arXiv:2501.02877 · Progress of Theoretical and Experimental Physics

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

Satoru Odake

Published Mar 12, 2025• 1 citation

Abstract

Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)($q$-)Hahn and 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.

DOI: 10.1093/ptep/ptaf041

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arXiv:2501.02877 · Progress of Theoretical and Experimental Physics

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

Satoru Odake

Published Mar 12, 2025• 1 citation

arXiv HTML PDF

Abstract

Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(

q

-)Hahn and 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.

DOI: 10.1093/ptep/ptaf041

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Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author [S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 (2008)]. The (q-) Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of qx (with x being the population) corresponding to the q-Racah polynomial.Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author [S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 (2008)]. The (q-) Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of qx (with x being the population) corresponding to the q-Racah polynomial.

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Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(

q

-)Hahn and 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.

\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}

, where

\ell_{\mathcal{D}}

is an even positive integer (

\mathcal{D}

: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

We consider dual polynomials of the multi-indexed (⁠|

q

|-)Racah orthogonal polynomials. The |

M

|-indexed (⁠|

q

|-)Racah polynomials satisfy the second-order difference equations and various |

1+2L

| (⁠|

L\geq M+1

|⁠) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various |

2L

|th-order difference equations. This means that the dual multi-indexed (⁠|

q

|-)Racah polynomials are ordinary orthogonal polynomials of Krall type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed (⁠|

q

|-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.

A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states

\sum_k α_{k} \ket{k}

coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure

μ(x)

. We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time

O(\text{polylog}(N, 1/ε))

, where

N

is the grid size and

ε

is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for

SU(2)

and

SU(1,1)

optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an

O\!\left(N\operatorname{polylog}((N+α+β+1)/ε)\right)

-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling

\mathrm{SU}(2)

representations with spins

(j_{1}, j_{2})

. Finally, we develop a new framework for Laguerre transforms for all orders

ν>0

by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.