Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials

Abstract

For the limit periodic J J -fraction K ( − a n / ( λ + b n ) ) K( - {a_n}/(\lambda + {b_n})) , a n {a_n} , b n ∈ C {b_n} \in \mathbb {C} , n ∈ N n \in \mathbb {N} , which is normalized such that it converges and represents a meromorphic function f ( λ ) f(\lambda ) on C ∗ := C ∖ [ − 1 , 1 ] {\mathbb {C}^{\ast } }: = \mathbb {C}\backslash [ - 1,1] , the numerators A n {A_n} and denominators B n {B_n} of its n n th approximant are explicitly determined for all n ∈ N n \in \mathbb {N} . Under natural conditions on the speed of convergence of a n {a_n} , b n {b_n} , n → ∞ n \to \infty , the asymptotic behaviour of the orthogonal polynomials B n {B_n} , A n + 1 {A_{n + 1}} (of first and second kind) is investigated on C ∗ {\mathbb {C}^{\ast } } and [ − 1 , 1 ] [ - 1,1] . An explicit representation for f ( λ ) f(\lambda ) yields continuous extension of f f from C ∗ {\mathbb {C}^{\ast } } onto upper and lower boundary of the cut ( − 1 , 1 ) ( - 1,1) . Using this and a determinant relation, which asymptotically connects both sequences A n {A_n} , B n {B_n} , one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences B n {B_n} , A n + 1 {A_{n + 1}} , n ∈ N n \in \mathbb {N} . This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for f ( λ ) f(\lambda ) yields meromorphic extension of f f from C ∗ {\mathbb {C}^{\ast } } across ( − 1 , 1 ) ( - 1,1) onto a region of a second copy of C \mathbb {C} which there is bounded by an ellipse, whose focal points ± 1 \pm 1 are first order algebraic branch points for f f . Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions K ( − a n ( z ) / ( λ ( z ) + b n ( z ) ) ) K( - {a_n}(z)/(\lambda (z) + {b_n}(z))) , where a n ( z ) {a_n}(z) , b n ( z ) {b_n}(z) , λ ( z ) \lambda (z) are holomorphic on a region in C \mathbb {C} . Finally, for T T -fractions T ( z ) = K ( − c n z / ( 1 + d n z ) ) T(z) = K( - {c_n}z/(1 + {d_n}z)) with c n → c {c_n} \to c , d n → d {d_n} \to d , n → ∞ n \to \infty , the exact convergence regions are determined for all c c , d ∈ C d \in \mathbb {C} . Again, explicit representations for T ( z ) T(z) yield continuous and meromorphic extension results. For all c c , d ∈ C d \in \mathbb {C} the regions (on Riemann surfaces) onto which T ( z ) T(z) can be extended meromorphically, are described explicitly.