A new system of rational functions is found which is biorthogonal on an elliptic grid. These functions are expressed in terms of modular hypergeometric functions appeared in a paper by Frenkel and Turaev on elliptic solutions of the Yang–Baxter equation. The functions, thus constructed, are the first explicit example beyond the frames of the famous Askey–Wilson scheme of orthogonal special functions. A new approach is proposed to construct explicit systems of orthogonal polynomials and functions. This approach is based on numerical algorithms like Lanczos and Bauer and related discrete integrable systems like the Toda chain. A new approach is proposed to construct Krall-type orthogonal polynomials which are eigensolutions of higher-order differential and difference operators.
Main publications:
Vinet L., Yermolayeva O., Zhedanov A. A method to study the Krall and q-Krall polynomials // J. Comput. Appl. Math., 2001, 133, 647–656.
Vinet L., A. Zhedanov. Generalized Little q-Jacobi Polynomials as Eigensolutions of Higher-Order q-Difference Operators // Proc. Amer. Math. Soc., 2001, 129, 1317–1327.
Spiridonov V., Zhedanov A. Classical biorthogonal rational functions on elliptic grids // C. R. Math. Acad. Sci. Canada, 2000, 22(2), 70–76.
Spiridonov V., Zhedanov A. Spectral transformation chains and some new biorthogonal rational functions // Comm. Math. Phys., 2000, 210, 49–83.
