Some results on the digamma function (original) (raw)
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The digamma function is defined for x>0 as a locally summable function on the real line by Ψ(x)=-γ+∫ 0 ∞ e -t -e -xt 1-e -t dt· In this paper we use the neutrix calculus to extend the definition for digamma function for the negative integers. Also, we consider some convolutions and neutrix convolutions of the digamma function and other distributions.
A family of pseudogamma functions: I
2008
Let RgeRsb0R \ge R\sb{0}RgeRsb0 with Rsb0R\sb{0}Rsb0 such that there are no zeros of the Riemann zeta function for Im(s)\Im(s)Im(s) less or equal to tfracRsb0+12\tfrac{R\sb{0}+1}{2}tfracRsb0+12 when Re(s)>tfrac12\Re(s) >\tfrac{1}{2}Re(s)>tfrac12. We construct a family of functions with respect to RRR, denoted by nabla(s)\nabla(s)nabla(s), which we call pseudogamma functions with RRR. Any such function shares some of the fundamental properties of Euler's Gamma function, including analyticity and absence of zeros and poles in the disks centered at s=tfrac12s =\tfrac{1}{2}s=tfrac12 of the radius R+3R +3R+3, in the complex plane. In this article, we construct a family of pseodugamma functions and give a formula towards estimating lower and upper bounds of each such pseudogamma function on the circle ∣s−tfrac12∣=R|s-\tfrac{1}{2}| =R∣s−tfrac12∣=R. This family of pseudogamma functions is a new tool crucial in our further work on the density hypothesis as in \cite{Dh7} and the Lindel\"of hypothesis as in \cite{Lh1}. The results in \cite{Dh7} and \cite{Lh1} constitute the key steps to prove the Riemann hypothesis in \cite{CGP}. For these applications, we compute lower bounds on log∣nabla(s)∣\log|\nabla(s)|log∣nabla(s)∣ in \cite{CMathematica}, give upper bounds of the Riemann Xi function via these pseudogamma functions in \cite{CAGP}, and obtain other estimates on pseudogamma functions in \cite{Lh0}.
A FAMILY OF PSEUDOGAMMA FUNCTIONS: II
Let R ≥ R 0 with R 0 such that there are no zeros of the Riemann zeta function for ℑ(s) less or equal to R0+1 2 when ℜ(s) > 1 2. We construct a family of functions with respect to R, denoted by ∇(s), which we call pseudogamma functions with R. Any such function shares some of the fundamental properties of Euler's Gamma function, including analyticity and absence of zeros and poles in the disks centered at s = 1 2 of the radius R + 3, in the complex plane. In this article, we construct a family of pseodugamma functions and give a formula towards estimating lower and upper bounds of each such pseudogamma function on the circle |s− 1 2 | = R. This family of pseudogamma functions is a new tool crucial in our further work on the density hypothesis as in [7] and the Lindelöf hypothesis as in [8]. The results in [7] and [8] constitute the key steps to prove the Riemann hypothesis in [18]. For these applications, we compute lower bounds on log |∇(s)| in [15], give upper bounds of the Riemann Xi function via these pseudogamma functions in [9], and obtain other estimates on pseudogamma functions in [16].
On Some Properties of the Trigamma Function
arXiv (Cornell University), 2023
In 1974, Gautschi proved an intriguing inequality involving the gamma function Γ. Precisely, he proved that, for z > 0, the harmonic mean of Γ(z) and Γ(1/z) can never be less than 1. In 2017, Alzer and Jameson extended this result to the digamma function ψ by proving that, for z > 0, the harmonic mean of ψ(z) and ψ(1/z) can never be less than −γ where γ is the Euler-Mascheroni constant. In this paper, our goal is to extend the results to the trigamma function ψ ′. We prove among other things that, for z > 0, the harmonic mean of ψ ′ (z) and ψ ′ (1/z) can never be greater than π 2 /6.
Two parameterized series representations for the digamma function
Applicable Analysis and Discrete Mathematics, 2022
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 series representations for the digamma function that seem interesting due to their independence from the given parameters. Among many particular cases of our two main findings, some are covered in the examples.
On Some Sums of Digamma and Polygamma functions
Many formula involving sums of digamma and polygamma functions, few of which appear in standard reference works or the literature, but which commonly arise in applications, are collected and developed. Along the way, a new evaluation for some particular members of the family of hypergeometric functions - 4F3(1) - is presented, and a connection is made with Euler sums. All proofs are sketched.
Some Inequalities for the p-Digamma Function
2014
Some inequalities involving the p-digamma function are presented. These results are the p-analogues of some recent results. Mathematics Subject Classification: 33B15, 26A48.
Construction of the Digamma Function by Derivative Definition
The Digamma and Polygamma functions are important tools in mathematical physics, not only for its many properties but also for the applications in statistical mechanics and stellar evolution. In many textbooks is found its develop almost by the same procedure. In this paper expressions for the Digamma and Polygamma functions, in terms of hypergeometric functions, are found through the derivative definition.