IORTHOGONALITY OF RI, RII POLYNOMIALS AND COMPLEMENTARY CHAIN SEQUENCES

Abstract

The classical theory of orthogonal polynomials has found several applications in recent years, particularly, in the areas of spectral theory and mathematical physics. Many advancements are obtained through the matrix representations and associated eigenvalue problems of the orthogonal polynomials as well as the continued fraction expansions of the special functions that arise in such studies. The underlying theme of the thesis on one hand is to explore both structural and qualitative aspects of perturbations of the continued fraction parameters in case of special functions and recursion coe cients in case of orthogonal polynomials and on the other hand to obtain biorthogonality relations of the related functions. The structural and qualitative aspects of two perturbations in the parameters of a g-fraction is studied. The rst perturbation is when a nite number of parameters gj are missing in which case we call the corresponding g-fraction a gap-g-fraction. Using one of the gap-g-fractions, a class of Pick functions is identi ed. The second case is replacing fgng1 n=0 by a new sequence fg( k) n g1 n=0 in which the jth term gj is replaced by g( k) j and the results are illustrated using the Schur and Carath eodory functions. The consequences of the map mn 7! 1 􀀀 mn, where mn, n 0, is the minimal parameter sequence of a chain sequence, are explored in case of polynomials orthogonal both on the real line and on the unit circle. In this context, the concept of complementary chain sequence is introduced. It is shown, in particular, how the map can be useful in characterizing chain sequences with a single parameter sequences. The map F( ) 7! F( 2), where F( ) is a general T-fraction, is used to de ne generalized Jacobi pencil matrices. The denominators of the approximants of a T- fraction satisfy a recurrence relation of RI type, with which is associated a sequence of Laurent polynomials. The biorthogonality relations of these Laurent polynomials are i discussed. This serves as the bridge between the two parts of the thesis and provides a gradual transition from perturbation theory to the concept of biorthogonality. The sequence fQn( )g1 n=0, where Qn( ) := Pn( )+ nPn􀀀1( ), n 2 Rnf0g, n 0, is considered with fPn( )g1 n=0 satisfying Pn+1( ) = n( 􀀀 n)Pn( ) + n( 􀀀 n)Pn􀀀1( ); n 1: A unique sequence f ng1 n=0 is constructed such that fQn( )g1 n=0 not only satis es mixed recurrence relations of RI and RII type but also Qn(1) = 0, 8n 1. The polynomials Qn( ), n 1, are shown to satisfy biorthogonality relations with respect to a discrete measure that follows from their eigenvalue representations. With certain additional conditions, a para-orthogonal polynomial is also obtained from Qn( ). The recurrence relation of RII type On+1( ) = n( 􀀀 n)On(z) + n( 􀀀 an)( 􀀀 bn)On􀀀1( ); n 1; is used to construct a sequence of orthogonal rational functions f'n( )g satisfying two properties. First, the related matrix pencil has the numerator polynomials On( ) as the characteristic polynomials and 'n( ) as components of the eigenvectors. Second, the orthogonal sequence f'n( )g is also biorthogonal to another sequence of rational functions. A Christo el type transformation of the orthogonal rational functions so constructed is also obtained, illustrating the di erences with the results available in the literature. There is a conscious e ort to give the results obtained in the thesis a proper context in the vast theory of orthogonal polynomials and biorthogonality. At the same time, future direction of research is also provided wherever possible.