• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.145-164
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• 2011

In the paper, we study with weighted sharing method the uniqueness of entire functions concerning nonlinear differential polynomials sharing one value and prove two uniqueness theorems, first one of which generalizes some recent results in [10] and [16]. Our second theorem will supplement a result in [17].

• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.345-355
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• 2010

In this paper, we investigate uniqueness problems of meromorphic functions sharing a small function with their differential polynomials, and give some results which are related to a conjecture of R. Br$\"{u}$ck, and also improve several previous results.

• Kyungpook Mathematical Journal
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• v.53 no.2
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• pp.191-205
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• 2013

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM. Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.155-166
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• 2022

In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of f(z)ⁿL(g) - a(z) and g(z)ⁿL(f) - a(z), where L(h) takes the derivatives h^(k)(z) or the shift h(z+c) or the difference h(z+c)-h(z) or the delay-differential h^(k)(z+c), where k is a positive integer, c is a non-zero constant and a(z) is a nonzero small function with respect to f(z) and g(z). The related uniqueness problems of complex delay-differential polynomials are also considered.

• Kyungpook Mathematical Journal
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• v.54 no.3
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• pp.349-363
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• 2014

In this paper, we shall utilize Nevanlinna value distribution theory to study the uniqueness problems on entire functions and differential polynomials sharing one value. Our theorems improve or generalize some results of Zhang and Lin, Chen, Zhang, Lin and Chen.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.439-449
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• 2019

In 2006, S. Lin and W. Lin introduced the definition of weakly weighted-sharing of meromorphic functions which is between "CM" and "IM". In this paper, using the notion of weakly weighted-sharing, we study the uniqueness of nonconstant homogeneous differential polynomials P[f] and P[g] generated by meromorphic functions f and g, respectively. Our results generalize the results due to S. Lin and W. Lin, and H.-Y. Xu and Y. Hu.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1145-1155
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• 2019

Let f be a transcendental meromorphic function, defined in the complex plane $\mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function T(r, f) in terms of the counting function of a homogeneous differential polynomial generated by f. Our result improves and generalizes some recent results.

• Honam Mathematical Journal
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• v.43 no.1
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• pp.141-151
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• 2021

The main object of the present paper is to introduce the generalized Laguerre matrix polynomials for two variables. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, generating functions and some recurrence relations are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.495-505
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• 2018

The uniqueness problems on entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study an entire function f(z) that shares a nonzero polynomial a(z) with $f^{(1)}(z)$, together with its linear differential polynomials of the form: $L=L(f)=a_1(z)f^{(1)}(z)+a_2(z)f^{(2)}(z)+{\cdots}+a_n(z)f^{(n)}(z)$, where the coefficients $a_k(z)(k=1,2,{\ldots},n)$ are rational functions and $a_n(z){\not{\equiv}}0$.