q-差分内积中的小q-Jacobi-Sobolev多项式

本文讨论了关于以下内积的正交多项式 :〈p(x) ,r(x)〉( u0 ,u(α,β) ) =∑∞k=0p(qk) r(qk) (qk-c) ak(b) k(q) k +λ∑∞k=0(Dqp) (qk) (Dqr) (qk) (aq) k(bq) k(q) k给出了它的一些代数性质以及和小 q-Jacobi多项式的关系 ,得到了在 C\(0 ,1 ]∪ H )的紧子集上Qn(x)P(α- 1,β- 1)n (x) n和 Pn(x)P(α- 1,β- 1)n (x) n的相对渐近性质 .其中 Qn(x)是 n次的小 q -Jacobi-Sobolev正交多项式 ,P(α- 1,β- 1)n (x)和 Pn(x)分别是关于线性泛函 u(α- 1,β- 1)和 u0 的首一的 n次正交多项式 .

Inner Product Involving q-differences: the Little q-Jacobi-Sobolev Polynomials

In this paper, polynomials which are orthogonal with respect to the inner product: 〈p(x), r(x)〉_ (u_0, u (α, β) ) =∑∞k=0p(q k)r(q k)(q k-c)a k(b)_k(q)_k+λ∑∞k=0(D_qp)(q k)(D_qr)(q k)(aq) k(bq)_k(q)_k are studied. For these polynomials, algebraic properties and their relation with the monic little q -Jacobi polynomials are obtained. The relative asymptotics Q_n(x)P (α-1,β-1) _n(x)_n and P_n(x)P (α-1,β-1) _n(x)_n on compact subsets of C\\ \∪H is also given, where Q_n(x) is the n -th degree little q -Jacobi-Sobolev polynomial, P (α-1,β-1) _n(x) and P_n(x) is the monic orthogonal polynomials of degree n with respect to the functionals u (α-1,β-1) and u_0, respectively.