C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function - Lamgouni, Lahcen (The Ramanujan journal, 2024)
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 ).