Some q-Exponential Formulas Involving the Double Lowering Operator ψ for a Tridiagonal Pair (Research)

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Let $$\mathbb {K}$$ denote an algebraically closed field and let V denote a vector space over $$\mathbb {K}$$ with finite positive dimension. Let A, A ∗ denote a tridiagonal pair on V . We assume that A, A ∗ belongs to a family of tridiagonal pairs said to have q-Racah type. Let $$\{U_i\}_{i=0}^d$$ and $$\{U_i^\Downarrow \}_{i=0}^{d}$$ denote the first and second split decompositions of V . In an earlier paper we introduced a double lowering operator ψ : V → V with the notable feature that both ψU i ⊆ U i−1 and $$\psi U_i^\Downarrow \subseteq U_{i-1}^\Downarrow $$ for 0 ≤ i ≤ d, where U −1 = 0 and $$U_{-1}^\Downarrow =0$$ . In the same paper, we showed that there exists a unique linear transformation Δ : V → V such that $$\Delta (U_i)\subseteq U_i^{\Downarrow }$$ and ( Δ − I)U i ⊆ U 0 + U 1 + ⋯ + U i−1 for 0 ≤ i ≤ d. In the present paper, we show that Δ can be expressed as a product of two linear transformations; one is a q-exponential in ψ and the other is a q −1-exponential in ψ. We view Δ as a transition matrix from the first split decomposition of V to the second. Consequently, we view the q −1-exponential in ψ as a transition matrix from the first split decomposition to a decomposition of V which we interpret as a kind of halfway point. This halfway point turns out to be the eigenspace decomposition of a certain linear transformation $$\mathcal {M}$$ . We discuss the eigenspace decomposition of $$\mathcal {M}$$ and give the actions of various operators on this decomposition.