• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.243-257
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• 2022

In this paper, we present and prove an new theorem on symmetric functions. By using this theorem, we derive some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q)-Jacobsthal Lucas numbers with Chebyshev polynomials of the first kind.

• 대한수학회보
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• 제43권3호
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• pp.611-617
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• 2006

In this paper we construct q-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the q-analogue of alternating sums of powers of consecutive integers due to Euler.

• Journal of applied mathematics & informatics
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• 제38권5_6호
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• pp.601-609
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• 2020

In this paper, we introduce degenerate generalized poly-Bernoulli numbers and polynomials with (p, q)-logarithm function. We find some identities that are concerned with the Stirling numbers of second kind and derive symmetric identities by using generalized falling factorial sum.

• 호남수학학술지
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• 제42권3호
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• pp.425-447
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• 2020

The family of q-Konhauser matrix polynomials have been extended to Konhauser matrix polynomials. The purpose of the present work is to show that an extension of the explicit forms, generating matrix functions, matrix recurrence relations and Rodrigues-type formula for these matrix polynomials are given, our desired results have been established and their applications are presented.

• 대한수학회보
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• 제50권3호
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• 호남수학학술지
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• 제34권2호
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• pp.183-190
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• 2012

The main purpose of this paper is to introduce a new type of $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$): $\tilde{E}^{({\alpha},{\omega})}_{n,q}$ and $\tilde{E}^{({\alpha},{\omega})}_{n,q}(x)$, respectively. By using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_p$, we can obtain some results and derive some recurrence identities for the $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$).

• 호남수학학술지
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• 제34권1호
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• pp.1-9
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• 2012

The purpose of this paper is to give a new construction of the extended q-Euler numbers and polynomials of higher-order with weight by using p-adic q-integral on $\mathbb{Z}_p$.

• 대한수학회보
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• 제33권1호
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• pp.135-170
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• 1996

Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\phi(\cdot,\cdot)$ defined by $$ (1.1) $\phi(p,q) := (p,q)_N = \sum_{k=0}^{N} \int_{R}p^(k) (x)q^(k) (x) d\mu_k, $$ where each $d\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].

• Korean Journal of Mathematics
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• 제19권3호
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• pp.263-272
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• 2011

In the paper [1], an explicit correspondence between certain cubic irreducible polynomials over $\mathbb{F}_q$ and cubic irreducible polynomials of special type over $\mathbb{F}_{q^2}$ was established. In this paper, we show that we can mimic such a correspondence for quintic polynomials. Our transformations are rather constructive so that it can be used to generate irreducible polynomials in one of the finite fields, by using certain irreducible polynomials given in the other field.

• 대한수학회보
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• 제29권2호
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• pp.277-283
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• 1992

Let GF(q) be a finite field with q elements where q=p$^{n}$ for a prime number p and a positive integer n. Consider an arbitrary function .phi. from GF(q) into GF(q). By using the Largrange's Interpolation formula for the given function .phi., .phi. can be represented by a polynomial which is congruent (mod x$^{q}$ -x) to a unique polynomial over GF(q) with the degree < q. In [3], Wells characterized all polynomial over a finite field which commute with translations. Mullen [2] generalized the characterization to linear polynomials over the finite fields, i.e., he characterized all polynomials f(x) over GF(q) for which deg(f) < q and f(bx+a)=b.f(x) + a for fixed elements a and b of GF(q) with a.neq.0. From those papers, a natural question (though difficult to answer to ask is: what are the explicit form of f(x) with zero terms\ulcorner In this paper we obtain the exact form (together with zero terms) of a polynomial f(x) over GF(q) for which satisfies deg(f) < p$^{2}$ and (1) f(x+a)=f(x)+c for the fixed nonzero elements a and c in GF(q).