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

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1243-1264
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• 2017

In this paper, we introduce w-Catalan polynomials as a generalization of Catalan polynomials and derive fourteen basic identities of symmetry in three variables related to w-Catalan polynomials and analogues of alternating power sums. In addition, specializations of one of the variables as one give us new and interesting identities of symmetry even for two variables. The derivations of identities are based on the p-adic integral expression for the generating function of the w-Catalan polynomials and the quotient of p-adic integrals for that of the analogues of the alternating power sums.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.603-618
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• 2015

We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the p-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.119-130
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• 2013

In this paper, we obtain some generalized symmetry identities involving a unified class of polynomials related to the generalized Bernoulli, Euler and Genocchi polynomials of higher-order attached to a Dirichlet character. In particular, we prove a relation between a generalized X version of the power sum polynomials and this unified class of polynomials.

• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.869-882
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• 2023

In this paper, we find induced differential equations to give explicit identities of these polynomials from the generating functions of 2-variable mixed-type Hermite polynomials. Moreover, we observe the structure and symmetry of the zeros of the 2-variable mixed-type Hermite equations.

• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.115-120
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• 2018

In this paper, we discover symmetric properties for generalized Carlitz's q-tangent polynomials.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.117-133
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• 2019

In the current investigation, we obtain the generating function for Hermite-based degenerate Bernoulli polynomials attached to ${\chi}$ of higher order using p-adic methods over the ring of integers. Useful identities, formulae and relations with well known families of polynomials and numbers including the Bernoulli numbers, Daehee numbers and the Stirling numbers are established. We also give identities of symmetry and additive property for Hermite-based generalized degenerate Bernoulli polynomials attached to ${\chi}$ of higher order. Results are supported by remarks and corollaries.

• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.597-614
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• 2015

In this paper, we study some properties of the generalized Apostol type Hermite-based polynomials. which extend some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of high order. Numerous properties of these polynomials and some relationships between $F_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ and $_HF_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

• Archives of design research
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• v.18 no.2 s.60
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• pp.173-188
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• 2005

Design has become a buzzword in the marketing community these days and has been considered as an important success factor for business. Unfortunately, however, little research to date has been conducted on consumer responses to brand logo designs, which visualize brand identities. This article reports the findings from an experimental study which examined consumer evaluations of logo designs through conjoint analysis. The study explored potential factors which would give rise to the differences of consumer preferences. The results indicate that in the case of cognitive brand concepts, search goods show high elaborateness and low symmetry, experience goods show high associativeness, and credence goods show low elaborateness and high symmetry in the logo design characteristics. Affective brand concepts reveal different results in the logo design characteristics: search goods indicate high elaborateness and low associativeness, experience goods show low naturalness, and credence goods show low elaborateness and high associativeness. Implications of the results are discussed from a theoretical and practical standpoint.