Branching Formula for $q$-Toda Function of Type B

TL;DR

This paper proves an explicit branching formula for the eigenfunctions of the $B_N$ $q$-Toda operator, connecting them to $A_{N-1}$ eigenfunctions through a recursion relation.

Contribution

It provides the first rigorous proof of the conjectured branching formula for $q$-Toda eigenfunctions of type B, using contigulation and recursion relations.

Findings

Established the explicit branching formula for $q$-Toda functions of type B.

Connected $B_N$ eigenfunctions to $A_{N-1}$ eigenfunctions via a new recursion relation.

Validated the conjecture proposed by the authors regarding the eigenfunction structure.

Abstract

We present a proof of the explicit formula for the asymptotically free eigenfunctions of the $B_N$ $q$-Toda operator which was conjectured by the first and third authors. This formula can be regarded as a branching formula from the $B_N$ $q$-Toda eigenfunction restricted to the $A_{N-1}$ $q$-Toda eigenfunctions. The proof is given by a contigulation relation of the $A_{N-1}$ Toda eigenfunctions and a recursion relation of the branching coefficients.