Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler’s -function. Let q be a positive integer greater than 1 and γ denote Euler’s constant. We show that all the numbers ψ(a/q)+ γ, (a, q)= 1, 1 a q, are transcendental. We also prove that at most one of the numbers γ, ψ(a/q), (a, q)= 1, 1 a q, is algebraic. © 2007 Elsevier Inc. All rights reserved. MSC: primary 11J81...

In this paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers Hn are established respectively, some results in [I. Muqattash and M. Yahdi, Infinite family of approximations of the Digamma function, Math. Comput. Modelling 43 (2006), 1329–1336.] are improved, and some remarks are given.

The table of Gradshteyn and Rhyzik contains some integrals that can be expressed in terms of the digamma function ψ(x) = d dx log Γ(x). In this note we present some of these evaluations.

We present some inequalities and a concavity property of the digamma function ψ = Γ′/Γ, where Γ denotes Euler’s gamma function. In particular, we offer a new characterization of Euler’s constant γ = 0.57721.... We prove that −γ is the minimum of the harmonic mean of ψ(x) and ψ(1/x) for x > 0. Mathematics Subject Classification (2010). 33B15, 39B62, 41A44.

Numerous series representations for various special functions and mathematical constants have been developed by many authors. The aim of this article is to establish two parameterized the digamma function that seem interesting due their independence from given parameters. Among particular cases our main findings, some are covered in examples.

The definite integral M(a) := 4 π ∫ π/2 0 x2 dx x2 + ln2(2e−a cosx) is related to the Laplace transform of the digamma function L(a) := ∫ ∞ 0 e−asψ(s+ 1) ds, by M(a) = L(a) + γ/a when a > ln 2. Certain analytic expressions for M(a) in the complementary range, 0 < a ≤ ln 2, are also provided.

If the 750 GeV resonance in the diphoton channel is confirmed, what are the measurements necessary to infer the properties of the new particle and understand its nature? We address this question in the framework of a single new scalar particle, called digamma (z). We describe it by an effective field theory, which allows us to obtain general and model-independent results, and to identify the mo...

In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bou...