• 대한수학회보
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• 제36권4호
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• pp.779-797
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• 1999

We find necessary and sufficient conditions for an orthogonal polynomial system to be compatible with another orthogonal polynomial system. As applications, we find new characterizations of semi-classical and clasical orthorgonal polynomials.

• 대한수학회보
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• 제42권1호
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• pp.179-188
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• 2005

We give a method to derive partial differential equations for the product of any two classical orthogonal polynomials in one variable and thus find several new differential equations. We also explain with an example that our method can be extended to a more general case such as product of two sets of orthogonal functions.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.787-796
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• 2005

We generalize a characterization of classical orthogonal polynomials and obtain a two-variable analogue. Also an example is given arising from a second order partial differential equation.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.581-588
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• 2007

We reconsider the classical orthogonal polynomials which are solutions to a second order differential equation of the form $$l_2(x)y'(x)+l_1(x)y'(x)={\lambda}_ny(x)$$. We investigate two characterization theorems of F. Marcellan et all and K.H.Kwon et al. which gave necessary and sufficient conditions on $l_1(x)\;and\;l_2(x)$ for the above differential equation to have orthogonal polynomial solutions. The purpose of this paper is to give a proof that each result in their papers respectively is equivalent.

• 대한수학회논문집
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• 제10권2호
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• pp.471-482
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• 1995

We give a simple unified proof of various characterizations of discrete classical orthogonal polynomials including two new ones.

• 대한수학회지
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• 제50권5호
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• pp.1067-1082
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• 2013

In this paper, we will find some recurrence relations of classical multiple OPS between the same family with different parameters using the generating functions, which are useful to find structure relations and their connection coefficients. In particular, the differential-difference equations of Jacobi-Pineiro polynomials and multiple Bessel polynomials are given.

• 대한수학회지
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• 제34권4호
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• pp.973-1008
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• 1997

We reconsider the problem of calssifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form $$ \ell_2(x)y"(x) + \ell_1(x)y'(x) = \lambda_n y(x). $$ We first obtain new (algebraic) necessary and sufficient conditions on the coefficients $\ell_1(x)$ and $\ell_2(x)$ for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classification of all classical orthogonal polynomials : up to a real linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi.cobi.

• 대한수학회논문집
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• 제32권3호
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• pp.677-688
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• 2017

The composition of Jacobi type finite classes of the classical orthogonal polynomials with two generalized Riemann-Liouville fractional derivatives are considered. The outcomes are expressed in terms of generalized Wright function or generalized hypergeometric function. Similar composition formulas are also obtained by considering the generalized Riemann-Liouville and $Erd{\acute{e}}yi-Kober$ fractional integral operators.

• 대한수학회보
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• 제48권3호
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• pp.445-454
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• 2011

In this paper, we first find a raising operator and a lowering operator for multiple Bessel polynomials and then give a differential equation having multiple Bessel polynomials as solutions. Thus the differential equations were found for all multiple orthogonal polynomials that are orthogonal with respect to the same type of classical weights introduced by Aptekarev et al.

• 대한수학회지
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• 제41권6호
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• pp.977-994
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• 2004

A pair of quasi-definite linear functionals {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair if monic orthogonal polynomials (equation omitted) relative to u$_{0}$ and u$_1$, respectively, satisfy a relation (equation omitted) where $\sigma$$_{n}$ and T$_{n}$ are arbitrary constants and $\Delta$p = p($\chi$＋1) - p($\chi$) is the difference operator. We show that if {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair, then u$_{0}$ and u$_{1}$ must be discrete-semiclassical linear functionals. We also find conditions under which either u$_{0}$ or u$_1$ is discrete-classical.ete-classical.