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On Stirling numbers and Euler sums Victor Adamchik

Summary: On Stirling numbers and Euler sums
Victor Adamchik
Wolfram Research Inc.,
100 Trade Center Dr.,
Champaign, IL 61820, USA
October 21, 1996
In this paper, we propose the another yet generalization of Stirling numbers of
the first kind for non­integer values of their arguments. We discuss the analytic
representations of Stirling numbers through harmonic numbers, the generalized hy­
pergeometric function and the logarithmic beta integral. We present then infinite
series involving Stirling numbers and demonstrate how they are related to Euler
sums. Finally we derive the closed form for the multiple zeta function i(p; 1; : : : ; 1)
for p ? 1.
1 Introduction and notations.
Throughout this article we will use the following definitions and notations. Stirling
numbers of the first kind are defined by the recurrence relation (see [1])
h n


Source: Adamchik, Victor - School of Computer Science, Carnegie Mellon University


Collections: Computer Technologies and Information Sciences