In this paper we consider the summation of some Polygamma[n, f(x)] function from interval x equals to N_1 to interval x equals to N_2, where the value of f(x) along this interval includes some ...

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers.

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers.

The polygamma function of order $n$ is defined as the $n+1$th derivative of the gamma function, $\Gamma(z)$. $$ \Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t} dt $$ The ...

We obtain a variety of series and integral representations of the digamma function \psi (a) ψ(a). These in turn provide representations of the evaluations \psi (p/q) ψ(p/q) at rational argument and ...

An illustration of a magnifying glass. An illustration of a magnifying glass.

In a recent paper, Sarikaya et al. introduced a new analogue of the classical Gamma function which they called, the conformable Gamma function. Motivated by their results, this paper seeks to ...

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Special functions occupy a central role in mathematical analysis, bridging pure theory and practical application across diverse scientific fields. Their intrinsic properties—such as recurrence ...