Study of an example of Markov chains involving Chebyshev polynomials

Abstract

In his paper, ‘Random walks and orthogonal polynomials: some challenges’, F. A. Grunbaum gave the polynomials $Q_n(x)$ orthogonal with respect to the weight $\frac{\sqrt{4pq}-x^2}{1-x^2}$ on $(-\sqrt{4pq},\sqrt{4pq})$ explicitly as $Q_n\left(x\right)={\left(\frac{q}{p}\right)}^{\frac{n}{2}}((2-2p)T_n\left(\frac{x}{2\sqrt{pq}}\right)+(2p-1)U_n\left(\frac{x}{2\sqrt{pq}}\right)),$ where $T_n$ and $U_n$ are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials $R_n(x)$ defined by $R_n\left(x\right)={\left(\frac{q}{p}\right)}^{\frac{n}{2}}((2-2p)V_n\left(\frac{x}{2\sqrt{pq}}\right)+(2p-1)W_n\left(\frac{x}{2\sqrt{pq}}\right)),$ where $V_n$ and $W_n$ are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials $R_n$. Second, we give the kernel $K_n^Q(x,\sqrt{4pq})$ where $K_n^Q(x,y)$ is the Christoffel–Darboux formula for the polynomials $Q_n$. Finally we give the integral of $Q_n$ function of $T_n$ and we show how we deduce that $Q_n(2\sqrt{pq}\tan (\frac{\pi }{4}x))$ is orthogonal with respect to the weight $\varphi (x)=\pi pq\frac{1+{\tan }^2(\frac{\pi }{4}x)}{1-4pq{\tan }^2(\frac{\pi }{4}x)}\sqrt{1-{\tan }^2(\frac{\pi }{4}x)}$ on $(-1,1)$.

Keywords:

• Chebychev polynomials
• three-term recurrence relation
• quadratic transformation
• kernel

Subject classifications:

• Primary 42C05
• Secondary 33C45

Acknowledgments

The authors are grateful to the editor and the two anonymous reviewers, whose insightful comments improved the paper considerably.

Disclosure statement

The authors reported no potential conflicts of interest relevant to this article.