Notes on three conjectures involving the digamma and generalized digamma functions
نویسندگان
چکیده
منابع مشابه
Complete Monotonicities of Functions Involving the Gamma and Digamma Functions
In the article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [Γ(x+α+1)]1/(x+α) [Γ(x+1)]1/x , [Γ(x+1)]1/x (x+1)α and [Γ(x+1)]1/x xα in x ∈ (−1,∞) for α ∈ R are obtained. In the final, three open problems are posed.
متن کاملMultivariate Digamma Distribution
where x = l , 2, . . . ; r > 0 , a > I (ar a + r > 0 , (a)~=a(a+l)...(a-kx--1), and r log F(z)/dz is the digamma or the psi function. The distribution (1.1) will be referred to as DGa (a, r). In this paper we extend DGa (a, r) to a multivariate distribution. The digamma distribution is closely related to the logarithmic series distribution LSr (0) defined by (1.2) P r [ X = x ; 0 ] = 0 ~ / ( l ...
متن کاملDigamma, what next?
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...
متن کاملTranscendental values of the digamma function
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...
متن کاملNotes on the Generalized Ramanujan Conjectures
Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigenvalues of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associa...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Inequalities and Applications
سال: 2018
ISSN: 1029-242X
DOI: 10.1186/s13660-018-1936-z