Hiperharmonik Fonksiyon Üzerine

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.

Anahtar Kelimeler:

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

On the Hyperharmonic Function

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,

PDF

___

[1] Abramowitz, M., Stegun, I. 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York, 1046s.
[2] Andrews, G. E., Askey, R., Roy, R. 2000. Special Functions, Cambridge University Press, 682s.
[3] Bak, J., Newman, D. J. 1997. Complex Analysis, Springer, 328s.
[4] Conway, J. H., Guy, R. K. 1996. The Book of Numbers, New York, Springer-Verlag, 310s.
[5] Dil, A., Mez˝o, I., Cenkci, M. 2017. Evaluation of Euler-like sums via Hurwitz zeta values. Turk. J. Math., 41(6), 1640-1655.
[6] Dil A, Boyadzhiev KN. 2015. Euler sums of hyperharmonic numbers. J. Number Theory, 147: 490-498.
[7] Gaboury S. 2014. Further Expansion and Summation Formulas Involving the Hyperharmonic Function. Commun. Korean Math. Soc., 29 (2): 269-83.
[8] Gradshteyn, I. S., Ryzhik, I. M. 2007. Table of Integrals, Series, and Products, Elsevier Academic Press, USA, 1163s.
[9] Medina, L. A., Moll, V. H. 2009. The Integrals in Gradshteyn and Ryzhik. Part 10: The Digamma Function. SCIENTIA, Series A: Mathematical Sciences, Vol. 17, 45–66.
[10] Mez˝o, I. 2009. Analytic Extension of Hyperharmonic Numbers. Online Journal of Analytic Combinatorics, Issue 4 (2009).
[11] Mez˝o, I., Dil, A. 2010. Hyperharmonic Series Involving Hurwitz Zeta Function. J. Number Theory, 130, 2: 360-369.
[12] Milne-Thomson, L. M. 1965. The Calculus of Finite Differences. MacMillan & Co., 558s.
[13] Rainville, E. D. 1960. Special Functions. MacMillan, New York, 365s.
[14] Sofo, A., Srivastava, H. M. 2015. A Family of Shifted Harmonic Sums. Ramanujan J. 37, 89-108.
[15] Sofo, A. 2014. Shifted Harmonic Sums of Order Two. Commun. Korean Math. Soc. 29 (2), 239-255.
[16] Wang, Z. X., Guo, D. R. 1989. Special Functions, World Scientific, 720s.
[17] Xu, C. 2018. Euler Sums of Generalized Hyperharmonic Numbers. J. Korean Math. Soc. 55, No. 5, 1207-1220.
[18] Xu, C. 2018. Computation and Theory of Euler Sums of Generalized Hyperharmonic Numbers. C. R. Acad. Sci. Paris, Ser. I 356, 243-252.
[19] Xu, C. 2017. Identities for the Shifted Harmonic Numbers and Binomial Coefficients. Filomat 31:19, 6071-6086.