On the asymptotic behavior of Jacobi polynomials with first varying parameter

Oleg Szehr, Rachid Zarouf

Abstract: We investigate the large n<math><mi is="true">n</mi></math> behavior of Jacobi polynomials with varying parameters Pn(an+α,bn+β)(1−2λ2)<math><mrow is="true"><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mspace width="0.16667em" is="true"></mspace><mi is="true">b</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> for a,b>−1<math><mrow is="true"><mi is="true">a</mi><mo is="true">,</mo><mi is="true">b</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></math> and λ∈(0,1)<math><mrow is="true"><mi is="true">λ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mn is="true">0</mn><mo is="true">,</mo><mspace width="0.16667em" is="true"></mspace><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow></math>. This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case b=0<math><mrow is="true"><mi is="true">b</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">0</mn></mrow></math>, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of Pn(an+α,β)(1−2λ2),a>−1<math><mrow is="true"><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mspace width="0.16667em" is="true"></mspace><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow><mo is="true">,</mo><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></math> in terms of two integrals. The integrals’ asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if a∈(2λ1−λ,∞)<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mfrac is="true"><mrow is="true"><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">−</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">,</mo><mi is="true">∞</mi><mo is="true">)</mo></mrow></mrow></math> then λanPn(an+α,β)(1−2λ2)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi></mrow></msup><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> shows exponential decay and we derive simple exponential upper bounds in this region. If a∈(−2λ1+λ,2λ1−λ)<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mfrac is="true"><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">+</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">,</mo><mspace width="0.16667em" is="true"></mspace><mfrac is="true"><mrow is="true"><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">−</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">)</mo></mrow></mrow></math> then the decay of λanPn(an+α,β)(1−2λ2)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi></mrow></msup><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> is O(n−1/2)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn><mo is="true">/</mo><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> and if a∈{−2λ1+λ,2λ1−λ}<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">{</mo><mfrac is="true"><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">+</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">,</mo><mspace width="0.16667em" is="true"></mspace><mfrac is="true"><mrow is="true"><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">−</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">}</mo></mrow></mrow></math> then λanPn(an+α,β)(1−2λ2)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi></mrow></msup><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> decays as O(n−1/3)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn><mo is="true">/</mo><mn is="true">3</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math>. A new phenomenon occurs in the parameter range a∈(−1,−2λ1+λ)<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mo is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mfrac is="true"><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">+</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">)</mo></mrow></mrow></math>, where we find that the behavior depends on whether or not an+α<math><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">α</mi></mrow></math> is an integer: If a∈(−1,−2λ1+λ)<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mo is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mfrac is="true"><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">+</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">)</mo></mrow></mrow></math> and an+α<math><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">α</mi></mrow></math> is an integer then λanPn(an+α,β)(1−2λ2)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi></mrow></msup><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> decays exponentially. If a∈(−1,−2λ1+λ)<math><mrow is="true"><mi is="true">a</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mo is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mfrac is="true"><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">+</mo><mi is="true">λ</mi></mrow></mfrac><mo is="true">)</mo></mrow></mrow></math> and an+α<math><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">α</mi></mrow></math> is not an integer then λanPn(an+α,β)(1−2λ2)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mi is="true">a</mi><mi is="true">n</mi></mrow></msup><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo is="true">)</mo></mrow></mrow></msubsup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> may increase exponentially depending on the proximity of the sequence (an+α)n<math><msub is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">a</mi><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> to integers.