On the Hyperharmonic Function

Year 2019, Volume: 23 Issue: Special [en], 187 - 193, 01.03.2019

Ayhan Dil

https://doi.org/10.19113/sdufenbed.453758

Cited By: 1

Abstract

In this paper we investigate some properties of Hyperharmonic function defined

$H_{z}^{(w)}=\frac{\left( z\right) _{w}}{z\Gamma\left( w\right) }\left(
\Psi\left( z+w\right) -\Psi\left( w\right) \right)$

where $\text{ \ \ }w\text{, }z+w\in\mathbb{C}\backslash\left( \mathbb{Z}^{-}\cup\left\{ 0\right\} \right).$

Using this definition we introduce harmonic numbers with complex index and
we give some series of these numbers. Also formulas for the calculation of
harmonic numbers with rational index are obtained. For the simplicity of
differentiation we reorganized representation of $H_{z}^{(w)}$. With the
help of this new form we get higher derivatives of Hyperharmonic function
more easily. Besides these, owing to the fact that the Hyperharmonic
function is composed of some important functions, we interested in properties
and connections of it. We get connections between Hyperharmonic function and
trigonometric functions. Infinite product representation, integral
representation and differentiation identities of this function also obtained.

Keywords

Harmonic numbers, Hyperharmonic numbers, Gamma function, Digamma function, Beta function

Hiperharmonik Fonksiyon Üzerine

Abstract

 Özet: Bu çalışmada

$H_{z}^{(w)}=\frac{\left( z\right) _{w}}{z\Gamma\left( w\right) }\left(
\Psi\left( z+w\right) -\Psi\left( w\right) \right)$

where $\text{ \ \ }w\text{, }z+w\in\mathbb{C}\backslash\left( \mathbb{Z}^{-}\cup\left\{ 0\right\} \right).$

eşitliği ile tanımlanan Hiperharmonik fonksiyonun bazı özellikleri araştırılmıştır. Bu tanımdan faydalanarak karmaşık indeksli harmonik sayılar tanıtılmış ve bu sayıların bazı serileri verilmiştir. Ayrıca rasyonel indeksli harmonik sayıların hesaplanması için formüller elde edilmiştir. $H_{z}^{(w)}$ fonksiyonunun türevlerinin daha kolay hesaplanabilmesi için, mevcut gösterim yeniden düzenlenmi¸stir. Bu yeni gösterim yardımıyla Hiperharmonik fonksiyonun yüksek mertebeli türevleri daha kolay hesaplanabilmektedir. Bunların yanı sıra, Hiperharmonik fonksiyonun özel bazı fonksiyonların birleşimi biçiminde ifade edilebildiği gerçeğinden hareketle, bazı özellikleri ve bağlantıları çalışılmıştır. Hiperharmonik fonksiyonun trigonometrik fonksiyonlarla ilişkileri elde edilmiş, sonsuz çarpım gösterimi, integral gösterimi ve bazı türevsel özdeşlikleri verilmiştir.

Keywords

Harmonik sayılar, Hiperharmonik sayılar, Gamma fonksiyonu, Digamma fonksiyonu, Beta fonksiyonu

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