Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

We consider the generalized eigenvalue problem Aψ = λBψ for two operators A,B. Self-similar closure of this problem under a simplest Darboux transformation gives rise to two possible types of regular algebras of dimension 2 with generators A,B. Realization of the operators A,B by tri-diagonal operators leads to a theory of biorthogonal rational functions. We find the general solution of this problem in terms of the ordinary and basic hypergeometric functions. In special cases we obtain general Padé interpolation tables for the exponential and power function on uniform and exponential grids. 1. Generalized eigenvalue problem and its Darboux transformations Let A,B be two operators in some linear space L (either finiteor infinite-dimensional) . The linear combination Y (λ) = A− λB is called a linear pencil [6]. Assume that a vector ψ(λ) belongs to a kernel space of the operator Y (λ). Clearly Aψ(λ) = λBψ(λ). (1.1) Thus the vector ψ(λ) is a solution of the generalized eigenvalue problem (GEVP) (1.1) [2], [18]. When B = I, where I is identical operator, then GEVP is reduced to the ordinary eigenvalue problem for then operator A: Aψ(λ) = λψ(λ). The GEVP arises in many problems in the theory of separation of variables in PDE and in the mechanical vibrations [2], [6]. Assume that a scalar product 〈ψ,χ〉 is introduced on the space L. Then it is possible to consider the conjugated GEVP Aψ(λ) = λBψ(λ), (1.2) for some vectors ψ(λ), where A, B are conjugated operators defined by 〈Aψ,χ〉 = 〈ψ,Aχ〉 (note that we define the conjugated operator without taking complex conjugates, Copyright c © 2005 by A Zhedanov 334 A Zhedanov so, e.g. in the finite-dimensional case the conjugated matrix coincides with the transposed matrix). It is then elementary to verify the biorthogonality property 〈ψ(λ), Bψ(μ)〉 = 0, if μ 6= λ (1.3) Equivalently, we can write 〈ψ(λ), φ(μ)〉 = 0, if μ 6= λ (1.4) where φ(μ) ≡ Bψ(μ) (1.5) We thus have two sets ψ(λ) and φ(λ) of biorthogonal vectors with respect to scalar product 〈ψ,χ〉. The GEVP (1.1) possesses an important property of projective invariance. Indeed, define a new pair of operators C = αA+ βB, D = γA+ δB, (1.6) where α, β, γ, δ are arbitrary complex parameters such that αδ− γβ 6= 0. Then the vector ψ(λ) satisfies GEVP Cψ(λ) = λ̃Dψ(λ), (1.7) where λ̃ = αλ+ β γλ+ δ . (1.8) This property allows one to choose arbitrary linear transformations of initial operators A,B which leads merely to the Möbius transformation (1.8) of the spectral parameter λ while the eigenvector ψ(λ) remains the same. Define the so-called Darboux transformations for GEVP (1.1) (for details see, e.g. [20]). Assume that four operators T (1), T (2), Ã, B̃ exist such ÃT (1) = T A, B̃T (1) = T B (1.9) Then it is elementary to verify that the vector ψ̃(λ) ≡ T ψ(λ) (1.10) satisfies Ãψ̃(λ) = λB̃ψ̃(λ). (1.11) Thus the operator T (1) transforms a generalized eigenvector ψ(λ) to a new generalized eigenvector ψ̃(λ) of a new GEVP (1.11) with the same eigenvalue λ. Applying this procedure step-by-step, we can construct a set of GEVP Anψn(λ) = λBnψn(λ), n = 0, 1, . . . (1.12) Regular algebras, the generalized eigenvalue problem and Padé interpolation 335 such that An+1T (1) n = T (2) n An, Bn+1T (1) n = T (2) n Bn, (1.13) where eigenvectors ψn are related as ψn+1(λ) = T (1) n ψn(λ). (1.14) Thus the operator T (1) n can be considered as the upward operator, i.e. n → n + 1 on the space of eigenvector ψn. Note that the vector T (2) n ∗ ψ n+1(λ) belongs to a kernel space of the operator Y ∗ n (λ) = A∗n−λB ∗ n. Thus the operator T (2) n ∗ can be considered as “backward” operator, i.e. n+1 → n (but acting on the space of conjugated eigenvectors ψ n). There is a special case of the Darboux transformations (1.9) when T1 = B, T2 = B̃. (1.15) Then the second relation in (1.9) holds automatically and we need the only condition ÃB = B̃A. (1.16) In this case ψ̃(λ) = Bψ(λ). (1.17) Repeating this procedure we obtain the chain of vectors ψn+1(λ) = Bnψn(λ), n = 0, 1, . . . satisfying GEVP (1.12) with the condition An+1Bn = Bn+1An (1.18) Consider a simple example of the Darboux transformation of the type (1.15). Let A = ∂N x + ∑N−1 k=0 ak(x)∂ k x be a differential operator of the order N with unity coefficient at the highest derivative. Consider GEVP of the form Aψ(x;λ) = λb(x)ψ(x;λ), (1.19) where the operator B coincides with multiplication by a function b(x). Relation (1.16) reads