• East Asian mathematical journal
• /
• v.31 no.1
• /
• pp.41-53
• /
• 2015

A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) ${\gamma}_k$ and the generalized Stieltjes constants ${\gamma}_k(a)$ have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants ${\gamma}_k(a)$ by choosing to use four known integral representations for the generalized zeta function ${\zeta}(s,a)$. As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants ${\gamma}_k$. Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.

• Honam Mathematical Journal
• /
• v.35 no.4
• /
• pp.639-645
• /
• 2013

We aim at presenting certain integral representations for the Riemann Zeta function ${\zeta}(s)$ at positive integer arguments by using some known integral representations of log ${\Gamma}(1+z)$ and ${\psi}(1+z)$.

• Honam Mathematical Journal
• /
• v.35 no.2
• /
• pp.137-146
• /
• 2013

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function.

• Journal of applied mathematics & informatics
• /
• v.40 no.1_2
• /
• pp.205-213
• /
• 2022

In this paper we give a simple and direct proof of an Euler integral representation for a special class of _q+1F_q,k k-hypergeometric functions for q ≥ 2. The values of certain ₃F_2,k and ₄F_3,k functions at $x=\frac{1}{k}$, some of which can be derived using other methods. We may conclude that for k = 1 the results are reduced to [3].

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1651-1672
• /
• 2023

In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u₀, u₁, . . . , u_m, v₀, v₁, v₂) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u₀, u₁, . . ., u_n, v) with coefficients in a set L_δ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is any non-trivial polynomial in the m + n + 2 variables u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is a distinguished polynomial in the n + 1 variables v₀, v₁, . . . , v_n. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.

• Journal of the Chungcheong Mathematical Society
• /
• v.26 no.4
• /
• pp.769-780
• /
• 2013

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways. Very recently, Choi [6] presented explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function. In the present sequel to the investigation [6], we evaluate the log-sine and log-cosine integrals involved in more complicated integrands than those in [6], by also using the Beta function.

• Honam Mathematical Journal
• /
• v.35 no.4
• /
• pp.701-706
• /
• 2013

Summation theorems for hypergeometric series $_2F_1$ and generalized hypergeometric series $_pF_q$ play important roles in themselves and their diverse applications. Some summation theorems for $_2F_1$ and $_pF_q$ have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series $_3F_2$(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

• East Asian mathematical journal
• /
• v.35 no.1
• /
• pp.23-30
• /
• 2019

Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati's work, we aim to investigate a generalized multi-index Mittag-Leffler function and, among possible numerous formulas, choose to present several formulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.