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.
展开▼
机译：已知在z +上的实J分数{dollar} {dollar} {lcub} ksb1的第{dollar} n {dollar}分母{dollar} Qsb {lcub} n {rcub}（z）{dollar} ellsb1 {rcub}-{lcub} ksb2 over z + ellsb2 {rcub}-{lcub} ksb3 over z + ellsb3 {rcub} -cdots'{dollar} {dollar}其中{dollar} ell sb {lcub} n {rcub} {dollar} {dollar} in {dollar} IR {dollar}，{dollar} n {dollar} {dollar} geq {dollar} 1和{dollar} ksb {lcub} n {rcub}> 0，ngeq { dollar} 2形成关于{dollar} IR {dollar}上的分布函数{dollar} psi（t）{dollar}的正交多项式序列（OPS），反过来，每个OPS都构成{dollar} {lcub} Qsb {lcub} n {rcub}（z）{rcub} {dollar}是实J分数的分母的序列。使用连续分数的单独收敛结果，我们获得了正交多项式和连续分数族的渐近性质。我们首先考虑一个带有{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}和{dollar} {dollar} bsb {lcub} n {rcub}：= {lcub} 1超过4lbrack 4（n + a）sp2-1rbrack {rcub} ，in IR leftlbrack- {lcub} 1over2 {rcub}，-{lcub} 3over2 {rcub}，-{lcub} 5over2 {rcub}，cdotsrightrbrack。{dollar} {dollar}首先证明了分母的显式封闭形式表达式。 ，然后证明{dollar} {lcub} zsp {lcub} -n {rcub} Qsb {lcub} n {rcub}（z）{rcub} {dollar}收敛到整个函数{dollar} Q（z）{dollar }。 {dollar} Q（z）{dollar}是根据Bessel和伽马函数编写的。继续使用已开发的方法，我们接下来考虑与Jacobi多项式有关的连续分数的两个参数族，其中{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}和{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}表示{dollar} alpha，请获得doubc，asb {lcub} n {rcub} not = 0 {dollar}。我们证明{dollar} {lcub}（2z / n）sp {lcub} n {rcub} Qsb {lcub} n {rcub}（n / 2z）{rcub} {dollar}收敛到{dollar} esp {lcub}- z {rcub} {dollar}。最后，对于doubc {dollar}中的{dollar} ksb {lcub} n {rcub}，ellsb {lcub} n {rcub}，{dollar} ksb {lcub} n {rcub} not = {dollar} 0，我们证明{ dollar} {dollar} limlimitssb {lcub} n到infty {rcub} left（在n + 1上的{lcub} z {right）qsb {lcub} n {rcub}上的左（{lcub} n +1在z {rcub上） }右} =1。{美元} {美元}示例包括Jacobi多项式，关联的Jacobi多项式和例外Jacobi多项式。
展开▼
