Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes

Abstract

We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)($q$-)Hahn, three kinds of $q$-Krawtchouk and $q$-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, $q$-Racah, Meixner, little $q$-Jacobi and little $q$-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.

35 pages. Typos are corrected. Comments and some data are added. To appear in PTEP

References in corpus (22)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• Infinitely many shape invariant potentials and new orthogonal polynomials
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Orthogonal Polynomials from Hermitian Matrices
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exactly Solvable Birth and Death Processes
• Orthogonal Polynomials from Hermitian Matrices II
• Dual Christoffel transformations
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• Birth and death processes and quantum spin chains
• Exactly solvable discrete time Birth and Death processes
• Fermionic logarithmic negativity in the Krawtchouk chain
• Markov Chains Generated by Convolutions of Orthogonality Measures
• Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials
• Exactly solvable inhomogeneous fermion systems
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• "Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme