EULERIAN FRACTIONS AND STIRLING, BERNOULLI AND EULER FUNCTIONS WITH COMPLEX ORDER PARAMETERS AND THEIR IMPACT ON THE POLYLOGARITHM FUNCTION

Applicable Analysis and Discrete Mathematics

article

Paul Butzer¹  Tian-Xiao He²  Clemens Markett¹ 

[1] Lehrstuhl A f¨ur Mathematik;Dept. Math., llinois Wesleyan University

关键词: Polylogarithm;    Eulerian polynomials;    Eulerian fractions;    Stirling functions;    Bernoulli functions;   

DOI  :  10.2298/AADM220506002B

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

We first study some generalizations of Eulerian fractions with complex orderparameter and investigate their interrelationship with likewise generalizedEulerian functions as well as Stirling functions. We apply the new approach topolylogarithms of non-integral order, for which only a few values are known inclosed form. In particular, we present a structural solution of the counterpartof an old conjecture of Mengoli and Euler in the polylogarithm case with theaid of Riemann’s zeta function and the Dirichlet eta and beta functions.

【 授权许可】

Unknown