On sieved orthogonal polynomials. III. Orthogonality on several intervals

Abstract

We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is { x : − 2 c / ( 1 + c ) ⩽ T k ( x ) ⩽ 2 c / ( 1 + c ) } \{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\} , where c c is a positive parameter. The weight function of the polynomials of the second kind is { 1 − ( ( 1 + c ) 2 / 4 c ) T k 2 ( x ) } 1 / 2 / | U k − 1 ( x ) | {\{ 1 - ({(1 + c)^2}/4\operatorname {c} )T_k^2(x)\} ^{1/2}}/|{U_{k - 1}}(x)| when c ⩾ 1 c \geqslant 1 . When c > 1 c > 1 we pick up discrete masses located at the zeros of U k − 1 ( x ) {U_{k - 1}}(x) . The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their q q -analogues are also treated. Their continuous spectra are also the above mentioned set. The q q -analogues include a sieved version of the Rogers q q -ultraspherical polynomials and another set of q q -ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.