Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes

Abstract

In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.

Keywords

• orthogonal polynomial
• weight function
• chain sequence
• birth and death process
• quadrature formula
• Romanovski–Hermite type polynomials