• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.695-708
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• 2022

In this paper, we present a new definition and generalization of the first and second kinds of Daehee numbers and polynomials with the higher order. Some new results for these polynomials and numbers are derived. Furthermore, some interesting special cases of the new generalized Daehee polynomials and numbers of higher order are deduced.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.343-353
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• 2016

A remarkably large number of integrals whose integrands are associated, in particular, with a variety of special functions, for example, the hypergeometric and generalized hypergeometric functions have been recorded. Here we aim at presenting certain (presumably) new and (potentially) useful integral formulas whose integrands are involved in a product of $_2F_1$, Srivastava polynomials, and $Kamp{\acute{e}}$ de $F{\acute{e}}riet$ functions. The main results are derived with the help of some known definite integrals obtained earlier by Qureshi et al. [4]. Some interesting special cases of our main results are also considered.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.831-845
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• 2014

In this paper, we obtain new generating functions involving families of pairs of inverse functions by using a generalization of the Srivastava's theorem [H. M. Srivastava, Some generalizations of Carlitz's theorem, Pacific J. Math. 85 (1979), 471-477] obtained by Tremblay and Fug$\grave{e}$ere [Generating functions related to pairs of inverse functions, Transform methods and special functions, Varna '96, Bulgarian Acad. Sci., Sofia (1998), 484-495]. Special cases are given. These can be seen as generalizations of the generalized Bernoulli polynomials and the generalized degenerate Bernoulli polynomials.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1427-1437
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• 2016

Cauchy polynomials are also called Bernoulli polynomials of the second kind and these polynomials are very important to study mathematical physics. D. S. Kim et al. have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus. T. Kim introduced the degenerate Cauchy numbers and polynomials which are derived from the degenerate function $e^t$. Recently J. Jeong, S. H. Rim and B. M. Kim studied on finite times degenerate Cauchy numbers and polynomials. In this paper we consider finite times degenerate higher-order Cauchy numbers and polynomials, and give some identities and properties of these polynomials.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.429-438
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• 2015

In this paper, we discuss how the operational calculus can be exploited to the theory of mixed generating functions. We use operational methods associated with multi-variable Hermite polynomials, Laguerre polynomials and Bessels functions to drive identities useful in electromagnetism, fluid mechanics etc. Certain special cases giving bilateral generating relations related to these special functions are also discussed.

• Journal of applied mathematics & informatics
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• v.38 no.1_2
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• pp.145-157
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• 2020

In this article, we define a generating function of the generalized (p, q)-poly-Bernoulli polynomials with variable a by using the polylogarithm function. From the definition, we derive some properties that is concerned with other numbers and polynomials. Furthermore, we construct a special functions and give some symmetric identities involving the generalized (p, q)-poly-Bernoulli polynomials and power sums of the first integers.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.413-423
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• 2022

In this paper, we prove some results concerning the zeros of Bi-complex polynomials. These results as special cases include Grace's theorem and related results.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.293-301
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• 2022

The present paper concerns with a study of certain generating functions and summation formulas of generalized Poisson-Charlier polynomials. Some special cases are also discussed.

• 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].

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.489-494
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• 2003

In this paper, we first establish an interesting theorem involving the $\bar{H}$-function introduced by Inayat-Hussain ([7], [8]). The convergence and existence condition, basic properties of this function were given by Buschman and Srivastava ([2]). Next, we obtain certain new integrals and an expansion formula by the application of our theorem. On account of the most general nature of the functions involved herein, our main findings are capable of yielding a large number of new, interesting and useful integrals, expansion formulae involving simple special functions and polynomials as their special cases. A known special case of our main theorem in also given ([11]).