| It is known that the {dollar}n{dollar}th denominators {dollar}Qsb{lcub}n{rcub}(z){dollar} of a real J-fraction{dollar}{dollar}{lcub}ksb1 over z + ellsb1{rcub}-{lcub}ksb2 over z + ellsb2{rcub}-{lcub}ksb3 over z + ellsb3{rcub}-cdots'{dollar}{dollar}where {dollar}ell sb{lcub}n{rcub}{dollar} {dollar}in{dollar} {dollar}IR{dollar}, {dollar}n{dollar} {dollar}geq{dollar} 1 and {dollar}ksb{lcub}n{rcub}>0,ngeq{dollar} 2, form an orthogonal polynomial sequence (OPS) with respect to a distribution function {dollar}psi(t){dollar} on {dollar}IR{dollar} and, conversely, every OPS, {dollar}{lcub}Qsb{lcub}n{rcub}(z){rcub}{dollar}, is the sequence of denominators of a real J-fraction. Using separate convergence results for continued fractions, we obtain asymptotic properties for families of orthogonal polynomials and continued fractions.; We begin by considering a one parameter family of continued fractions with{dollar}{dollar}eqalign{lcub}&ksb1 := bsb1,qquad ksb{lcub}n{rcub} := bsb{lcub}2n-2{rcub} bsb{lcub}2n-1{rcub},qquad n geq 2,cr&ellsb1 := bsb2,qquad ellsb{lcub}n{rcub} := bsb{lcub}2n-1{rcub} + bsb{lcub}2n{rcub},qquad n geq 2,cr{rcub}{dollar}{dollar}and{dollar}{dollar}bsb{lcub}n{rcub} := {lcub}1 over 4lbrack 4(n + a)sp2 - 1rbrack {rcub}, a in IR leftlbrack-{lcub}1over2{rcub},-{lcub}3over2{rcub},-{lcub}5over2{rcub},cdotsrightrbrack.{dollar}{dollar}First proving an explicit closed form expression for the denominators, we then prove {dollar}{lcub}zsp{lcub}-n{rcub}Qsb{lcub}n{rcub}(z){rcub}{dollar} converges to an entire function {dollar}Q(z){dollar}. {dollar}Q(z){dollar} is written in terms of Bessel and gamma functions.; Continuing with developed methods, we next consider the two parameter family of continued fractions related to Jacobi polynomials where{dollar}{dollar}eqalign{lcub}&ksb1 := asb1,qquad ksb{lcub}n{rcub} := asb{lcub}2n-2{rcub}a sb{lcub}2n-1{rcub},qquad ngeq 2,cr&ellsb1 := asb2,qquad ellsb{lcub}n{rcub} := asb{lcub}2n-1{rcub}+asb{lcub}2n{rcub},qquad ngeq 2,cr{rcub}{dollar}{dollar}and {dollar}asb1 := 1{dollar}, {dollar}{dollar}eqalign{lcub}asb{lcub}2n{rcub}&:= {lcub}-(beta-alpha+n-{lcub}1over2{rcub})(alpha+n-1)over(beta+2n-2)(beta+2n-1){rcub},cr asb{lcub}2n+1{rcub}&:= {lcub}-(beta-alpha+n)(alpha+n-{lcub}1over2{rcub})over(beta+2n-1)(beta+2n){rcub}, ngeq 1,cr{rcub}{dollar}{dollar}for {dollar}alpha,betain doubc, asb{lcub}n{rcub}not=0{dollar}. We prove {dollar}{lcub}(2z/n)sp{lcub}n{rcub}Qsb{lcub}n{rcub}(n/2z){rcub}{dollar} converges to {dollar}esp{lcub}-z{rcub}{dollar}.; Lastly, for {dollar}ksb{lcub}n{rcub},ellsb{lcub}n{rcub} in doubc{dollar}, {dollar}ksb{lcub}n{rcub} not={dollar} 0, we prove{dollar}{dollar}limlimitssb{lcub}n to infty{rcub}left({lcub}z over n + 1{rcub}right)Qsb{lcub}n{rcub}left({lcub}n + 1 over z{rcub}right) = 1.{dollar}{dollar}Examples include Jacobi polynomials, associated Jacobi polynomials and exceptional Jacobi polynomials. |
