Biorthogonal Rational Functions Research Articles

Bispectrality and biorthogonality of the rational functions of q-Hahn type

We introduce families of rational functions that are biorthogonal with respect to the q-hypergeometric distribution. A triplet of q-difference operators X, Y, Z is shown to play a role analogous to the pair of bispectral operators of orthogonal polynomials. The recurrence relation and difference equation take the form of generalized eigenvalue problems involving the three operators. The algebra generated by X, Y, Z is akin to the algebras of Askey–Wilson type in the case of orthogonal polynomials. The actions of these operators in three different bases are presented. Connections with Wilson's ϕ910 biorthogonal rational functions are also discussed.

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

A generalized inverse eigenvalue problem and m-functions

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil (zJ[0,n]−H[0,n]) of (n+1) by (n+1) matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix H[0,n], characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of J[0,n] which is often an assumption in the direct problem is also isolated. Further, the reconstruction of H[0,n] is viewed through the inverse of the pencil (zJ[0,n]−H[0,n]) which involves the concept of m-functions.

About families of orthogonal polynomials satisfying Heun’s differential equation

We consider special families of orthogonal polynomials satisfying differential equations. Besides known hypergeometric cases, we look especially for Heun’s differential equations. We show that such equations are satisfied by orthogonal polynomials related to some classical weight functions modified by Dirac weights or by division of powers of binomials. An appropriate set of biorthogonal rational functions, or 2-point Padé approximations, is also described.

A unified algebraic underpinning for the Hahn polynomials and rational functions

An algebra denoted mH with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on mH. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of mH in U(sl2) is presented. A Padé approximation table for the binomial function is obtained as a by-product.

An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.

Biorthogonal rational functions of $R_{II}$-type

In this work, a sequence of orthonormal rational functions that is also biorthogonal to another sequence of rational functions arising from recurrence relations of $R_{II}$ type is constructed. The biorthogonality is proved by a procedure which we call Zhedanov method. A particular case of a sequence of orthonormal rational functions having denominators of special form is considered to motivate the general case. The particular case provides a Christoffel type transformation of the generalized eigenvalue problem with a reformulation different from the existing literature.

The Heun operator of Hahn-type

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.

Rahman\u2019s Biorthogonal Rational Functions and Superconformal Indices

We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman’s biorthogonal rational {}_{10}W_9-functions or their degenerate cases. We also give new bilateral extensions of the Jackson and q-Saalschütz summation formulas and new continuous and discrete biorthogonality measures for Rahman’s functions.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved.First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info.In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm.

On Dirichlet, Poncelet and Abel problems

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in\Bbb Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Pad\'e interpolation.

Generalized factorial Schur functions, biorthogonal rational functions and the RII chain

Generalized factorial Schur functions are introduced through a Newton-type interpolation. They are applied to construct a class of biorthogonal rational functions and a special solution to its spectral transformation chain, the RII chain. A two-parameter deformation of the Schur-type functions is also discussed.

The linear pencil approach to rational interpolation

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils z B − A of two tridiagonal matrices A , B , following Spiridonov and Zhedanov. In the present paper, as well as revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Padé approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to ∞ , we compute the spectrum and the numerical range of the underlying linear pencil explicitly.

Integrable discrete time chains for the Frobenius-Stickelberger-Thiele polynomials

The notion of Frobenius-Stickelberger-Thiele (FST) polynomials is introduced. Spectral transformations for these polynomials analogous to the Christoffel and Geronimus transformations for orthogonal polynomials are constructed. They yield an integrable discrete time chain (the FST chain) related to the generalized e-algorithm. Relations of the FST polynomials to the Pade interpolation problem and to general and symmetric biorthogonal rational functions are considered in detail.

Extensions of two q-series expansions with applications to biorthogonal rational functions

In this paper, we derive two most general possible q-hypergeometric expansion formulae for 12 Φ 11( q) and r Φ s ( z), respectively. The results are unique in the sense that no such results are available in the literature beyond 4 Φ 3( q) so far and have the advantage that these results are also applicable to the top-level 10 Φ 9( q) biorthogonal rational functions.

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.

Spectral Transformation Chains and Some New Biorthogonal Rational Functions

A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series 9 F 8, basic series 10ϕ9 and its elliptic analogue 10 E 9. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.

Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of RII type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.

Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised 10 ϕ 9 {_{10} \phi _{9}} basic hypergeometric series that is known to satisfy a transformation. We call this Φ \Phi and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for Φ \Phi we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle’s theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson’s results for rational biorthogonality, Watson’s q q -analogue of Ramanujan’s Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new q q -series identities are also obtained. One is an important three-term transformation for Φ \Phi ’s which generalizes all the known two- and three-term 8 ϕ 7 {_{8} \phi _{7}} transformations. Others are new and unexpected quadratic identities for these very-well-poised 8 ϕ 7 {_{8} \phi _{7}} ’s.

Q-Hermite Polynomials and Classical Orthogonal Polynomials

AbstractWe use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.