• Journal of applied mathematics & informatics
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• v.41 no.2
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• pp.439-448
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• 2023

We introduce several higher-order (p, q)-differential equation of which are related to (p, q)-Bernoulli polynomials. We also find some relations between (p, q)-Bernoulli, (p, q)-Euler, and (p, q)-Genocchi polynomials.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.381-402
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• 1999

Let {Rn($\chi$)}{{{{ { } atop {n=0} }}}} be a discrete Sobolev orthogonal polynomials (DSOPS) relative to a symmetric bilinear form (p,q)={{{{ INT _{ } }}}} pqd$\mu$0 +{{{{ INT _{ } }}}} p qd$\mu$1, where d$\mu$0 and d$\mu$1 are signed Borel measures on . We find necessary and sufficient conditions for {Rn($\chi$)}{{{{ { } atop {n=0} }}}} to satisfy a second order difference equation 2($\chi$) y($\chi$)+ 1($\chi$) y($\chi$)= ny($\chi$) and classify all such {Rn($\chi$)}{{{{ { } atop {n=0} }}}}. Here, and are forward and backward difference operators defined by f($\chi$) = f($\chi$+1) - f($\chi$) and f($\chi$) = f($\chi$) - f($\chi$-1).

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1529-1561
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• 2018

Let f and g be nonconstant meromorphic (entire, respectively) functions in the complex plane such that f and g are of finite order, let a and b be nonzero complex numbers and let n be a positive integer satisfying $n{\geq}21$ ($n{\geq}12$, respectively). We show that if the difference polynomials $f^n(z)+af(z+{\eta})$ and $g^n(z)+ag(z+{\eta})$ share b CM, and if f and g share 0 and ${\infty}$ CM, where ${\eta}{\neq}0$ is a complex number, then f and g are either equal or at least closely related. The results in this paper are difference analogues of the corresponding results from.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.541-552
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• 2013

This research is a continuation of a recent paper due to the first author in [9]. Different from previous results, we investigate the value distribution of difference polynomials of moromorphic functions in this paper. In particular, we are interested in the existence of zeros of $f(z)^n({\lambda}f(z+c)^m+{\mu}f(z)^m)-a$, where f is a moromorphic function, n, m are two non-negative integers, and ${\lambda}$, ${\mu}$ are non-zero complex numbers. However, the proof here is obviously different to the one in [9]. We also study difference polynomials of entire functions sharing a common value, which improves the result in [10, 13].

• Journal of Integrative Natural Science
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• v.7 no.2
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• pp.145-147
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• 2014

Let $T_n(x)$ be the Chebyshev polynomials of first kind of degree n. In this paper, we show that for a > 1, the polynomial with integer coefficients $\frac{T_n(z)-T_n(a)}{z-a}$ has all its roots in $|z|{\leq}a$.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.447-458
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• 2018

In this paper, we investigate the uniqueness problems of certain types of difference polynomials sharing a small function. The results of the paper improve and generalize the recent results due to H.P. Waghamore [Tbilisi Math. J. 11(2018), 1-13], P. Sahoo and B. Saha [App. Math. E-Notes. 16(2016), 33-44].

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.411-427
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• 2015

In this article, we main study the uniqueness problem of meromorphic function which difference polynomials sharing common values. We consider the entire function $(f^n(f^m-1)\prod_{j=1}^{s}f(z+c_j)^{{\mu}j})^{(k)}$ and the meromorphic function $f^n(f^m-1)\prod_{j=1}^{s}f(z+c_j)^{{\mu}j}$ to get the main results which extend Theorem 1.1 in paper[5] and theorem 1.4 in paper[6].

• Journal of the Korean Mathematical Society
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• v.50 no.5
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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.

• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.543-564
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• 1999

Consider a symmetric bilinear form defined on $\prod\times\prod$ by $_{\lambda\mu}$ = $<\sigma,fg>\;+\;\lambdaL[f](a)L[g](a)\;+\;\muM[f](b)m[g](b)$ ,where $\sigma$ is a quasi-definite moment functional, L and M are linear operators on $\prod$, the space of all real polynomials and a,b,$\lambda$ , and $\mu$ are real constants. We find a necessary and sufficient condition for the above bilinear form to be quasi-definite and study various properties of corresponding orthogonal polynomials. This unifies many previous works which treated cases when both L and M are differential or difference operators. finally, infinite order operator equations having such orthogonal polynomials as eigenfunctions are given when $\mu$=0.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.37-50
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• 2022

The purpose of the paper is to study the uniqueness problems of certain type of difference polynomials sharing a small function. With the concept of weakly weighted sharing and relaxed weighted sharing we obtain some results which extend and generalize some results due to P. Sahoo and G. Biswas [Tamkang Journal of Mathematics, 49(2)(2018), 85-97].