 + w)]^x$

So,

$\dfrac{a^k G(k, v) G(k, w)}{k!}$ $= e^{-a(v + w)} \dfrac{[a(1 + v)(1 + w)]^k}{k!}$

Now, summing over k varying from 0 to ∞ we get that

$\displaystyle\sum_{k=0}^{\infty} \dfrac{a^k G(k, v) G(k, w)}{k!}$ $ = e^{-a(v + w)} \displaystyle\sum_{k=0}^{\infty} \dfrac{[a(1 + v)(1 + w)]^k}{k!}$

⇒ $\displaystyle\sum_{k=0}^{\infty} \dfrac{a^k G(k, v) G(k, w)}{k!}$ $=e^{-a(v+w)} \times e^{a(1+v)(1+w)}$

⇒ $\displaystyle\sum_{k=0}^{\infty} \dfrac{a^k G(k, v) G(k, w)}{k!}$ $=e^{a+avw}=e^a e^{avw}$

By (∗∗), we deduce that

$\displaystyle \sum_{k=0}^{\infty} \frac{a^{k}}{k!}\sum_{m,n=0}^{\infty} P_{m}(k) P_{n}(k) v^{m} w^{n}$ $=\displaystyle\sum_{n=0}^{\infty} \dfrac{e^a a^n (vw)^n}{n!}$ ⇒ $\displaystyle \sum_{m,n=0}^{\infty} \sum_{k=0}^{\infty} P_{m}(k) P_{n}(k) \dfrac{a^{k}}{k!} v^{m} w^{n}$ $=\displaystyle\sum_{n=0}^{\infty} \dfrac{e^a a^n (vw)^n}{n!}$.

Now comparing the co-efficients of v^mwⁿ on both sides, we obtain that

$\displaystyle\sum_{k=0}^{\infty} P_m(k) P_n(k) \dfrac{a^k}{k!}$ $= \begin{cases} \dfrac{e^{a} a^{n}}{n!}, & \text{if } m = n \\ 0, & \text{if } m \ne n. \end{cases}$

This shows that the sequence {P_n(x)} of Charlier polynomials satisfies an orthogonal relation with respect to the discrete mass distribution having mass a^k/k! at the point k (k=0, 1, 2, …).

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Source: An Introduction to Orthogonal Polynomials by T. S. Chihara