Publication: Inner products involving q-differences: the little q-Laguerre-Sobolev polynomials
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2000-06
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Elsevier
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Abstract
In this paper, polynomials which are orthogonal with respect to the inner product $$\multline\langle p,r\rangle_S=\sum^\infty_{k=0}p(q^k)r(q^k) {(aq)^k(aq;q)_\infty\over(q;q)_k}\\ +\lambda\sum^\infty_{k=0} (D_qp)(q^k)(D_qr)(q^k){(aq)^k(aq;q)_\infty\over(q;q)_k},\endmultline$$ where $D_q$ is the $q$-difference operator, $\lambda\geq0,\ 0<q<1$ and $0<aq<1$, are studied. For these polynomials, algebraic properties and $q$-difference equations are obtained as well as their relation with the monic little $q$-Laguerre polynomials. Some properties of the zeros of these polynomials are also deduced. Finally, the relative asymptotics $\{Q_n(x)/p_n(x;a )\}_n$ on compact subsets of ${\bf C}\sbs[0,1]$ is given, where $Q_n(x)$ is the $n$th degree monic orthogonal polynomial with respect to the above inner product and $p_n(x;a )$ denotes the monic little $q$-Laguerre polynomial of degree $n$.
Description
22 pages, no figures.-- MSC codes: Primary 33C25; Secondary 33D45.-- Issue title: "Higher transcendental functions and their applications".
MR#: MR1765938 (2001d:33018)
Zbl#: Zbl 0957.33008
MR#: MR1765938 (2001d:33018)
Zbl#: Zbl 0957.33008
Keywords
Orthogonal polynomials, Sobolev orthogonal polynomials, Little q-Laguerre polynomials
Bibliographic citation
Journal of Computational and Applied Mathematics, 2000, vol. 118, n. 1-2, p. 1-22