• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.63-79
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• 1997

Consider an oparator equation of the form : $$ (1.1) H[y](x) = \alpha(x)\delta^2 y(x) + \beta(x)\delta y(x) = \lambda_n y(x), $$ where $\alphs(x)$ and $\beta(x)$ are polynomials of degree at most two and one respectively, $\lambda_n$ is the eigenvalue parameter, and $\delta$ is Hahn's operator $$ (1.2) \delta f(x) = \frac{(q - 1)x + \omega}{f(qx + \omega) - f(x)}, $$ for real constants $q(\neq \pm 1)$ and $\omega$.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.935-950
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• 1997

When $\tau$ is a quasi-definite moment functional on P, the vector space of all real polynomials, we consider a symmetric bilinear form $\phi(\cdot,\cdot)$ on $P \times P$ defined by $$ \phi(p,q) = \lambad p(a)q(a) + \mu p(b)q(b) + <\tau,p'q'>, $$ where $\lambda,\mu,a$, and b are real numbers. We first find a necessary and sufficient condition for $\phi(\cdot,\cdot)$ and show that such orthogonal polynomials satisfy a fifth order differential equation with polynomial coefficients.