C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function

arXiv2212.04383MaRDI QIDQ6419984

Publication date: 8 December 2022

Abstract: Let

f (t) = s u m_{n = 0}^{+ i n f t y} f r a c C_{f, n} n! t^{n}

be an analytic function at

0

, and let

be the sequence of Appell polynomials, referred to as

e x t i t C - p o l y n o m i a l s a s s o c i a t e d t o f

, constructed from the sequence of coefficients

C_{f, n}

. We also define

P_{f, n} (x)

as the sequence of C-polynomials associated to the function

p_{f} (t) = f (t) (e^{t} - 1) / t

, called

e x t i t P - p o l y n o m i a l s a s s o c i a t e d t o f

. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on

f

, we introduce and study the complex-variable function

, which generalizes the

s^{z}

function and is denoted by

s^{(z, f)}

. Thirdly, the paper's significant contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz's formula, by constructing a novel class of functions defined by

L (z, f) = s u m_{n = n_{f}}^{+ i n f t y} n^{(- z, f)}

, which are intrinsically linked to C-polynomials and referred to as

e x t i t L C - f u n c t i o n s a s s o c i a t e d t o f

(the constant

n_{f}

is a positive integer dependent on the choice of

f

). This research offers a detailed analysis of C-polynomials, P-polynomials, and LC-functions associated to a given analytic function

f

, thoroughly examining their interrelations and introducing unexplored research directions for a novel and expansive class of LC-functions possessing a functional equation equivalent to that of the Riemann zeta function, thereby highlighting the potential applications and implications of the findings.

Mathematics Subject Classification ID

Other Dirichlet series and zeta functions (11M41) Other functions defined by series and integrals (33E20) Special sequences and polynomials (11B83)

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