Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Let 𠑤 𠜆 ( ð ‘¥ ) ∶ = ( 1 − ð ‘¥ 2 ) 𠜆 − 1 / 2 and 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) be the ultraspherical polynomials with respect to 𠑤 𠜆 ( ð ‘¥ ) . Then, we denote the Stieltjes polynomials with respect to 𠑤 𠜆 ( ð ‘¥ ) by ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) satisfying ∫ 1 − 1 𠑤 𠜆 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) ð ‘¥ ð ‘š ð ‘‘ ð ‘¥ = 0 , 0 ≤ ð ‘š < ð ‘› + 1 , ∫ 1 − 1 𠑤 𠜆 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) ð ‘¥ ð ‘š ð ‘‘ ð ‘¥ ≠0 , ð ‘š = ð ‘› + 1 . In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) and the product ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) . Especially, we estimate the even-order derivative values of ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) and ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) at the zeros of ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) and the product ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) , respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) and ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) at the zeros of ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) and ð ¸ ð œ† , ð ‘› + 1 ( ð ‘¥ ) 𠑃 𠜆 , ð ‘› ( ð ‘¥ ) on a closed subset of ( − 1 , 1 ) , respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.