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I have read in various places the following objects are related:

  • orthogonal polynomials
  • birth-death processes
  • Lattice paths
  • continued fractions

After a lot of searching online, I found sketches about the various relationships

  • the moments of orthogonal polynomials count weighted lattice paths
  • the Kolmogoroff forward/backward equations of birth-death process can be solved using orthogonal polynomials of a given weight.

However the papers have been hard to decipher, such as this one by Karlin and McGregor, or the first few pages in this paper by Fabrice Gullemin.

The birth-death process on

$$ 1 \rightleftharpoons 2 \rightleftharpoons 3 \rightleftharpoons 4 \rightleftharpoons \dots $$

where $\mathrm{Rate}(n \to n+1)=\lambda_n, \mathrm{Rate}(n+1 \to n)=\mu_n, \mathrm{Rate}(n \circlearrowleft)=-(\mu_n + \lambda_n)$

The Karlin and McGregor showed you can find orthogonal polynomials $Q_n(x)$ with respect to a measure $\mu(dx)$ that solve the transition probabilities (the odds of starting at m and winding up at n at time t)

$$ \mathbb{P}(\Lambda(t) = n | \Lambda(0) = m) = \pi_n \int_0^\infty e^{-tx} Q_n(x)Q_m(x) \mu(dx)$$

They even figured out a way to construct the polynomials

$$ Q_{-1}(x)=0, Q_0(x)=1, \lambda_n Q_{n+1}(x) +(x - \lambda_n - \mu_n) Q_n(x) + \mu_n Q_{n-1}(x) = 0 $$

And then one notices the 3-term relation is the "magic box" recurrence formula for continued fractions.

I first read about these relationships in regards to the Rogers-Ramanujan continued fraction.

\[ r(q) = \cfrac{1}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \dots}}} \quad\quad\quad\quad = 1 - q + q^2 - q^4 + \dots \]

However the birth and death process is rather complicated.

What is the birth-death process associated to the Hermite polynomials or the Legendre polynomials?

or maybe in general

What do we reconstruct the birth-death processes correspond to the classical orthogonal polynomials?

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I don't understand the question. You already seem to have everything you need to determine the answer: given a family of orthogonal polynomials, you find the coefficients of their 3-term recurrence relations, and that tells you the rate constants of the birth-death process. – Qiaochu Yuan Dec 12 '13 at 9:29

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