The Dedekind sum can be defined by \[ s(h,k) := {1 \over {4k}} \sum_{ r=1 }^{ k-1 } \cot \left( {{\pi r} \over k} \right) \cot \left( {{\pi hr} \over k} \right) , \] where \(h\) and \(k\) are positive, relatively prime integers. Dedekind sums first appeared in Dedekind’s work on the \(\eta\)-function and have been generalized in various ways, e.g., by D. Zagier [Math. Ann. 202, 149–172 (1973; Zbl 0237.10025)] who considered an arbitrary number of cotangent factors, and by A. Bayad and A. Raouj [J. Number Theory 132, No. 2, 332–347 (2012; Zbl 1247.11058)] who replaced the cotangent by cotangent derivatives.
The paper under review continues the development of Dedekind sums in function fields [Y. Hamahata, Proc. Japan Acad., Ser. A 84, No. 7, 89–92 (2008; Zbl 1225.11055); S. Okada, J. Number Theory 130, No. 8, 1750–1762 (2010; Zbl 1197.11155); A. Bayad and Y. Hamahata, Acta Arith. 152, No. 1, 71–80 (2012; Zbl 1301.11047)]. Let \(A:= {\mathbb F}_q[x]\) and \(\Lambda\) be an \(A\)-lattice, i.e., a finitely generated \(A\)-module satisfying certain discreteness conditions for a completion of the quotient field of \(A\), and let \[ e_\Lambda (z) := z \prod_{ \lambda \in \Lambda } \!{}^\prime \left( 1 - {z \over \lambda} \right) \] where the \('\) indicates that the sum/product is only taken over nonzero and nonsingular values. Note that \(e_\Lambda (z)^{ -1 } \) is a natural analogue of the cotangent function in this setting, and in [Zbl 1301.11047] the authors studied \[ s_\Lambda \left( a_n; a_1, a_2, \dots, a_{ n-1 } \right) = {{(-1)^{ n-1 }} \over { a_n }} \sum_{ \lambda \in \Lambda/a_n \Lambda } \!\!\!\!\!\!{}^\prime \;\;e_\Lambda \left( {{ a_1 \lambda } \over { a_n }} \right)^{ -1 } \cdots \;e_\Lambda \left( {{ a_{n-1} \lambda } \over { a_n }} \right)^{ -1 } , \] a function-field analogue of Zagier’s higher-dimensional Dedekind sum mentioned above. In the present paper the authors replace the \((-1)\)-powers by arbitrary negative powers, giving rise to an analogue of cotangent derivatives. The main result is a reciprocity theorem: when the arguments are coprime, a certain sum of multiple function-field Dedekind sums equals an expression involving analogues of Eisenstein series in this setting. The authors also prove a Petersson-Knopp identity for the multiple function-field Dedekind sums, in analogy of such identities for Dedekind sums [M. I. Knopp, J. Number Theory 12, 2–9 (1980; Zbl 0423.10015)] and their multivariate versions [M. Beck, Acta Arith. 109, No. 2, 109–130 (2003; Zbl 1061.11043)]. They finish by exhibiting an appearance of the multiple function-field Dedekind sums in connection with the Goss \(L\)-function [D. Goss, Basic structures of function field arithmetic. Berlin: Springer (1998; Zbl 0892.11021)].