Overview
This research investigates the analytic structure and properties of the parametric harmonic zeta function, denoted as $\zeta_{H}\left(s, a, b\right)$. This function is characterized as a Dirichlet series associated with generalized harmonic numbers. The study focuses on establishing its meromorphic continuation across the complex plane and identifying the residues at its poles. A Taylor expansion for $\zeta_{H}\left(s, a, b+t\right)$ is derived, which facilitates the development of harmonic analogues and extensions of existing mathematical identities, including those attributed to Landau, Singh-Verma, and Srivastava.
Further, the paper constructs a systematic theoretical framework for the associated harmonic Stieltjes constants. This involves deriving explicit formulas for these constants, some of which are noted as previously unknown. A limit representation is also established specifically for the first-order harmonic Stieltjes constants. The research additionally introduces a harmonic analogue of the classical digamma function, outlining its primary analytic properties and its relationship to the harmonic Stieltjes constants. The study concludes by presenting Raabe-type formulas relevant to the parametric harmonic zeta function, the harmonic Stieltjes constants, and the harmonic digamma function.
Research Context
The study is grounded in the analysis of the parametric harmonic zeta function, $\zeta_{H}\left(s, a, b\right)$, which is defined as a Dirichlet series linked to generalized harmonic numbers. This function serves as the central object of inquiry, building upon established mathematical concepts such as Dirichlet series and the analytical properties of zeta functions. The research aims to extend the understanding of these functions by developing harmonic analogues of classical mathematical identities and functions, including those from Landau, Singh-Verma, and Srivastava, as well as the classical digamma function.
Approach
The research approach involves several analytical steps. Initially, the meromorphic continuation of $\zeta_{H}\left(s, a, b\right)$ to the complex plane is established. This process includes determining the residues at all identified poles of the function. Subsequently, a Taylor expansion for the function in the form $\zeta_{H}\left(s, a, b+t\right)$ is derived. This expansion serves as a basis for developing harmonic analogues and extensions of classical identities.
The systematic theory of harmonic Stieltjes constants is developed through the derivation of explicit formulas. This part of the approach also involves establishing a limit representation for the first-order constants. The construction of the harmonic analogue of the classical digamma function is undertaken, with its principal analytic properties and its connection to the harmonic Stieltjes constants being determined. The final step involves obtaining Raabe-type formulas for the parametric harmonic zeta function, the harmonic Stieltjes constants, and the harmonic digamma function.
Findings
- The parametric harmonic zeta function, $\zeta_{H}\left(s, a, b\right)$, possesses a meromorphic continuation to the complex plane.
- The residues at all poles of $\zeta_{H}\left(s, a, b\right)$ have been determined.
- A Taylor expansion for $\zeta_{H}\left(s, a, b+t\right)$ has been derived.
- Harmonic analogues and extensions of classical identities by Landau, Singh-Verma, and Srivastava have been established.
- A systematic theory for harmonic Stieltjes constants has been developed, including explicit formulas and identification of previously unknown cases.
- A limit representation for the first-order harmonic Stieltjes constants has been obtained.
- The harmonic analogue of the classical digamma function has been constructed, and its principal analytic properties and connection with harmonic Stieltjes constants have been established.
- Raabe-type formulas for the parametric harmonic zeta function, harmonic Stieltjes constants, and the harmonic digamma function have been obtained.