Orthogonal polynomials on the unit circle satisfying a second-order differential equation with varying polynomial coefficients

ABSTRACT

Consider the linear second-order differential equation (1.1) $A_n\left(z\right)y^{\mathrm{\prime \prime }}+B_n\left(z\right)y^{\mathrm{\prime }}+C_{n}y=0,$(1.1) where $A_n\left(z\right)=a_{2,n}{z}^2+a_{1,n}z+a_{0,n}$ with $a_{2,n}\ne 0,a_{1,n}^2-4a_{2,n}{a}_{0,n}\ne 0,\mathrm{\forall }n\in \mathbb{N}$ or $a_{2,n}=0,a_{1,n}\ne 0,\mathrm{\forall }n\in \mathbb{N}$, $B_n\left(z\right)=b_{1,n}+b_{0,n}z$ are polynomials with complex coefficients and $C_n\in \mathbb{C}$. Under some assumptions over a certain class of lowering and raising operators, we show that for a sequence of polynomials $({\phi }_n{)}_{n=0}^{\mathrm{\infty }}$ orthogonal on the unit circle to satisfy the differential equation (1.1(2.23) $T_{m_n,2}\left[{\phi }_{m_{n-1}}\right(z\left)\right]=\frac{A_{m_{n-1}}\left(z\right)}{z}\frac{{\kappa }_{m_n-1}}{{\kappa }_{m_{n-1}-1}}\frac{{\phi }_{m_{n-1}}\left(0\right)}{{\phi }_{m_n}\left(0\right)}{\phi }_{m_n}\left(z\right),$(2.23) ), the polynomial ${\phi }_n$ must be of a specific form involving and extension of the Gauss and confluent hypergeometric series.

KEYWORDS:

• Orthogonal polynomials on the unit circle
• differential equations
• special functions
• complex analysis

AMS SUBJECT CLASSIFICATION:

• 42C05
• 34M03
• 33C05
• 33C15
• 30Axx

Acknowledgments

We thank the anonymous reviewer for the careful reading of our article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author acknowledges the financial support of FAPESP of Brazil, under (grant no. 2012/21042–0). The second author acknowledges the support of CNPq (grant no. 305073/2014–1) and FAPESP (grant no. 2009/13832–9).