• Journal of the Korean Mathematical Society
• /
• v.47 no.3
• /
• pp.645-657
• /
• 2010

In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences.

• Journal of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.1045-1073
• /
• 2014

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359].

• Communications of the Korean Mathematical Society
• /
• v.20 no.2
• /
• pp.395-403
• /
• 2005

The authors aim mainly at giving fifteen three-term contiguous relations for the basic hypergeometric series $series\;_2{\phi}_1$ corresponding to Gauss's contiguous relations for the hypergeometric series $series\;_2F_1$ given in Rainville([6], p.71). They also apply them to obtain two summation formulas closely related to a known q-analogue of Kummer's theorem.

• Journal of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1159-1172
• /
• 2019

A semigroup S is called a weakly abundant semigroup if its every $\tilde{\mathcal{L}}$-class and every $\tilde{\mathcal{R}}$-class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set E(S) of a weakly abundant semigroup S being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type Q semigroup which is a left quasi-abundant semigroup having the PC condition.

• Journal of the Korean Mathematical Society
• /
• v.43 no.1
• /
• pp.111-131
• /
• 2006

The goal of this paper is to define p-adic Hardy sums and p-adic q-higher-order Hardy-type sums. By using these sums and p-adic q-higher-order Dedekind sums, we construct p-adic continuous functions for an odd prime. These functions contain padic q-analogue of higher-order Hardy-type sums. By using an invariant p-adic q-integral on $\mathbb{Z}_p$, we give fundamental properties of these sums. We also establish relations between p-adic Hardy sums, Bernoulli functions, trigonometric functions and Lambert series.

• Journal of applied mathematics & informatics
• /
• v.36 no.5_6
• /
• pp.567-574
• /
• 2018

Let $f:f(z)=z+{\sum^{{\infty}}_{n=2}}a_nz^n$ be analytic in the open unit disc E. Then f is said to belong to the class $M_{\alpha}$ of alpha-convex functions, if it satisfies the condition ${\Re}\{(1-{{\alpha})}{\frac{zf^{\prime}(z)}{f(z)}}+{{\alpha}}{\frac{(zf^{\prime}(z))^{\prime})}{f^{\prime}(z)}}\}$ > 0, ($z{\in}E$). In this paper, we introduce and study q-analogue of the class $M_{\alpha}$ by using concepts of Quantum Analysis. It is shown that the functions in this new class $M(q,{\alpha})$ are q-starlike. A problem related to q-Bernardi operator is also investigated.

• Journal of the Korean Mathematical Society
• /
• v.40 no.1
• /
• pp.1-28
• /
• 2003

The Iwahori-Hecke algebra $H_{k}$ ( $q^2$) of type A acts on the k-fold tensor product space of the natural representation of the quantum superalgebra (equation omitted)$_{q}$(gl(m, n)). We show the Hecke algebra $H_{k}$ ( $q^2$) and the quantum superalgebra (equation omitted)$_{q}$(gl(m n)) have commuting actions on the tensor product space, and determine the centralizer of each other. Using this result together with Gyoja's q-analogue of the Young symmetrizers, we construct highest weight vectors of irreducible summands of the tensor product space.

• Communications of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.65-94
• /
• 2016

In 1812, Gauss obtained fifteen contiguous functions relations. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gauss's 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper [16] published in Computer & Mathematics with Applications 61 (2011), 620.629. In this paper, first we obtained 15 q-contiguous functions relations due to Henie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q-contiguous functions relations. These q-contiguous functions relations have wide applications.

• Nonlinear Functional Analysis and Applications
• /
• v.26 no.2
• /
• pp.331-345
• /
• 2021

Let p(z)be a polynomial of degree n. Then Bernstein's inequality [12,18] is $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;n\;{\max_{{\mid}z{\mid}=1}{\mid}(z){\mid}}$$. For q > 0, we denote $${\parallel}p{\parallel}_q=\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}$$, and a well-known fact from analysis [17] gives $${{\lim_{q{\rightarrow}{{\infty}}}}\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}={\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. Above Bernstein's inequality was extended by Zygmund [19] into L^q norm by proving ║p'║_q ≤ n║p║_q, q ≥ 1. Let p(z) = a₀ + ∑ⁿ_𝜈=𝜇 a_𝜈z^𝜈, 1 ≤ 𝜇 ≤ n, be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then for 0 < r ≤ R ≤ k, Aziz and Zargar [4] proved $${\max\limits_{{\mid}z{\mid}=R}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{nR^{{\mu}-1}(R^{\mu}+k^{\mu})^{{\frac{n}{\mu}}-1}}{(r^{\mu}+k^{\mu})^{\frac{n}{\mu}}}\;{\max\limits_{{\mid}z{\mid}=r}}\;{\mid}p(z){\mid}}$$. In this paper, we obtain the L^q version of the above inequality for q > 0. Further, we extend a result of Aziz and Shah [3] into L^q analogue for q > 0. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Advances in Traditional Medicine
• /
• v.10 no.4
• /
• pp.239-253
• /
• 2010

Coenzyme $Q_{10}$ ($CoQ_{10}$, or ubiquinone) is an electron carrier of the mitochondrial respiratory chain (electron transport chain) with antioxidant properties. In view of the involvement of $CoQ_{10}$ in oxidative phosphorylation and cellular antioxidant protection a deficiency in this quinone would be expected to contribute to disease pathophysiology by causing a failure in energy metabolism and antioxidant status. Indeed, a deficit in $CoQ_{10}$ status has been determined in a number of neuromuscular and neurodegenerative disorders. Primary disorders of $CoQ_{10}$ biosynthesis are potentially treatable conditions and therefore a high degree of clinical awareness about this condition is essential. A secondary loss of $CoQ_{10}$ status following HMG-CoA reductase inhibitor (statins) treatment has been implicated in the pathophysiology of the myotoxicity associated with this pharmacotherapy. $CoQ_{10}$ and its analogue, idebenone, have been widely used in the treatment of neurodegenerative and neuromuscular disorders. These compounds could potentially play a role in the treatment of mitochondrial disorders, Parkinson's disease, Huntington's disease, amyotrophic lateral sclerosis, Friedreich's ataxia, and other conditions which have been linked to mitochondrial dysfunction. This article reviews the physiological roles of $CoQ_{10}$, as well as the rationale and the role in clinical practice of $CoQ_{10}$ supplementation in different neurological diseases, from primary $CoQ_{10}$ deficiency to neurodegenerative disorders. These will help in future for treatment of patients suffering from neurodegenerative disease.