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

• Published in: The Ramanujan journal Vol. 65; no. 2; pp. 821 - 855
• Main Author: Lamgouni, Lahcen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2024; Springer Nature B.V
• Subjects: Analytic functions; Bivariate analysis; Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; Polynomials; 11M99; 33B99; P-polynomials; 33E20; Appell polynomials; Analytic continuation; Primary 11M41; Faulhaber’s formula generalization; Hurwitz’s formula generalization; 11B83; C-polynomials; Multiplication formula generalization; Secondary 11M35; Hurwitz zeta function generalization; LC-functions; Euler–Maclaurin formula generalization; 11M38
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-024-00919-1

Let f ( t ) = ∑ n = 0 + ∞ C f , n n ! t n be an analytic function at 0, and let C f , n ( x ) = ∑ k = 0 n n k C f , k x n - k be the sequence of Appell polynomials, referred to as C-polynomials associated to 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 P-polynomials associated to 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 bivariate complex function P f ( s , z ) = ∑ k = 0 + ∞ z k P f , k s z - k , which generalizes the s z function and is denoted by s ( z , f ) . Thirdly, the paper’s main 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 ) = ∑ n = n f + ∞ n ( - z , f ) , which are intrinsically linked to C-polynomials and referred to as LC-functions associated to f (the constant n f is a positive integer dependent on the choice of f ).